On symplectic integration in Lie groups and manifolds

Preface This thesis is submitted in partial fulfilment of the requirements for the degree of philosophiae doctor (PhD) at the Norwegian University of Science and Technology (NTNU). The research was funded by the Department of Mathematical Sciences and the Research Council of Norway, and was carried out at the department, as well as at Massey University in Palmerston North, New Zealand, and La Trobe University in Melbourne, Australia. Writing this thesis has been a roller coaster of emotions for me and would have been impossible without all the help and support I have received. I would first and foremost like to thank my advisor Professor Brynjulf Owren for not giving up on me. He has always given me the push I needed, when I needed it, and our many discussions always left me feeling reinvigorated. I would also like to thank him for introducing me to many of the big names of our field, and I truly feel like a part of the Norwegian geometric integration community which meets yearly at the Manifolds and Geometric Integration Colloquia (MaGIC). Many thanks also go to my co-authors Professor Elena Celledoni and Geir Bogfjellmo for their insightful ideas and hard work. Professor Robert McLachlan and Professor Reinout Quispel made me feel welcome during my four month research visit to New Zealand and Australia in 2012. Thanks go to Assistant Professor Klas Modin for some very helpful discussions. The technical and administrative staff at the department have been indispensable, and deserve thanks for their help. I would also like to thank the rest of my friends and colleagues at the department for creating such a friendly and open atmosphere. On a more personal level, I would like to thank my parents, Aina Wroldsen and Dag Marthinsen, as well as Yaman Umuro˘glu for their unconditional support and love. Special thanks go to all my friends in Studentorchesteret Dei Taktlause. They have truly saved my social life in Trondheim. Finally, many thanks to the rest of my family and friends for their support. Håkon Marthinsen Trondheim, 4 September 2014

iii

Contents 1 Introduction

1

1.1 Structure preserving numerical solution of ordinary differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Lie group integrators . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The generic presentation of ODEs on manifolds . . . . . . 1.2.2 Runge–Kutta–Munthe-Kaas methods . . . . . . . . . . . . 1.2.3 Crouch–Grossman methods . . . . . . . . . . . . . . . . . 1.2.4 Homogeneous spaces . . . . . . . . . . . . . . . . . . . . . 1.3 Preservation of first integrals . . . . . . . . . . . . . . . . . . . . . 1.4 Autonomous mechanics . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . 1.4.2 Hamiltonian mechanics . . . . . . . . . . . . . . . . . . . . 1.4.3 Hamilton–Pontryagin mechanics . . . . . . . . . . . . . . 1.5 Variational integrators . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Variational Lie group integrators . . . . . . . . . . . . . . . 1.6 Non-autonomous Hamiltonian mechanics . . . . . . . . . . . . . 1.7 Canonical transformations as integrators . . . . . . . . . . . . . . 1.8 Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 4 4 5 6 6 7 7 8 8 9 9 11 11 12 13 15

2 An introduction to Lie group integrators – basics, new devel. and appl. 23

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2.2 Lie group integrators . . . . . . . . . . . . . . . . . . 2.2.1 Generalizing Runge–Kutta methods . . . . . 2.2.2 A plenitude of group actions . . . . . . . . . 2.2.3 Isotropy – challenges and opportunities . . 2.3 Applications to nonlinear problems of evolution mechanics . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Rigid body and rod dynamics . . . . . . . . .

. . . . . . . . . . in . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . classical . . . . . . . . . . . .

23 24 30 34 38 39 41

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Contents 2.4 Applications to problems of data analysis and statistical signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Gradient-based optimization on Riemannian manifolds . 2.4.2 Principal component analysis . . . . . . . . . . . . . . . . 2.4.3 Independent component analysis . . . . . . . . . . . . . . 2.4.4 Computation of Lyapunov exponents . . . . . . . . . . . . 2.5 Symplectic integrators on the cotangent bundle of a Lie group . 2.6 Discr. gradients and integral preserving methods on Lie groups . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 High order symplectic partitioned Lie group methods

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Motivation and background . . . . . . . . . . . . . . . . . . 3.1.2 ODEs on Lie groups . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Hamilton–Pontryagin mechanics . . . . . . . . . . . . . . 3.1.4 Variational integrators . . . . . . . . . . . . . . . . . . . . . 3.2 From a Lie group method to a var. integrator on the cotangent bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Group structure and Hamiltonian ODEs on G × g∗ . . . . 3.2.2 General format for our integrators . . . . . . . . . . . . . . 3.3 First and second order integrators . . . . . . . . . . . . . . . . . . 3.4 Higher order integrators . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Variational Runge–Kutta–Munthe-Kaas integrators . . . . 3.4.2 Variational Crouch–Grossman integrators . . . . . . . . . 3.5 Order analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Order of VRKMK integrators . . . . . . . . . . . . . . . . . 3.5.2 Order of VCG integrators . . . . . . . . . . . . . . . . . . . 3.6 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Order tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Long time behaviour . . . . . . . . . . . . . . . . . . . . . . 3.7 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Geometric integration of non-autonomous Hamiltonian problems

44 45 47 50 52 53 57 63 73

73 73 76 76 77 78 78 80 83 87 88 92 94 94 100 105 106 109 109 112 113 119

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.2 Four classes of problems . . . . . . . . . . . . . . . . . . . . . . . . 121

vi

Contents 4.3 Autonomous and non-autonomous Hamiltonian mechanics . . 4.3.1 Autonomous Hamiltonian systems . . . . . . . . . . . . . 4.3.2 Non-autonomous Hamiltonian systems . . . . . . . . . . 4.3.3 Canonical transformations . . . . . . . . . . . . . . . . . . 4.4 Canonical transf. and integrators for non-auton. Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Constructing a canonical transformation from a given map ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Constructing a canonical transformation from a given map ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Long-time performance . . . . . . . . . . . . . . . . . . . . 4.5.2 Symmetric methods and canonical transformations . . . 4.5.3 Canonical, symmetric, and exponential methods . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122 122 123 125 129 129 130 132 133 135 136 140 141

vii

1 Introduction Laws of physics often originate from geometric principles, such as variational principles or symmetries of space-time. These symmetries give rise to conservation laws. For example, in Newtonian mechanics we have conservation of mass, linear and angular momentum, and energy. From the principles, we can deduce equations of motion, which take the form of ordinary differential equations (ODEs). In practice, these equations almost never have solutions that can be expressed in closed form, so we have to resort to numerical methods. There are two ways we can approach this situation, either by constructing methods that solve the ODEs numerically without considering the underlying preservation principles (e.g. general Runge–Kutta (RK) methods and linear multistep methods), or by incorporating a selection of the principles into the methods (e.g. symplectic methods and Lie group methods). The second of these approaches is called geometric numerical integration [11, 17, 31] and has proven to exhibit excellent properties, particularly in cases where we are interested in the long-term qualitative behaviour of the equations. Examples where structure preserving methods have been successful include molecular dynamics [45], celestial mechanics [15], and general Hamiltonian mechanics [25, 44]. The main topic of the thesis is symplectic integration on manifolds and Lie groups, both for autonomous and non-autonomous Hamiltonian problems. In this introduction we explain the key concepts that are needed for the three papers constituting the main part of the thesis. The three papers are summarized in Section 1.8.

1.1 Structure preserving numerical solution of ordinary differential equations Classical integrators, like RK methods and linear multistep methods [18] all focus on solving ODEs in Rn . We can formulate this as approximating the

1

1 Introduction solution of y˙ = f (y),

y(0) = y 0 ∈ Rn .

From the geometry of the ODE, we can often derive certain qualitative facts about the structure of the exact solution without actually solving the ODE [2, 4, 40]. For example, symmetries of the exact solution may give rise to conservation laws that we want to ensure are kept true also for the numerical solution. McLachlan and Quispel [33] give a systematic classification of geometric structure in ODEs, and also of structure preserving integrators. There are many examples of preserved structure in ODEs and geometric integrators. A few of these are briefly discussed below. We will for the most part consider Lie group integrators and symplectic integrators in this thesis. One example of structure preservation is the symplectic structure in Hamiltonian mechanics, where the symplectic form ω0 = dp i ∧dq i (expressed in canonical coordinates) is conserved along the flow of the ODE (see Section 1.4). Symplectic integrators conserve ω0 along the numerical solution. There are several known classes of symplectic integrators, e.g. symplectic (partitioned) Runge–Kutta methods, methods based on generating functions, variational integrators (see Section 1.5), and symplectic splitting methods [17, Chapter VI], [25, 32, 44]. Using the machinery of backward error analysis [17, Chapter IX], it is possible to explain why symplectic integrators perform very well in long-time simulations of Hamiltonian mechanics. Symplectic structure.

A second example is when div f (y) = 0 for all y. This is the divergence-free case, i.e. the case where volume in phase space is preserved. Chartier and Murua [10] and Iserles, Quispel and Tse [22] showed that for the general divergence-free case, the only volume-preserving B-series method [17, Section III.1] is the exact solution. This result rules out many methods, but nevertheless, several volume-preserving methods have been found, see e.g. Xue and Zanna [47] and the references therein. Hamiltonian systems form a subset of the divergence-free systems. Volume preservation.

A third example are the ρ-reversible ODEs, i.e. ODEs where there exists a nonsingular linear map ρ such that ρ ◦ f = − f ◦ ρ for all y. The flow φt for such an ODE satisfies ρ ◦ φt = φ−t ◦ ρ = φ−1 t ◦ ρ. It turns out that symmetric methods (which have numerical flow Φh = Φ−1 ) are also ρ-reversible, and −h Reversibility.

2

1.1 Structure preserving numerical solution of ordinary differential equations

Figure 1.1: Lack of structure preservation using the Euler method. can be applied to these ODEs (see [17, Section V.1]). There are several ways of constructing symmetric methods of arbitrarily high order, e.g. by composing lower order methods, and their adjoints, with carefully chosen time-steps [17, Section V.3]. A fourth example is the preservation of first integrals. A system which preserves a first integral I (y) can be written in the skew-gradient form y˙ = S(y)∇I (y), where S(y) is a skew-symmetric matrix depending on y. Hamiltonian systems fit into this framework, with I = H , and £ 0 I¤ S = J := −I 0 , where I is the identity matrix. An introduction to methods which preserve first integrals can be found in [17, Section IV.1–4]. Discrete gradient methods is a class of integral-preserving methods which was popularized by the works of Gonzalez [16] and McLachlan, Quispel and Robidoux [34]. See Section 1.3 for a short introduction to these methods. Further developments of integral-preserving methods can be found in [14, 39, 43]. Preservation of first integrals.

Finally, we consider the case when the exact solution evolves on a manifold M which may be different from Rn . In order to solve this ODE with a classical method, we can embed M into Euclidean space Rm , and then apply the integrator. With this approach, nothing restricts the numerical solution to M . The solution is likely to drift away from the manifold, and the numerical solution will display anomalous behaviour. A typical example of this is if we apply the Euler method to the ODE describing the path Manifold and Lie group structure.

3

1 Introduction of a particle going around in a circle at constant speed, x˙ = −y, y˙ = x. The result is a numerical solution that spirals outward, as seen in Figure 1.1. Methods on manifolds are designed to work intrinsically on M , and can ensure that we will never drift away (disregarding computer rounding errors). In the case of the particle, we could let M be the circle and apply a manifold-based method, e.g. the Lie–Euler method [17, Equation IV.8.2]. This would give a solution that is simply a time-reparameterization of the exact solution. Many ODEs which evolve on manifolds can be expressed using the concept of a Lie group acting on a manifold. This action is the central concept of Lie group integrators, and can be used to update the numerical solution without leaving the manifold. We explore Lie group integrators a bit more in Section 1.2.

1.2 Lie group integrators In Lie group integrators, we no longer advance the numerical solution by adding a vector in Rn to the current iterate (as in RK methods), but instead follow the exact flow of a simpler vector field which is guaranteed to stay on the correct manifold (disregarding rounding errors). This simpler vector field is calculated using Lie group actions, as we will see below. The flow of such a vector field should be computable in a reasonably simple manner in order to retain efficiency. For a comprehensive introduction to Lie group methods, see Iserles et al. [21] and Christiansen, Munthe-Kaas and Owren [11, Section 2].

1.2.1 The generic presentation of ODEs on manifolds Consider a Lie group G with identity element e and associated Lie algebra g, acting on a manifold M by the Lie group action Λ:G × M → M ,

Λg := Λ(g , ·).

This gives rise to a Lie algebra action λ defined by λ: g × M → M ,

λ(v, x) := Λ(exp v, x),

λv := λ(v, ·).

Since exp is a local diffeomorphism, there is a one-to-one correspondence between the Lie group action and the Lie algebra action near the identity. It is worth mentioning that instead of using exp, it is possible to pick any local

4

1.2 Lie group integrators diffeomorphism φ: g → G which satisfies φ(0) = e. Given an element v ∈ g, we define a vector field on M by ¯ d¯ λ∗ : g → X(M ), (λ∗ v)(x) := ¯¯ λ(t v, x) ∈ Tx M . (1.1) dt t =0 Here, X(M ) is the set of vector fields on M . We see that λt v is the flow of the vector field λ∗ v. If Λ is locally transitive, i.e. given any x ∈ M , the set Λ(G, x) contains a neighbourhood of x, then we can write any ODE on M in the form ¡ ¢ y˙ = λ∗ f (y) (y),

(1.2)

where the map f : M → g specifies the ODE. This formulation of ODEs on manifolds is called the generic presentation of ODEs on manifolds [35].

1.2.2 Runge–Kutta–Munthe-Kaas methods One important class of Lie group methods are the Runge–Kutta–MuntheKaas (RKMK) methods, developed by Munthe-Kaas [35–37] and Munthe-Kaas and Zanna [38]. The main idea here is that we transform the ODE (1.2) into a new ODE in g (which is a linear space), solve it using a Runge–Kutta method, and then transform back to the original space. In the neighbourhood of a point x ∈ M , we can express the exact solution y(t ) (with y(0) = x) via a curve σ(t ) in the Lie algebra as ¡ ¢ y(t ) = λ σ(t ), x ,

σ(0) = 0.

By differentiating this equations with respect to t and comparing with (1.2), we find that we can find a solution of the original ODE if we solve ³ ¡ ¢´ ˙ ) = dexp−1 σ(t ◦ f λ σ(t ), x , σ(t ) and then transform back to M . Here, dexp−1 u v :=

∞ B X k adku v, k! k=0

where Bk are the Bernoulli numbers and ad is the adjoint endomorphism (i.e. matrix commutator for matrix Lie algebras). We have converted the ODE to an ODE in the linear space g and proceed by truncating the series expansion

5

1 Introduction of dexp−1 to an appropriate number of terms (written as dexpinv), taking one step with an RK method, and then transforming back to M . If the RK method has s stages and is given by the coefficients a i j and b i , one step of RKMK can be written as Ã µ ¶! X k i = dexpinvh P j ai j k j ◦ f λ h a i j k j , y 0 , j

µ s ¶ X y1 = λ h bi ki , y 0 .

(1.3)

i =1

We see that we can interpret RKMK as taking a step of length h along the flow P of the vector field λ∗ is=1 b i k i . As long as we include enough terms of the expansion of dexp−1 in dexpinv, the order of the RKMK method will be of the same order as the underlying RK method [17, Theorem IV.8.5].

1.2.3 Crouch–Grossman methods Another important class of Lie group methods is the Crouch–Grossman (CG) methods [12], which were presented using the language of frames and frozen vector fields. These are also a Lie group generalization of RK methods, and using group actions they take the form ³ ¡ ¢´ k i = f Λ exp(ha i ,s k s ) · · · exp(ha i ,1 k 1 ), y 0 , (1.4) ¡ ¢ y 1 = Λ exp(hb s k s ) · · · exp(hb 1 k 1 ), y 0 . The complete order conditions for CG methods were found by Owren and Marthinsen [42]. CG methods can be generalized to commutator-free Lie group methods [8], for which Owren [41] gave a complete set of order conditions.

1.2.4 Homogeneous spaces We say that a Lie group action Λ:G × M → M is transitive, or that G acts transitively on M , if for any x, y ∈ M , there exists a g ∈ G such that y = Λ(g , x). In this case, we call M a homogeneous space. If the isotropy subgroup G x := {g ∈ G | Λ(g , x) = x} is nontrivial, i.e. if G x 6= {e}, then the map λ∗ from (1.1) is no longer injective. Thus, there may be more than one element of g corresponding to the same vector field in X(M ). This complication can be exploited to obtain better numerical solutions of the ODE [28]. As we will see in Paper 2, for variational integrators, isotropy may cause the minimization procedure of Section 1.5 to

6

1.3 Preservation of first integrals fail, resulting in an update equation with no solution. This is the main reason why we in Paper 2 only consider the situation where a Lie groups acts on itself. In this case, λ∗ is always injective.

1.3 Preservation of first integrals First integrals are functions which are preserved along the exact solution of the ODE in question. There are many systems in classical mechanics where e.g. energy, mass, or electric charge is conserved. These are examples of first integrals. An ODE with a first integral I (y) may be written in the form y˙ = S(y)∇I (y), where S(y) is a skew-symmetric matrix. Discrete gradient (DG) methods [16, 34] are numerical integrators which guarantee that I is preserved along the numerical solution. The key idea is to replace ∇I (y) with a discrete gradient ∇I : Rn × Rn → Rn , which is a continuous map satisfying ∇I (y, y) = ∇I (y),

for all y ∈ Rn ,

I (y) − I (x) = ∇I (x, y)T (y − x),

for all x 6= y.

˜ Rn × Rn → Rn×n , which satisfies S(y, ˜ y) = S(y) Next, we replace S(y) by a map S: n for all y ∈ R . The DG method y n 7→ y n+1 is then defined for a step-length h as y n+1 − y n ˜ n , y n+1 )∇I (y n , y n+1 ). = S(y h

(1.5)

It is a simple matter to prove that this integrator exactly preserves I along the numerical solution, see e.g. the introduction of Dahlby [13].

1.4 Autonomous mechanics In this section, we present an overview of autonomous mechanics, i.e. mechanical systems which do not explicitly depend on time. Hamiltonian mechanics, which is a source of ODEs with symplectic structure, are found here. We summarize some of the basic results which can be found in Abraham and Marsden [1], Arnol’d [3], Marsden and Ratiu [29] and Yoshimura and

7

1 Introduction Marsden [48]. The behaviour of autonomous mechanical systems is encoded as a path q(t ) in configuration space Q (which is a smooth n-dimensional manifold).

1.4.1 Lagrangian mechanics In Lagrangian mechanics, the behaviour of the mechanical system in question is fully determined by its associated Lagrangian L: TQ → R and the initial configuration. Hamilton’s principle postulates that the system follows the path q: [0, T ] → Q which extremizes the action integral SH =

T

Z 0

¡ ¢ ˙ ) dt , L q(t ), q(t

(1.6)

where q(0) and q(T ) are kept fixed. It can be shown that this principle is equivalent to the Euler–Lagrange equations d ∂L ∂L = . dt ∂q˙ i ∂q i

1.4.2 Hamiltonian mechanics Let p i := ∂L/∂q˙ i . We say that the Lagrangian is hyperregular if the Legendre transform (q i , q˙ i ) 7→ (q i , p i ) is a diffeomorphism. Given a hyperregular Lagrangian, it can be shown that the Euler–Lagrange equations are equivalent to Hamilton’s equations ∂H ∂H q˙ i = , p˙i = − i , (1.7) ∂p i ∂q ˙ − L(q, q). ˙ where H : T∗Q → R is the Hamiltonian H (q, p) := 〈p, q〉 Hamilton’s equations can be expressed more compactly and coordinate-free as i X H ω0 = −dH , ˙ p), ˙ where X H is the vector field (q, ω0 = dp i ∧ dq i is a symplectic 2-form expressed in canonical (or Darboux) coordinates, and i X H ω0 := ω0 (X H , ·). A standard result in Hamiltonian mechanics is that ω0 and H are both preserved along the exact solution.

8

1.5 Variational integrators

1.4.3 Hamilton–Pontryagin mechanics An alternative variational principle, called the Hamilton–Pontryagin (HP) principle [48], states that the system follows the path q: [0, T ] → Q which extremizes the action integral Z T ¢ ¡ L(q, v) + 〈p, q˙ − v〉 dt , S HP = 0

where q(0) and q(T ) are kept fixed, but where p ∈ T∗q Q and v ∈ Tq Q can be varied arbitrarily. By setting the variational derivative of S HP equal to zero, we obtain the Hamilton–Pontryagin equations q˙ = v,

p˙ =

∂L , ∂q

p=

∂L . ∂v

(1.8)

The HP principle unifies Lagrangian and Hamiltonian mechanics, since it automatically provides the Legendre transform.

1.5 Variational integrators Variational integrators form a class of symplectic integrators which are based on the idea of approximating the action integral (1.6) by quadrature. This yields an action sum which which is then extremized, analogous to the continuous case. This procedure ensures that the resulting integrators are automatically symplectic [30]. To construct a variational integrator, we first define the discrete Lagrangian L h :Q × Q → R as an approximation to the action integral over a small time step h, Z hk ¡ ¢ ˙ ) dt . L h (q k−1 , q k ) ≈ L q(t ), q(t h(k−1)

This allows us to approximate the action integral as S H ≈ S h :=

N X

L h (q k−1 , q k ),

k=1

h :=

T . N

By extremizing S h while keeping q 0 and q N fixed, we obtain the discrete Euler– Lagrange equations D1 L h (q k , q k+1 ) + D2 L h (q k−1 , q k ) = 0,

1 ≤ k ≤ N − 1,

9

1 Introduction where Di L h signifies the partial derivative of L h with respect to the i th argument. The discrete conjugate momenta are defined via the discrete Legendre transforms as p k := −D1 L h (q k , q k+1 ),

p k+1 := D2 L h (q k , q k+1 ).

We immediately see that if these two equations are consistent with each other, then the discrete Euler–Lagrange equations are satisfied. If the first of these equations can be solved for q k+1 , we can find p k+1 from the second. Thus, we have obtained the variational integrator (q k , p k ) 7→ (q k+1 , p k+1 ). For a thorough introduction to variational integrators, see Marsden and West [30]. Newer developments can be found in Hall and Leok [20] and Leok and Shingel [27]. Symplectic partitioned Runge–Kutta methods are variational integrators [17, Section VI.6.3],[46]. They originate from the discrete Lagrangian Example.

L h (q 0 , q 1 ) = h

s X

b i L(Q i , Q˙ i ),

i =1

with constraints Q i = q0 + h By setting

s X

a i j Q˙ j ,

j =1

P˙i = D1 L(Q i , Q˙ i ),

q1 = q0 + h

s X

b i Q˙ i .

i =1

P i = D2 L(Q i , Q˙ i ),

and performing the extremization procedure with the constraints enforced by Lagrange multipliers, we get p1 = p0 + h Pi = p0 + h

s X i =1 s X j =1

b i P˙i , aî j P˙ j ,

q1 = q0 + h Q i = q0 + h

s X i =1 s X

b i Q˙ i , a i j Q˙ j ,

j =1

where aî j = b j − b j a j i /b i . This is equivalent to applying a partitioned Runge– Kutta method with coefficients a i j , b i , aî j = b j −b j a j i /b i , bˆ i = b i , to Hamilton’s equations (1.7).

10

1.6 Non-autonomous Hamiltonian mechanics

1.5.1 Variational Lie group integrators The application of variational integrators to mechanics on Lie groups has recently received much attention. In this case, special care has to be taken when approximating the action integral, since we now want the path q(t ) to evolve on a Lie group. One possibility is to use a Lie group method (see Section 1.2) as quadrature; another is to consider the Lie group as a level set of a continuously differentiable function on a Euclidean space, and then use Lagrange multipliers to restrict the solution to the Lie group. The latter approach does not exploit the Lie group structure of the manifold, so in Paper 2, we will focus on the former. For recent developments on variational Lie group integrators, see Bou-Rabee and Marsden [7], Hall and Leok [19], Lee, Leok and McClamroch [23, 24] and Leok and Shingel [26].

1.6 Non-autonomous Hamiltonian mechanics In Paper 3, we consider Hamiltonian mechanics with time as an explicit variable, i.e. non-autonomous mechanics. This setting complicates the symplectic structure we are familiar with from the autonomous case, since we now have an odd number of dimensions, while a symplectic form can only be defined on an even number of dimensions. One way to resolve this problem is to add an auxiliary variable denoted by u which is conjugate to time t , so that we get back to an even number of dimensions. This enlarged space is called extended phase space T∗(Q × R), with coordinates (q, p, t , u). We then introduce the extended Hamiltonian K (q, p, t , u) := u + H (q, p, t ), and the symplectic form

Ω0 = dp i ∧ dq i + du ∧ dt ,

(1.9)

expressed in canonical coordinates. Analogous to the autonomous case, we can write the extended Hamilton’s equations as i X K Ω0 = −dK ,

˙ p, ˙ t˙, u), ˙ in canonical coordinates, where X K is the vector field (q, q˙ i =

∂K ∂H = , ∂p i ∂p i

p˙i = −

∂K ∂q i

=−

∂H ∂q i

,

t˙ =

∂K = 1, ∂u

u˙ = −

∂K ∂H =− . ∂t ∂t (1.10)

11

1 Introduction Furthermore, Ω0 and K are preserved along the exact solution. The original Hamiltonian H is not preserved in the non-autonomous case. Observe that the original Hamilton’s equations (1.7) are contained in the extended variant of Hamilton’s equations (1.10).

1.7 Canonical transformations as integrators In the autonomous case, symplectic integrators are the natural choice for solving Hamiltonian problems numerically, but this choice is not as straightforward for the non-autonomous case. The symplectic form ω0 is no longer preserved along the flow of the exact solution, however Ω0 from (1.9) is. Thus, we should consider methods which preserve Ω0 . In this section, we will transition from symplectic to canonical integrators, which are examples of canonical transformations as defined by Asorey, Cariñena and Ibort [5]. A canonical transformations is a pair (ψ, ϕ) of diffeomorphisms, ψ on T∗(Q × ¢ ¡ ¯ R) and ϕ on T∗Q × R, such that ψ(q, p, t , u) = ϕ(q, p, t ), ϕ(q, p, t , u) for some ¯ T∗(Q × R) → R, and ψ is a symplectic map (i.e. ψ∗ Ω0 = Ω0 ). We are map ϕ: particularly interested in the case where we advance time by a constant steplength h, since this is common for integrators. This framework is the basis of the integrators for non-autonomous Hamiltonian systems that we consider in Paper 3.

12

1.8 Summary of papers

1.8 Summary of papers The layout, bibliography, and typography of the three papers have been unified, and a few grammatical errors and typos have been corrected. In order to fit the B5 format of this thesis, a few sentences had to be rephrased, and a few equations had to be reformatted. No other substantial modifications have been made to the papers.

Paper 1: An introduction to Lie group integrators – basics, new developments and applications Elena Celledoni, Håkon Marthinsen and Brynjulf Owren Published in Journal of Computational Physics [9] This paper is a combination of a survey aimed at physicists and some new results. The first part of the paper consists of a short introduction to the basic concepts of Lie group integrators, a summary of how one can generalize Runge– Kutta methods to the Lie group setting, and different aspects of how the choice of group action affects the integrators. Difficulties with the case of a non-trivial isotropy group are touched upon. Many applications of Lie group integrators are considered: Rigid body and rod dynamics, gradient-based optimization on Riemannian manifolds, principal and independent component analysis, and computation of Lyapunov exponents. Inspired by the work of Bou-Rabee and Marsden [7], we show that we can formulate symplectic Lie group integrators on the cotangent bundle of a Lie group by introducing a group structure on the cotangent bundle. This is the basis of Paper 2. Finally, we show that discrete gradient methods (1.5) can be extended to the Lie group setting and apply this to attitude rotation of a free rigid body.

Paper 2: High order symplectic partitioned Lie group methods Geir Bogfjellmo and Håkon Marthinsen Submitted to Foundations of Computational Mathematics [6] Symplectic Lie group integrators have been studied earlier from the viewpoint of variational integrators, e.g. by Bou-Rabee and Marsden [7]. In this paper, we approach symplectic Lie group integrators from the “Lie group integrator side” by formulating Hamilton–Pontryagin mechanics (1.8) on the cotangent

13

1 Introduction bundle of a Lie group G using the generic presentation of ODEs on manifolds (1.2). We present a way to extend Lie group integrators operating on G to symplectic integrators on T∗G. In particular, we demonstrate that the integrators we obtain based on Crouch–Grossman (CG) methods (1.4) and Runge–Kutta–Munthe-Kaas (RKMK) methods (1.3) are generalizations of symplectic partitioned Runge–Kutta methods. We show that arbitrarily high order can be reached with the methods based on CG and RKMK methods. Furthermore, we give the order conditions up to third order for the CG based methods as well as the complete set of order conditions for the RKMK based methods. We verify correct order numerically on a test problem and observe small energy error over long time, as expected for symplectic integrators. In order to avoid isotropy problems, which may cause trouble with solvability for the integrators, we restrict ourselves to the setting where the Lie group acts on itself.

Paper 3: Geometric integration of non-autonomous Hamiltonian problems Håkon Marthinsen and Brynjulf Owren To be submitted For autonomous Hamiltonian problems, we usually apply symplectic methods because of their excellent long-time behaviour. In this paper, we take a closer look at non-autonomous Hamiltonian problems (1.10), in particular the linear ones. We show that canonical transformations as defined by Asorey, Cariñena and Ibort [5] (see Section 1.7) can be used to extend the notion of symplectic integrators to non-autonomous mechanics. Conditions for the existence of canonical transformations with constant time-step are provided. We also show which integrators operating on T∗Q×R can be extended to canonical integrators on T∗(Q × R), and vice versa. Finally, we perform numerical experiments on a toy problem, which indicate that canonicity and symmetricity are important for good long time behaviour of exponential integrators.

14

1.8 Summary of papers

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Paper 1 An introduction to Lie group integrators – basics, new developments and applications

Elena Celledoni, Håkon Marthinsen and Brynjulf Owren Published in Journal of Computational Physics 257, part B (2014), pp. 1040–1061

2 An introduction to Lie group integrators – basics, new developments and applications We give a short and elementary introduction to Lie group methods. A selection of applications of Lie group integrators are discussed. Finally, a family of symplectic integrators on cotangent bundles of Lie groups is presented and the notion of discrete gradient methods is generalized to Lie groups. Abstract.

2.1 Introduction The significance of the geometry of differential equations was well understood already in the nineteenth century, and in the last few decades such aspects have played an increasing role in numerical methods for differential equations. Nowadays, there is a rich selection of integrators which preserve properties like symplecticity, reversibility, phase volume and first integrals, either exactly or approximately over long times [30]. Differential equations are inherently connected to Lie groups, and in fact one often sees applications in which the phase space is a Lie group or a manifold with a Lie group action. In the early nineties, two important papers appeared which used the Lie group structure directly as a building block in the numerical methods. Crouch and Grossman [22] suggested to advance the numerical solution by computing flows of vector fields in some Lie algebra. Lewis and Simo [45] wrote an influential paper on Lie group based integrators for Hamiltonian problems, considering the preservation of symplecticity, momentum and energy. These ideas were developed in a systematic way throughout the nineties by several authors. In a series of three papers, Munthe-Kaas [54–56] presented what are now known as the Runge–Kutta–Munthe-Kaas methods. By the turn of the millennium, a survey paper [35] summarized most of what was known by then about Lie group integ-

23

2 An introduction to Lie group integrators – basics, new devel. and appl. rators. More recently a survey paper on structure preservation appeared with part of it dedicated to Lie group integrators [20]. The purpose of the present paper is three-fold. First, in Section 2.2 we give an elementary, geometric introduction to the ideas behind Lie group integrators. Secondly, we present some examples of applications of Lie group integrators in Section 2.3 and 2.4. There are many such examples to choose from, and we give here only a few teasers. These first four sections should be read as a survey. But in the last two sections, new material is presented. Symplectic Lie group integrators have been known for some time, derived by Marsden and West [49] by means of variational principles. In Section 2.5 we consider a group structure on the cotangent bundle of a Lie group and derive symplectic Lie group integrators using the model for vector fields on manifolds defined by Munthe-Kaas in [54]. In Section 2.6 we extend the notion of discrete gradient methods as proposed by Gonzalez [29] to Lie groups, and thereby we obtain a general method for preserving first integrals in differential equations on Lie groups. We would also like to briefly mention some of the issues we are not pursuing in this article. One is the important family of Lie group integrators for problems of linear type, including methods based on the Magnus and Fer expansions. An excellent review of the history, theory and applications of such integrators can be found in [2]. We will also skip all discussions of order analysis of Lie group integrators. This is a large area by itself which involves technical tools and mathematical theory which we do not wish to include in this relatively elementary exposition. There have been several new developments in this area recently, in particular by Lundervold and Munthe-Kaas, see e.g. [47].

2.2 Lie group integrators The simplest consistent method for solving ordinary differential equations is the Euler method. For an initial value problem of the form y˙ = F (y),

y(0) = y 0 ,

one takes a small time increment h, and approximates y(h) by the simple formula y 1 = y 0 + hF (y 0 ),

24

2.2 Lie group integrators advancing along the straight line coinciding with the tangent at y 0 . Another way of thinking about the Euler method is to consider the constant vector field F y 0 (y) := F (y 0 ) obtained by parallel translating the vector F (y 0 ) to all points of phase space. A step of the Euler method is nothing else than computing the exact h-flow of this simple vector field starting at y 0 . In Lie group integrators, the same principle is used, but allowing for more advanced vector fields than the constant ones. A Lie group generalization of the Euler method is called the Lie–Euler method, and we shall illustrate its use through an example [22]. Example, the Duffing equation.

Consider the system in R2

x˙ = y,

y˙ = −ax − bx 3 ,

a ≥ 0, b ≥ 0,

(2.1)

a model used to describe the buckling of an elastic beam. Locally, near a point (x 0 , y 0 ) we could use the approximate system x˙ = y, ¡ ¢ y˙ = − a + bx 02 x,

x(0) = x 0 ,

(2.2)

y(0) = y 0 ,

which has the exact solution ¯ ) = x 0 cos ωt + x(t

y0 sin ωt , ω

¯ ) = y 0 cos ωt − ωx 0 sin ωt , y(t

ω=

q

a + bx 02 .

(2.3)

Alternatively, we may consider the local problem x˙ = y,

y˙ = −ax − bx 03 ,

having exact solution y0 cos αt − 1 sin αt + b x 03 , α α2 sin αt ¯ ) = y 0 cos αt − αx 0 sin αt − b x 03 y(t , α

¯ ) = x 0 cos αt + x(t

α=

p

a.

¯ ¯ In each of the two cases, one may take x 1 = x(h), y 1 = y(h) as the numerical approximation at time t = h. The same procedure is repeated in subsequent steps. A common framework for discussing these two cases is provided by the

25

2 An introduction to Lie group integrators – basics, new devel. and appl. 2

2

1

1

0

0

−1

−1

−2 −2

−1

0

1

2

−2 −2

−1

0

1

2

Figure 2.1: (Rd , +)-frozen vector field (left) and sl(2)-frozen vector field (right) for the Duffing equation. Both are frozen at (x 0 , y 0 ) = (0.75, 0.75). The thin black curve in each plot shows the flows of the frozen vector fields 0 ≤ t ≤ 20. The thicker curve in each plot is the exact flow of the Duffing equation. use of frames, i.e. a set of of vector fields which at each point is spanning the tangent space. In the first case, the numerical method applies the frame · ¸ · ¸ y 0 X= =: y ∂x, Y = =: x ∂y. (2.4) x 0 Taking the negative Jacobi–Lie bracket (also called the commutator) between X and Y yields the third element of the standard basis for the Lie algebra sl(2), i.e. H = −[X , Y ] = x ∂x − y ∂y, (2.5)

so that the frame may be augmented to consist of {X , Y , H }. In the second case, the vector fields E 1 = y ∂x − ax ∂y and E 2 = ∂y can be used as a frame, but again we choose to augment these two fields with the commutator E 3 = −[E 1 , E 2 ] = ∂x to obtain the Lie algebra of the special Euclidean group SE(2) consisting of translations and rotations in the plane. The situation is illustrated in Figure 2.1. In the left part, we have considered the constant vector field corresponding to the Duffing vector field evaluated at (x 0 , y 0 ) = (0.75, 0.75), and the exact flow of this constant field is just the usual Euler method, a straight line. In the right part, we have plotted the vector field defined in (2.2) with the same (x 0 , y 0 ) along with its flow (2.3). The exact flow of (2.1) is shown in both plots (thick curve).

26

2.2 Lie group integrators In general, a way to think about Lie group integrators is that we have a manifold M where there is such a frame available; {E 1 , . . . , E d } such that at any point p ∈ M one has span{E 1 (p), . . . , E d (p)} = Tp M . Frames with this property are said to be locally transitive. The frame may be a linear space or in many cases even a Lie algebra g of vector fields. In the example with Duffing’s equation, the set {X , Y , H } is locally transitive on R2 \ {0} and {E 1 , E 2 , E 3 } is locally transitive on R2 . Given an arbitrary vector field F on M , then at any point p ∈ M there exists a vector field F p in the span of the frame vector fields such that F p (p) = F (p). An explicit way of writing this is by using a set of basis vector fields E 1 , . . . , E d for g, such that any smooth vector field F has a representation F (y) =

d X

f k (y)E k (y),

(2.6)

k=1

for some functions f k : M → R. The vector fields F p ∈ g, called vector fields with frozen coefficients by Crouch and Grossman [22], are then obtained as F p (y) =

d X

f k (p)E k (y).

k=1

In the example with the Duffing equation we took E 1 = X , E 2 = Y , f 1 (x, y) = 1 and f 2 (x, y) = −(a + bx 2 ). The Lie–Euler method reads in general y n+1 = exp(hF y n )y n ,

(2.7)

where exp denotes the flow of a vector field. A more interesting example, also found in [22] is obtained by choosing M = 2 S , the 2-sphere. A suitable way to induce movements of the sphere is that of rotations, that is, by introducing the Lie group SO(3) consisting of orthogonal matrices with unit determinant. The corresponding Lie algebra so(3) of vector fields are spanned by E 1 (x, y, z) = −z ∂y + y ∂z,

E 2 (x, y, z) = z ∂x − x ∂z,

E 3 (x, y, z) = −y ∂x + x ∂y.

27

2 An introduction to Lie group integrators – basics, new devel. and appl. We note that xE 1 (x, y, z) + yE 2 (x, y, z) + zE 3 (x, y, z) = 0, showing that the functions f k in (2.6) are not unique. A famous example of a system whose solution evolves on S 2 is the free rigid body Euler equations µ ¶ µ ¶ µ ¶ 1 1 1 1 1 1 x˙ = − y z, y˙ = − xz, z˙ = − x y, (2.8) I3 I2 I1 I3 I2 I1 where x, y, z are the coordinates of the angular momentum relative to the body, and I 1 , I 2 , I 3 are the principal moments of inertia. A choice of representation (2.6) is obtained with f 1 (x, y, z) = −

x , I1

f 2 (x, y, z) = −

y , I2

f 3 (x, y, z) = −

z , I3

so that the ODE vector field can be expressed in the form       0 z −y x   y   z   −z − 0 − x . F (x, y, z) = − I1 I2 I3 y −x 0

We compute the vector field with coefficients frozen at p 0 = (x 0 , y 0 , z 0 ),    0 x F p 0 (x, y, z) = F p 0  y  := −z 0 /I 3 z y 0 /I 2

z 0 /I 3 0 −x 0 /I 1

  −y 0 /I 2 x x 0 /I 1   y  . z 0

The h-flow of this vector field is the solution of a linear system of ODEs and ¡ ¢ can be expressed in terms of the matrix exponential expm hF p 0 . The Lie–Euler method can be expressed as follows: p 0 ← (x 0 , y 0 , z 0 ) for n ← 0, 1, . . . do ¡ ¢ p n+1 ← expm hF p n p n end for Notice that the matrix to be exponentiated belongs to the matrix group so(3) of real skew-symmetric matrices. The celebrated Rodrigues’ formula expm(A) = I +

sin α 1 − cos α 2 A+ A , α α2

α2 = kAk22 = 12 kAk2F ,

provides an inexpensive way to compute this.

28

A ∈ so(3),

2.2 Lie group integrators Whereas the notion of frames was used by Crouch and Grossman in their pioneering work [22], a different type of notation was used in a series of papers by Munthe-Kaas [54–56], see also [47] for a more modern treatment. Let G be a finite dimensional Lie group acting transitively on a manifold M . A Lie group action is generally a map from G × M into M , having the properties that e · m = m, ∀m ∈ M ,

g · (h · m) = (g · h) · m, ∀g , h ∈ G, m ∈ M ,

where e is the group identity element, and the first · in the right hand side of the second identity is the group product. Transitivity means that for any two points m 1 , m 2 ∈ M there exists a group element g ∈ G such that m 2 = g · m 1 . We denote the Lie algebra of G by g. For any element ξ ∈ g there exists a vector field on M ¯ d ¯¯ X ξ (m) = exp(t ξ) · m =: λ∗ (ξ)(m). (2.9) dt ¯t =0

Munthe-Kaas introduced a generic representation of a vector field F ∈ X(M ) by a map f : M → g such that ¡ ¢ F (m) = λ∗ f (m) (m). (2.10)

The corresponding frame is E i = λ∗ (e i ) where {e 1 , . . . , e d } is some basis for g P and one chooses the functions f i : M → R such that f (m) = di=1 f i (m)e i . The map λ∗ is an anti-homomorphism of the Lie algebra g into the Lie algebra of vector fields X(M ) under the Jacobi–Lie bracket, meaning that λ∗ ([X m , Ym ]g ) = −[λ∗ (X m ), λ∗ (Ym )]JL . This separation of the Lie algebra g from the manifold M allows for more flexibility in the way we represent the frame vector fields. For instance, in the example with Duffing’s equation and the use of sl(2), we could have used the matrix Lie algebra with basis elements · ¸ · ¸ · ¸ 0 1 0 0 1 0 , Ym = , Hm = Xm = , 0 0 1 0 0 −1 rather than the basis of vector fields (2.4), (2.5). The group action by g ∈ SL(2) on a point m ∈ R2 would then be simply g · m, matrix-vector multiplication, and the exp in (2.9) would be the matrix exponential. The map f (x, y) would in this case be · ¸ 0 y f : (x, y) 7→ , −(a + bx 2 ) 0

29

2 An introduction to Lie group integrators – basics, new devel. and appl. but note that since the dimension of the manifold is just two whereas the dimension of sl(2) is three, there is freedom in the choice of f . In the example we chose not to use the third basis element H .

2.2.1 Generalizing Runge–Kutta methods In order to construct general schemes, as for instance a Lie group version of the Runge–Kutta methods, one needs to introduce intermediate stage values. This can be achieved in a number of different ways. They all have in common that when the methods are applied in Euclidean space where the Lie group is (Rm , +), they reduce to conventional Runge–Kutta schemes. Let us begin by studying the simple second order Heun method, sometimes called the improved Euler method. k 1 = F (y n ),

k 2 = F (y n + hk 1 ),

y n+1 = y n + 12 h(k 1 + k 2 ).

Geometrically, we may think of k 1 and k 2 as constant vector fields, coinciding with the exact ODE F (y) at the points y n and y n + hk 1 respectively. The update y n+1 can be interpreted in at least three different ways, ¶ µ µ ¶ µ ¶ µ ¶ µ ¶ h h h h h exp (k 1 + k 2 ) · y n , exp k 1 · exp k 2 · y n , exp k 2 · exp k 1 · y n . 2 2 2 2 2 (2.11) The first is an example of a Runge–Kutta–Munthe-Kaas method and the second is an example of a Crouch–Grossman method. All three fit into the framework of commutator-free Lie group methods. All three suggestions above are generalizations that will reduce to Heun’s method in (Rm , +). In principle we could extend the idea to Runge–Kutta methods with several stages y n+1 = y n + h

s X i =1

b i F (Yi ),

Yi = y n + h

s X j =1

a i j F (Y j ),

i = 1, . . . , s,

by for instance interpreting the summed expressions as vector fields with frozen coefficients whose flows we apply to the point y n ∈ M . But it is unfortunately not true that one in this way will retain the order of the Runge–Kutta method when applied to cases where the acting group is non-abelian. Let us first describe methods as proposed by Munthe-Kaas [54], where one may think of the method simply as a change of variable. As before, we assume that the action of G on M is locally transitive. Since the exponential mapping

30

2.2 Lie group integrators is a local diffeomorphism in some open set containing 0 ∈ g, it is possible to represent any smooth curve y(t ) on M in some neighbourhood of a point p ∈ M by means of a curve σ(t ) through the origin of g as follows ¡ ¢ y(t ) = exp σ(t ) · p, σ(0) = 0, (2.12) though σ(t ) is not necessarily unique. We may differentiate this curve with respect to t to obtain µ ³ ¶ ¡ ¢¡ ¢ ¡ ¢ ¡ ¢ ´ ¡ ¢ ˙ ) = λ∗ dexpσ(t ) σ(t ˙ ) y(t ) = F y(t ) = λ∗ f exp σ(t ) · p y(t ) . (2.13) y(t The details are given in [54] and the map dexpσ : g → g was derived by Hausdorff [32] as an infinite series of commutators £ ¤ dexpσ (v) = v + 12 [σ, v] + 16 σ, [σ, v] + · · · ∞ X 1 = adkσ v (2.14) (k + 1)! k=0 ¯ exp(z) − 1 ¯¯ v, = ¯ z z=adσ

with the usual definition of adu (v) as the commutator [u, v]. The map λ∗ does not have to be injective, but a sufficient condition for (2.13) to hold is that ³ ¡ ¢´ ˙ = dexp−1 σ σ f exp(σ) · p . This is a differential equation for σ(t ) on a linear space, and one may choose any conventional integrator for solving it. The map dexp−1 σ : g → g is the inverse of dexpσ and can also be obtained by differentiating the logarithm, i.e. the inverse of the exponential map. From (2.14) we find that one can write dexp−1 σ (v) as ¯ ¯ £ ¤ z −1 1 ¯ dexpσ (v) = v = v − 12 [σ, v] + 12 σ, [σ, v] + · · · . (2.15) ¯ exp(z) − 1 z=adσ

The coefficients appearing in this expansion are scaled Bernoulli numbers Bk!k , and B 2k+1 = 0 for all k ≥ 1. One step of the resulting Runge–Kutta–Munthe-Kaas method is then expressed in terms of evaluations of the map f as follows µ s ¶ X y 1 = exp h bi ki · y 0 , i =1

P k i = dexp−1 h j ai j k j

Ã

! µ ¶ X f exp h a i j k j · y 0 , j

i = 1, . . . , s.

31

2 An introduction to Lie group integrators – basics, new devel. and appl. This is not so surprising seen from the perspective of the first alternative in (2.11), the main difference is that the stages k i corresponding to the frozen vector fields λ∗ (k i ) need to be “corrected” by the dexp−1 map. Including this map in computational algorithms may seem awkward, however, fortunately truncated versions of (2.15) may frequently be used. In fact, by applying some clever tricks involving graded free Lie algebras, one can in many cases replace dexp−1 with a low order Lie polynomial while retaining the convergence order of the original Runge–Kutta method. Details of this can be found in [9, 57]. There are also some important cases of Lie algebras for which dexp−1 σ can be computed exactly in terms of elementary functions, among those is so(3) reported in [16]. Notice that the representation (2.12) does not depend on the use of the exponential map from g to G. In principle, one can replace this map with any local diffeomorphism ϕ, where one usually scales ϕ such that ϕ(0) = e and T0 ϕ = Idg . An example of such map is the Cayley transformation [25] which can be used for matrix Lie groups of the type G P = {X ∈ Rn×n | X T P X = P } for a nonsingular n × n-matrix P . These include the orthogonal group O(n) = G I and the linear symplectic group Sp(n) = G J where J the skew-symmetric matrix of the standard symplectic form. Another possibility is to replace the exponential map by canonical coordinates of the second kind [61]. We present here the well-known Runge–Kutta–Munthe-Kaas method based on the popular fourth order method of Kutta [38], having Butcher tableau

0 1 2

1 2

1 2

0

1 2

1

0

0

1

1 6

1 3

1 3

(2.16)

1 6

In the Lie group method, the dexp−1 map has been replaced by the optimal Lie

32

2.2 Lie group integrators polynomials. k 1 = h f (y 0 ), ´ ³ ¢ ¡ k 2 = h f exp 12 k 1 · y 0 , ´ ³ ¢ ¡ k 3 = h f exp 12 k 2 − 18 [k 1 , k 2 ] · y 0 , ¡ ¢ k 4 = h f exp(k 3 ) · y 0 , ³ ¡ ¢´ y 1 = exp 16 k 1 + 2k 2 + 2k 3 + k 4 − 12 [k 1 , k 4 ] · y 0 . An important advantage of the Runge–Kutta–Munthe-Kaas schemes is that it is easy to preserve the convergence order when extending them to Lie group integrators. This is not the case with for instance the schemes of Crouch and Grossman [22, 62], where it is necessary to develop order conditions for the nonabelian case. This is also true for the commutator-free methods developed by Celledoni, Marthinsen and Owren [14]. In fact, these methods include those of Crouch and Grossman. The idea here is to allow compositions of exponentials or flows instead of commutator corrections. With stages k 1 , . . . , k s in the Lie algebra, one includes expressions of the form µ ¶ µ ¶ µ ¶ X i X i X i exp β J k i · · · exp β2 k i · exp β1 k i · y 0 , i

i

i

both in the definition of the stages and the update itself. In some cases it is also possible to reuse flow calculations from one stage to another, and thereby lower the computational cost of the scheme. An extension of (2.16) can be obtained as follows, setting k i = h f (Yi ) for all i , Y1 = y 0 ,

¢ ¡ Y2 = exp 21 k 1 · y 0 , ¡ ¢ Y3 = exp 12 k 2 · y 0 ¡ ¢ Y4 = exp k 3 − 12 k 1 · Y2 , ¡1 ¢ y 1 = exp 12 (3k 1 + 2k 2 + 2k 3 − k 4 ) · y 0 , 2 ¡1 ¢ y 1 = exp 12 (−k 1 + 2k 2 + 2k 3 + 3k 4 ) · y 1 . 2

Note in particular in this example how the expression for Y4 involves Y2 and thereby one exponential calculation has been saved.

33

2 An introduction to Lie group integrators – basics, new devel. and appl.

2.2.2 A plenitude of group actions We saw in the first examples with Duffing’s equation that the manifold M , the group G and even the way G acts on M can be chosen in different ways. It is not obvious which action is the best or suits the purpose in the problem at hand. Most examples we know from the literature are using matrix Lie groups G ⊆ GL(n), but the choice of group action depends on the problem and the objectives of the simulation. We give here several examples of situations where Lie group integrators can be used. G acting on G . In the case M = G, it is natural to use either left or right multiplication as the action L g (m) = g · m

or R g (m) = m · g ,

g , m ∈ G.

The correspondence between the vector field F ∈ X(M ) and the map (2.10) is then just the tangent map of left or right multiplication ¡ ¢ ¡ ¢ F (g ) = Te L g f (g ) or F (g ) = Te R g f˜(g ) ,

g ∈ G.

When working with matrices, this simply amounts to setting F (g ) = g · f (g ) or F (g ) = f˜(g ) · g . Note that f˜(g ) is related to f (g ) through the adjoint representation of G, Ad:G → Aut(g), f˜(g ) = Adg f (g ),

Adg = Te L g ◦ Te R g−1 .

Lie group integrators can also be used for approximating the solution to partial differential equations, the most obvious choice of PDE model being the semilinear problem The affine group and its use in semilinear PDE methods.

u t = Lu + N (u),

(2.17)

where L is a linear differential operator and N (u) is some nonlinear map, typically containing derivatives of lower order than L. After discretizing in space, (2.17) is turned into a system of n d ODEs, for some large n d , L becomes an n d × n d -matrix, and N : Rnd → Rnd a nonlinear function. We may now as in [54] introduce the action on Rnd by some subgroup of the affine group represented

34

2.2 Lie group integrators as the semidirect product G = GL(n d ) n Rnd . The group product, identity, and inverse are given as (A 1 , b 1 ) · (A 2 , b 2 ) = (A 1 A 2 , A 1 b 2 + b 1 ),

e = (I, 0),

(A, b)−1 = (A −1 , −A −1 b).

The action on Rnd is (A, b) · x = Ax + b,

(A, b) ∈ G, x ∈ Rnd ,

and the Lie algebra and commutator are given as

g = (ξ, c), ξ ∈ gl(n d ), c ∈ Rnd , [(ξ1 , c 1 ), (ξ2 , c 2 )] = ([ξ1 , ξ2 ], ξ1 c 2 − ξ2 c 1 + c 1 ). In many interesting PDEs, the operator L is constant, so it makes sense to consider the n d + 1-dimensional subalgebra gL of g consisting of elements (αL, c) where α ∈ R, c ∈ Rnd , so that the map f : Rnd → gL is given as ¡ ¢ f (u) = L, N (u) . One parameter subgroups are obtained through the exponential map as follows ¡ ¢ ¡ ¢ exp t (L, b) = exp(t L), φ(t L)t b . ¡ ¢ Here the entire function φ(z) = exp(z) − 1 /z familiar from the theory of exponential integrators appears. As an example, one could now consider the Lie–Euler method (2.7) in this setting, which coincides with the exponential Euler method ³ ¡ ¢´ u 1 = exp h L, N (u 0 ) · u 0 = exp(hL)u 0 + hφ(hL)N (u 0 ). There is a large body of literature on exponential integrators, going almost half a century back in time, see [34] and the references therein for an extensive account. Lie group integrators for this interesting case were studied by Engø and Faltinsen [27]. Suppose G is a Lie group and the manifold under consideration is the dual space g∗ of its Lie algebra g. The coadjoint action by G on g∗ is denoted Ad∗g defined for any g ∈ G as ∗ ® Adg µ, ξ = 〈µ, Adg ξ〉, ∀ξ ∈ g, µ ∈ g∗ , (2.18) The coadjoint action and Lie–Poisson systems.

35

2 An introduction to Lie group integrators – basics, new devel. and appl. for a duality pairing 〈·, ·〉 between g∗ and g. It is well known (see e.g. Section 13.4 in [48]) that mechanical systems formulated on the cotangent bundle T∗G with a left or right invariant Hamiltonian can be reduced to a system on g∗ given as µ˙ = ± ad∗∂H µ, ∂µ

where the negative sign is used in case of right invariance. The solution to this system preserves coadjoint orbits, which makes it natural to suggest the group action g · µ = Ad∗g −1 µ, so that the resulting Lie group integrator also respects this invariant. For Euler’s equations for the free rigid body, the Hamiltonian is left invariant and the coadjoint orbits are spheres in g∗ ∼ = R3 . The situation when G acts on itself by left of right multiplication is a special case of a homogeneous space [59], where the assumption is only that G acts transitively and continuously on some manifold M . Homogeneous spaces are isomorphic to the quotient G/G x where G x is the isotropy group for the action at an arbitrarily chosen point x ∈ M

Homogeneous spaces and the Stiefel and Grassmann manifolds.

G x = {h ∈ G | h · x = x}. Note that if x and z are two points on M , then by transitivity of the action, z = g ·x for some g ∈ G. Therefore, whenever h ∈ G z it follows that g −1 ·h ·g ∈ G x so isotropy groups are isomorphic by conjugation [6]. Therefore the choice of x ∈ M is not important for the construction of the quotient. For a readable introduction to this type of construction, see [6], in particular Lecture 3. A much encountered example is the hypersphere M = S d −1 corresponding to the left action by G = SO(d ), the Lie group of orthogonal d × d matrices with unit determinant. One has S d −1 = SO(d )/ SO(d − 1). We have in fact already discussed the example of the free rigid body (2.8) where M = S 2 . The Stiefel manifold St(d , k) can be represented by the set of d × k-matrices with orthonormal columns. An action on this set is obtained by left multiplication by G = SO(d ). Lie group integrators for Stiefel manifolds are extensively studied in the literature, see e.g. [17, 37] and some applications involving Stiefel

36

2.2 Lie group integrators manifolds are discussed in Section 2.4. An important subclass of the homogeneous spaces is the symmetric spaces, also obtained through a transitive action by a Lie group G, where M = G/G x , but here one requires in addition that the isotropy subgroup is an open subgroup of the fixed point set of an involution of G [58]. A prominent example of a symmetric space in applications is the ¡ ¢ Grassmann manifold, obtained as SO(d )/ SO(k) × SO(d − k) . In isospectral integration one considers dynamical systems evolving on the manifold of d × d -matrices sharing the same Jordan form. Considering the case of symmetric matrices, one can use the transitive group action by SO(d ) given as Isospectral flows.

g · m = g mg T . This action is transitive, since any symmetric matrix can be diagonalized by an appropriately chosen orthogonal matrix. If the eigenvalues are distinct, then the isotropy group is discrete and consists of all matrices in SO(d ) which are diagonal. Lie group integrators for isospectral flows have been extensively studied, see for example [7, 8]. See also [10] for an application to the KdV equation.

For mechanical systems the natural phase space will often be the tangent bundle TM as in the Lagrangian framework or the cotangent bundle T∗M in the Hamiltonian framework. The seminal paper by Lewis and Simo [45] discusses several Lie group integrators for mechanical systems on cotangent bundles, deriving methods which are symplectic, energy and momentum preserving. Engø [26] suggested a way to generalize the Runge–Kutta–Munthe-Kaas methods into a partitioned version when M is a Lie group. Marsden and collaborators have developed the theory of Lie group integrators from the variational viewpoint over the last two decades. See [49] for an overview. For more recent work pertaining to Lie groups in particular, see [3, 41, 67]. In Section 2.5 we present what we believe to be the first symplectic partitioned Lie group integrators on T∗G phrased in the framework we have discussed here. Considering trivialized cotangent bundles over Lie groups is particularly attractive since there is a natural way to extend action by left multiplication from G to G × g∗ via (2.25). Tangent and cotangent bundles.

37


2.2.3 Isotropy – challenges and opportunities An issue which we have already mentioned a few times is that the map λ∗ : g → X(M ) defined in (2.9) is not necessarily injective. This means that the choice of ¢ ¡ f : M → g is not unique. In fact, if g : M → g is any map satisfying λ∗ g (m) (m) = 0 for all m ∈ M , then we could replace the map f by f + g in (2.10) without altering the vector field F . But such a modification of f will have an impact on the numerical schemes that we consider. This freedom in the setup of the methods makes it challenging to prove general results for Lie group methods, it might seem that some restrictions should apply to the isotropy choice for a more well defined class of schemes. However, the freedom can of course also be taken advantage of to obtain approximations of improved quality. An illustrative example is the two-sphere S 2 acted upon linearly by the special orthogonal group SO(3). Representing elements of the Lie algebra so(3) by vectors in R3 , and points on the sphere as unit length vectors in R3 , we may facilitate (2.10) as ¡ ¢ F (m) = f (m) × m = f (m) + α(m)m × m,

for any scalar function α: S 2 → R. Using for instance the Lie–Euler method one would get ¡ ¢ (2.19) m 1 = exp f (m 0 ) + α(m 0 )m 0 m 0 , where the exp is the matrix exponential of the 3 × 3 skew-symmetric matrix associated to a vector in R3 via the hat-map (2.20). Clearly the approximation depends on the choice of α(m). The approach of Lewis and Olver [44] was to use the isotropy to improve certain qualitative features of the solution. In particular, they studied how the orbital error could be reduced by choosing the isotropy in a clever way. In Figure 2.2 we illustrate the issue of isotropy for the Euler free rigid body equations. The curve drawn from the initial point z 0 to z 1 is the exact solution, i.e. the momenta in body coordinates. The broken line shows the terminal points using the Lie–Euler method for α varying between 0 and 25. Another potential difficulty with isotropy is the increased computational complexity when the group G has much higher dimension than the manifold M . This could for instance be the case with the Stiefel manifold St(d , k) if d À k. Linear algebra operations used in integrating differential equations on the Stiefel manifold should preferably be of complexity O(d k 2 ). But solving a

38

2.3 Applications to nonlinear problems of evolution in classical mechanics

1 α = 25

0.5 z0 0

z1

−0.5

α = 11.4

α=0

−1 −0.5

0

0.5

0−0.5 0.5 1

Figure 2.2: The effect of isotropy on S 2 for Euler’s free rigid body equations. The curve drawn from the initial point z 0 to z 1 is the exact solution, i.e. the momenta in body coordinates. The broken line shows the terminal points using the Lie–Euler method for α(z 0 ) (as in (2.19)) varying between 0 and 25.

corresponding problem in the Lie algebra so(d ) would typically require linear algebra operations of complexity O(d 3 ), see for example [17] and references therein. By taking advantage of the many degrees of freedom provided by the isotropy, it is actually possible to reduce the cost down to the required O(d k 2 ) operations as explained in for instance [16] and [37].

2.3 Applications to nonlinear problems of evolution in classical mechanics The emphasis on the use of Lie groups in modelling and simulation of engineering problems in classical mechanics started in the eighties with the pioneering and fundamental work of J. C. Simo and his collaborators. In the case of rod dynamics, for example, models based on partial differential equations were considered where the configuration of the centreline of the rod is parameter-

39


φ t1 φr t2

t3

e1 e2

e3

Figure 2.3: Geometric rod model. Here φ is the line of centroids and a cross section is identified by the frame Λ = [t 1 , t 2 , t 3 ], φr is the initial position of the line of centroids. ized via arc-length, and the movement of a rigid frame attached to each of the cross sections of the rod is considered (see Figure 2.3). This was first presented in a geometric context in [70]. In robot technology, especially robot locomotion and robot grasping, the occurrence of non-holonomically constrained models is very common. The motion of robots equipped with wheels is not always locally controllable, but is often globally controllable. A classical example is the parking of a car that cannot be moved in the direction orthogonal to its wheels. The introduction of Lie groups and Lie brackets to describe the dynamics of such systems, has been considered by various authors, see for example [60]. The design of numerical integration methods in this context has been addressed in the paper of Crouch and Grossman [22]. These methods have had a fundamental impact to the successive developments in the field of Lie group methods. The need for improved understanding of non-holonomic numerical integration has been for example advocated in [52]. Recent work in this field has led to the construction of low order non-holonomic integrators based on a discrete Lagrange–d’Alembert principle, [21, 50]. The use of Lie group integrators in this context has been considered in [42, 50].

40

2.3 Applications to nonlinear problems of evolution in classical mechanics We have already mentioned the relevance of rigid body dynamics to the numerical discretization of rod models. There are many other research areas in which the accurate and efficient simulation of rigid body dynamics is crucial: molecular dynamics, satellite dynamics, and celestial mechanics just to name a few, [43]. In some of these applications, it is desirable to produce numerical approximations which are accurate possibly to the size of roundoff. The simulations of interest occur over very long times and/or a large number of bodies and this inevitably causes propagation of errors even when the integrator is designed to be very accurate. For this reason accurate symplectic rigid body integrators are of interest because they can guarantee that the roundoff error produced by the accurate computations can stay bounded also in long time integration. This fact seems to be of crucial importance in celestial mechanics simulations, [39]. A symplectic and energy preserving Lie group integrator for the free rigid body motion was proposed in [45]. The method computes a time re-parameterization of the exact solution. Some recent and promising work in this field has been presented in [11, 19, 31, 53]. The control of rigid bodies with variational Lie group integrators was considered in [42]. In the next section we illustrate the use of Lie group methods in applications on a particular case study, the pipe-laying process from ships to the bottom of the sea.

2.3.1 Rigid body and rod dynamics The simulation of deep-water risers, pipelines and drill rigs requires the use of models of long and thin beams subject to large external forces. These are complicated nonlinear systems with highly oscillatory components. We are particularly interested in the correct and accurate simulation of the pipe-laying process from ships on the bottom of the sea, see Figure 2.4. The problem comprises the modelling of two interacting structures: a long and thin pipe (modelled as a rod) and a vessel (modelled as a rigid body). The system is subject to environmental forces (such as sea and wind effects). The control parameters for this problem are the vessel position and velocity, the pay-out speed and the pipe tension while the control objectives consist in determining the touchdown position of the pipe as well as ensuring the integrity of the pipe and to avoid critical deformations, [36, 68]. Pipe-laying problem.

The vessel rigid body equations determine the boundary conditions of the

41

2 An introduction to Lie group integrators – basics, new devel. and appl. Pipeline vessel x z

Overbend

Departure angle Inflection point

Fireline Stinger Stinger tip

Pipe

Touchdown point

Sagbend

Seabed

Figure 2.4: The pipe-laying process. rod. They are expressed in six degrees of freedom as ˙ +C (ν)ν + D(ν)ν + g (η) = τ, Mν where M is the system inertia matrix, C (ν) the Coriolis-centripetal matrix, D(ν) the damping matrix, g (η) the vector of gravitational and buoyancy forces and moments, and τ the vector of control inputs and environmental disturbances such as wind, waves and currents (see [64] for details). The vector ν contains linear and angular velocity and η is the position vector. It has been shown in [36] that the rigid body vessel equations are input-output passive. The equations can be integrated numerically with a splitting and composition technique where the vessel equations are split into a free rigid body part and a damping and control part. The free rigid body equations can be solved with a method proposed in [19] where the angular momentum is accurately and efficiently computed by using Jacobi elliptic functions, the attitude rotation is obtained using a Runge–Kutta–Munthe-Kaas Lie group method, and the control and damping part is solved exactly. Simulations of the whole pipe-lay problem with local parameterizations of the pipe and the vessel based on Euler angles have been obtained in [36].

42

2.3 Applications to nonlinear problems of evolution in classical mechanics At fixed time each cross section of the pipe is the result of a rigid rotation in space of a reference cross section, and analogously, for each fixed value of the space variable the corresponding cross section evolves in time as a forced rigid body, see Figure 2.3. In absence of external forces the equations are Rod dynamics.

∂t π +

¡

ρ A ∂t t φ = ∂S n, S ∈ [0, L], ¢ × π = ∂S m + (∂S φ) × n,

I ρ−1 π

t ≥ 0,

here φ = φ(S, t ) is the line of centroids of the rod, m and n are the stress couple and stress resultant, π is the angular momentum density, I ρ is the inertia in spatial coordinates, and ρ A = ρ A (S) is the mass per unit length of the rod (see [71] and [18]). The kinematic equations for the attitude rotation matrix are ∂t Λ = wˆ Λ,

ˆ ∂S Λ = MΛ,

where Λ(S, t ) = [t 1 , t 2 , t 3 ], I ρ−1 π = w , M = C 2 ΛT m and C 2 is a constant diagonal matrix. We denote by “ ˆ " the hat-map identifying R3 with so(3):     v1 0 −v 3 v2 0 −v 1  . v = v 2  7→ vˆ =  v 3 (2.20) v3 −v 2 v1 0 With no external forces one assumes pure displacement boundary conditions providing φ and Λ on the boundaries S = 0 and S = L. In [71], partitioned Newmark integrators, of Lie group type, and of moderate order were considered for this problem. While classical Newmark methods are variational integrators and as such are symplectic when applied to Hamiltonian systems [49], the particular approach of [71] involves the use of exponentials for the parameterization of the Lie group SO(3), and the geometric properties of this approach are not trivial to analyse. Moreover, since the model is a partial differential equation, space and time discretizations should be designed so that the overall discrete equations admit solutions and are stable. It turns out that conventional methods perform poorly on such problem in long time simulations. To obtain stable methods reliable in long-time simulation, an energy-momentum method was proposed for the rod problem in [72]. Later, this line of thought has been further developed in [66]. The Hamiltonian formulation of this model allows one to derive natural structure preserving discretizations into systems of coupled rigid bodies [46].

43

2 An introduction to Lie group integrators – basics, new devel. and appl. Following the geometric space-time integration procedure proposed in [28], a multi-Hamiltonian formulation1 of these equations has been proposed in [18], using the Lie group of Euler parameters. The design of corresponding multisymplectic Lie group discretizations is still under investigation.

2.4 Applications to problems of data analysis and statistical signal processing The solution of the many-body Schrödinger eigenvalue problem Hˆ Ψ = E Ψ,

(2.21)

where the so called electronic ground state (the smallest eigenstate) is sought, is an important problem of computational chemistry. The main difficulty is the curse of dimensionality. Since Hˆ is a differential operator in several space dimensions, a realistic simulation of (2.21) would require the numerical discretization and solution of a partial differential equation in several space dimensions. The number of space dimensions grows with the number of electrons included in the simulation. The eigenvalue problem admits an alternative variational formulation. Instead of looking for the smallest eigenvalue and eigenfunction of the Schrödinger equation, one minimizes directly the ground state energy. After appropriate spatial discretization, the problem becomes a minimization problem on a Riemannian manifold M , min φ(x), x∈M

(2.22)

where φ: M → R is a smooth discrete energy function to be minimized on M (see [1]). The discrete energy φ considered here is the so called Kohn–Sham energy. For background on density functional theory see for example [63]. For a related application of Lie group techniques in quantum control, see [23]. The general optimization problem giving rise to (2.22) appears in several applied fields, ranging from engineering to applied physics and medicine. Some specific examples are principal component/subspace analysis, eigenvalue and generalized eigenvalue problems, optimal linear compression, noise reduction, signal representation and blind source separation. 1 For a definition of the multi-symplectic structure of Hamiltonian partial differential equations,

see [4].

44

2.4 Applications to problems of data analysis and statistical signal processing

2.4.1 Gradient-based optimization on Riemannian manifolds Let M be a Riemannian manifold with metric 〈·, ·〉M and φ: M → R be a smooth cost function to be minimized on M . We want to solve (2.22). The optimization method based on gradient flow – written for the minimization problem only, for the sake of easy reading – consists in setting up the differential equation on the manifold, ¡ ¢ ˙ ) = − grad φ x(t ) , x(t (2.23)

with appropriate initial condition x(0) = x 0 ∈ M . The equilibria of (2.23) are the critical points of the function φ. In the above equation, the symbol grad φ denotes the Riemannian gradient of the function φ with respect to the chosen metric. Namely, grad φ(x) ∈ Tx M and Tx φ(v) = 〈grad φ(x), v〉M for all v ∈ Tx M . The solution of (2.23) on M may be locally expressed in terms of a curve on the tangent space Tx0 M using a retraction map R. Retractions are tangent space parameterizations of M , and allow us to write ¡ ¢ x(t ) = R x0 σ(t ) , σ(t ) ∈ Tx0 M , t ∈ [0, t f ], for small enough t f , see [69] for a precise definition. In most applications of interest, see for example [5, 33], M is a matrix manifold endowed with a Lie group action and there is a natural way to define a metric and a retraction. In fact, let M be a manifold acted upon by a Lie group G, with a locally transitive group action Λ(g , x) = Λx (g ). Let us also consider a coordinate map ψ, ψ: g → G,

and ρ x := T0 (Λx ◦ ψ).

One can prove that if there exists a linear map a x : Tx M → g such that ρ x ◦ a x = IdTx M , then R x , given by R x (v) := (Λx ◦ ψ ◦ a x )(v), is a retraction, see [17]. The existence of a x is guaranteed, at least locally, by the transitivity of the action and the fact that ψ is a local diffeomorphism. The approach is analogous to the one described for differential equations in Section 2.2.1. Therefore, we can construct retractions using any coordinate map from the Lie algebra g to the group. Any metric on g, 〈·, ·〉g induces a metric on M by 〈v x , w x 〉M = 〈a x (v x ), a x (w x )〉g .

45

2 An introduction to Lie group integrators – basics, new devel. and appl. Also, we may define the image of the tangent space under the map a x :

mx := a x (Tx M ) ⊂ g. The set mx is a linear subspace of the Lie algebra g, often of lower dimension. Parameterizations of the solution of (2.23) involving the whole Lie algebra are in general more computationally intensive than those restricted to mx , but, if the isotropy is chosen suitably, they might lead to methods which converge faster to the optimum. For the sake of illustration, we consider the minimization on a two-dimensional torus T 2 = S 1 × S 1 . Here we denote by S 1 the circle, i.e. S 1 = {g (α)e1 ∈ R2 | g (α) ∈ SO(2)}, · ¸ 0 −1 g (α) = exp(αE ), E = , 0 ≤ α < 2π, 1 0 where e1 is the first canonical vector and SO(2) is the commutative Lie group of planar rotations. Any element in T 2 is of the form x0 ∈ T 2 ,

¡ ¢ x 0 = g (θ)e1 , g (ϕ)e1 ,

g (θ), g (ϕ) ∈ SO(2).

The Lie group acting on T 2 is SO(2) × SO(2), its corresponding Lie algebra is so(2) × so(2), which has dimension d = 2 and basis {(E ,O), (O, E )}, where O is the zero element in so(2). The Lie group action is ¡ ¢ Λx0 (h 1 , h 2 ) = h 1 g (θ)e1 , h 2 g (ϕ)e1 ,

(h 1 , h 2 ) ∈ SO(2) × SO(2),

and ψ = exp. Any v x0 ∈ Tx0 T 2 can be written as v x0 = (αE e1 , βE e1 ), for some α, β ∈ R, so

a x0 (v x0 ) = (αE , βE ).

Assume the cost function we want to minimize is simply the distance from a fixed plane in R3 , say the plane with equation y = 8. This gives ¯ ¡ ¢ ¯¡ ¢ φ g (θ)e1 , g (ϕ)e1 = ¯ 1 + cos(θ) sin(ϕ) − 8¯,

46

2.4 Applications to problems of data analysis and statistical signal processing and the minimum is attained in θ = 0 and ϕ = π/2.2 In Figure 2.5 we plot − grad φ, the negative gradient vector field for the given cost function. The Riemannian metric we used is 〈(α1 E e1 , β1 E e1 ), (α2 E e1 , β2 E e1 )〉T 2 = α1 α2 + β1 β2 , and (α1 E e1 , β1 E e1 ) ∈ T(e1 ,e1 ) T 2 . This metric can be easily interpreted as a metric on the Lie algebra g = so(2) × so(2): 〈(α1 E , β1 E ), (α2 E , β2 E )〉g = α1 α2 + β1 β2 . ¡ ¢ At the point p 0 = g (θ0 )e1 , g (ϕ0 )e1 ∈ T 2 , the gradient vector field can be represented by ¡ ¢ γE g (θ0 )e1 , δE g (ϕ0 )e1 ,

where γ and δ are real values given by γ = −C sin(θ0 ) sin(ϕ0 ), and

¡ ¢ δ = C 1 + cos(θ0 ) cos(ϕ0 ),

³¡ ´ ¢ C = 2 1 + cos(θ0 ) sin(ϕ0 ) − 8 .

Gradient flows are not the only type of differential equations which can be used to solve optimization problems on manifolds. Alternative equations have been proposed in the context of neural networks [12, 13]. Often they arise naturally as the Euler–Lagrange equations of a variational problem.

2.4.2 Principal component analysis Data reduction techniques are statistical signal processing methods that aim at providing efficient representations of data. A well-known data compression 2 We have used a parameterization of T 2 in R3 in angular coordinates, obtained applying the

following mapping ¡ ¢ T ¡ ¢  x = 1 + eT  1 g (θ)e1 e1 g (ϕ)e1 = 1 + cos(θ) cos(ϕ),   ¡ ¢ ¡ ¢ T ¡ ¢ g (θ)e1 , g (ϕ)e1 7→ y = 1 + eT 1 g (θ)e1 e2 g (ϕ)e1 = 1 + cos(θ) sin(ϕ),    z = eT = sin(θ), 2 g (θ)e1

with 0 ≤ θ, ϕ < 2π. This is equivalent to the composition of two planar rotations and one translation in R3 .

47


2

1

−2

0

−1 y

0

z

−1

1 2

1

0

−1

−2

x

¡ ¢ Figure 2.5: The gradient vector field of the cost function φ g (θ)e1 , g (ϕ)e1 = ³¡ ´2 ¢ 1 + eT1 g (θ)e1 eT2 g (ϕ)e1 − 8 on the torus. The vector field points towards the two minima, the global minimum is marked with a black spot in the middle of the picture.

48

2.4 Applications to problems of data analysis and statistical signal processing technique consists of mapping a high-dimensional data space into a lower dimensional representation space by means of a linear transformation. It requires the computation of the data covariance matrix and then the application of a numerical procedure to extract its eigenvalues and the corresponding eigenvectors. Compression is then obtained by representing the signal in a basis consisting only of those eigenvectors associated with the most significant eigenvalues. In particular, principal component analysis (PCA) is a second-order adaptive statistical data processing technique that helps removing the secondorder correlation among given random signals. Let us consider a stationary multivariate random process x(t ) ∈ Rn and suppose its covariance mat£ ¤ rix A = E (x − E[x])(x − E[x])T exists and is bounded. Here the symbol E[·] denotes statistical expectation. If A ∈ Rn×n is not diagonal, then the components of x(t ) are statistically correlated. One can remove this redundancy by partially diagonalizing A, i.e. computing the operator F formed by the eigenvectors of the matrix A corresponding to its largest eigenvalues. This is possible since the covariance matrix A is symmetric (semi) positive-definite, and F ∈ St(n, p). To compute F and the corresponding p eigenvalues of the n × n symmetric and positive-definite matrix A, we consider the maximization of the function φ(X ) = 12 tr(X T AX ), on the Stiefel manifold, and solve numerically the corresponding gradient flow with a Lie group integrator. As a consequence the new random signal defined by ¡ ¢ y(t ) := F T x(t ) − E[x(t )] ∈ Rp

has uncorrelated components, with p ≤ n properly selected. The component signals of y(t ) are the so called principal components of the signal x(t ), and £ ¤ their relevance is proportional to the corresponding eigenvalues σ2i = E y i2 ¡ 2 ¢ which here are arranged in descending order σi ≥ σ2i +1 . Thus, the data stream y(t ) is a compressed version of the data stream x(t ). After the reduced-size data has been processed (i.e. stored, transmitted), it needs to be recovered, that is, it needs to be brought back to the original structure. However, the principal-component-based data reduction technique ˆ ) ∈ Rn of the original data stream is not lossless, thus only an approximation x(t ˆ ) = F y(t )+E[x]. Such may be recovered. An approximation of x(t ) is given by x(t

49

2 An introduction to Lie group integrators – basics, new devel. and appl. ¤ £ ˆ 22 = approximate data stream minimizes the reconstruction error E kx − xk Pp σ2 . i =n+1 i For a scalar or a vector-valued random variable x ∈ Rn endowed with a probability density function p x : x ∈ Rn → p x (x) ∈ R, the expectation of a function β: Rn → R is defined as Z E[β] := β(x)p x (x) dn x. Rn

Under the hypothesis that the signals whose expectation is to be computed are ergodic, the actual expectation (ensemble average) may be replaced by temporal-average on the basis of the available signal samples, namely E[β] ≈

T ¡ ¢ 1 X β x(t ) . T t =1

2.4.3 Independent component analysis An interesting example of a problem that can be tackled via statistical signal processing is the cocktail-party problem. Let us suppose n signals x 1 (t ), . . . , x n (t ) were recorded from n different positions in a room where there are p sources or speakers. Each recorded signal is a linear mixture of the voices of the sources s 1 (t ), . . . , s p (t ), namely x 1 (t ) = a 1,1 s 1 (t ) + · · · + a 1,p s p (t ), .. .

x n (t ) = a n,1 s 1 (t ) + · · · + a n,p s p (t ), where the np coefficients a i , j ∈ R denote the mixing proportions. The mixing matrix A = (a i , j ) is unknown. The cocktail party problem consists in estimating signals s 1 (t ), . . . , s p (t ) from only the knowledge of their mixtures x 1 (t ), . . . , x n (t ). The main assumption on the source signals is that s 1 (t ), . . . , s p (t ) are statistically independent. This problem can be solved using independent component analysis (ICA). Typically, one has n > p, namely, the number of observations exceeds the number of actual sources. Also, a typical assumption is that the source signals are spatially white, which means E[ss T ] = Ip , the p × p identity matrix. The aim of independent component analysis is to find estimates y(t ) of signals in s(t ) by

50

2.4 Applications to problems of data analysis and statistical signal processing constructing a de-mixing matrix W ∈ Rn×p and by computing y(t ) := W T x(t ). Using statistical signal processing methods, the problem is reformulated into an optimization problem on homogeneous manifolds for finding the de-mixing matrix W . The geometrical structure of the parameter space in ICA comes from a signal pre-processing step named signal whitening, which is operated on the observ˜ ) ∈ Rp in such a way that the components of the signal x(t ˜ ) able signal x(t ) → x(t are uncorrelated and have variances equal to 1, namely E[x˜ x˜ T ] = Ip . This also means that redundant observations are eliminated and the ICA problem is brought back to the smallest dimension p. This can be done by computing E[xx T ] = V DV T , with V ∈ St(n, p) and D ∈ Rp×p diagonal invertible. Then 1

˜ ) := D − 2 V T x(t ), x(t 1

and with A˜ := D − 2 V T A we have E[x˜ x˜ T ] = A˜ E[ss T ] A˜ T = A˜ A˜ T = Ip . After observable signal whitening, the de-mixing matrix may be searched for such that it solves the optimization problem max φ(W ).

W ∈O(p)

As explained, after whitening, the number of projected observations in the ˜ ) equals the number of sources. However, in some applications it signal x(t is known that not all the source signals are useful, so it is sensible to analyse only a few of them. In these cases, if we denote by p ¿ p the actual number of independent components that are sought after, the appropriate way to cast the optimization problem for ICA is max φ(W ),

W ∈St(n,p)

with p ¿ p.

The corresponding gradient flows obtained in this case are differential equations on the orthogonal group or on the Stiefel manifold, and can be solved numerically by Lie group integrators. As a possible principle for reconstruction, the maximization or minimization of non-Gaussianity is viable. It is based on the notion that the sum of independent random variables has distribution closer to Gaussian than the distributions of the original random variables. A measure of non-Gaussianity is the kurtosis, defined for a scalar signal z ∈ R as kurt(z) := E[z 4 ] − 3 E2 [z 2 ].

51

2 An introduction to Lie group integrators – basics, new devel. and appl. If the random signal z has unitary variance, then the kurtosis computes as kurt(z) = E[z 4 ] − 3. Maximizing or minimizing kurtosis is thus a possible way of estimating independent components from their linear mixtures, see [12] and references therein for more details.

2.4.4 Computation of Lyapunov exponents The Lyapunov exponents of a continuous dynamical system x˙ = F (x), x(t ) ∈ Rn , provide a qualitative measure of its complexity. They are numbers related to the linearization A(t ) of x˙ = F (x) along a trajectory x(t ). Consider the solution U of the matrix problem U˙ = A(t )U ,

U (t ) ∈ Rn×n .

U (0) = U0 ,

The logarithms of the eigenvalues of the matrix ¡ ¢1 Λ = lim U (t )TU (t ) 2t , t →∞

are the Lyapunov exponents for the given dynamical system. In [24] a procedure for computing just k of the n Lyapunov exponents of a dynamical system is presented. The exponents are computed by solving an initial value problem on St(n, k) and computing a quadrature of the diagonal entries of a k × k matrixvalued function. The initial value problem is defined as follows: Q˙ = (A −QQ T A +QSQ T )Q, with random initial value in St(n, k) and  T  k > j,  (Q AQ)k, j , S k, j =

0, k = j,   −(Q T AQ) , k < j , j ,k

k, j = 1, . . . , p.

It can be shown that the i -th Lyapunov exponent λi can be obtained as 1 t →∞ t

λi = lim and

52

t

Z 0

B i ,i (s) ds,

B = Q T AQ − S.

i = 1, . . . , k,

(2.24)

2.5 Symplectic integrators on the cotangent bundle of a Lie group One could use for example the trapezoidal rule to approximate the integral (2.24) and compute λi (i = 1, . . . , k). We refer to [24] for further details on the method, and to [15] for the use of Lie group integrators on this problem. Lie group methods for ODEs on Stiefel manifolds have also been considered in [14, 17, 37]. We have here presented a selection of applications that can be dealt with by solving differential equations on Lie groups and homogeneous manifolds. For these problems, Lie group integrators are a natural choice. We gave particular emphasis to problems of evolution in classical mechanics and problems of signal processing. This is by no means an exhaustive survey; other interesting areas of application are for example problems in vision and medical imaging, see for instance [40, 73].

2.5 Symplectic integrators on the cotangent bundle of a Lie group In this section we shall assume that the manifold is the cotangent bundle T∗G of a Lie group G. Let R g :G → G be the right multiplication operator such that R g (h) = h · g for any h ∈ G. The tangent map of R g is denoted R g ∗ := TR g . Any cotangent vector p g ∈ T∗g G can be associated to µ ∈ g∗ by right trivialization as follows: Write v g ∈ Tg G in the form v g = R g ∗ ξ where ξ ∈ g, so that 〈p g , v g 〉 = ® 〈p g , R g ∗ ξ〉 = R g∗ p g , ξ , where we have used R g∗ for the dual map of R g ∗ , and 〈·, ·〉 is a duality pairing. We therefore represent p g ∈ T∗g G by µ = R g∗ p g ∈ g∗ . Thus, we may use as phase space G × g∗ rather than T∗G. For applying Lie group integrators we need a transitive group action on G × g∗ and this can be achieved by lifting the group structure of G and using left multiplication in the extended group. The semidirect product structure on G := G n g∗ is defined as ¡ ¢ (g 1 , µ1 ) · (g 2 , µ2 ) = g 1 · g 2 , µ1 + Ad∗g −1 µ2 , (2.25) 1

where the coadjoint action Ad∗ is defined in (2.18). Similarly, the tangent map of right multiplication extends as ¡ ¢ TR (g ,µ) (R h∗ ζ, ν) = R hg ∗ ζ, ν − ad∗ζ Ad∗h −1 µ , g , h ∈ G, ζ ∈ g, µ, ν ∈ g∗ . Of particular interest is the restriction of TR (g ,µ) to Te G ∼ = g × g∗ , ¡ ¢ Te R (g ,µ) (ζ, ν) = R g ∗ ζ, ν − ad∗ζ µ .

53

2 An introduction to Lie group integrators – basics, new devel. and appl. The natural symplectic form on T∗G (which is a differential two-form) is defined as ¡ ¢ Ω(g ,p g ) (δv 1 , δπ1 ), (δv 2 , δπ2 ) = 〈δπ2 , δv 1 〉 − 〈δπ1 , δv 2 〉, and by right trivialization it may be pulled back to G and then takes the form ¡ ¢ ω(g ,µ) (R g ∗ ξ1 , δν1 ), (R g ∗ ξ2 , δν2 ) = 〈δν2 , ξ1 〉−〈δν1 , ξ2 〉−〈µ, [ξ1 , ξ2 ]〉, ξ1 , ξ2 ∈ g. (2.26) ∗ The presentation of differential equations on T G as in (2.10) is now achieved via the action by left multiplication, meaning that any vector field F ∈ X(G) is expressed by means of a map f :G → Te G, ¡ ¢ F (g , µ) = Te R (g ,µ) f (g , µ) = R g ∗ f 1 , f 2 − ad∗f 1 µ , (2.27) where f 1 = f 1 (g , µ) ∈ g, f 2 = f 2 (g , µ) ∈ g∗ are the two components of f . We are particularly interested in the case that F is a Hamiltonian vector field which means that F satisfies the relation iF ω = dH ,

(2.28)

for some Hamiltonian function H : T∗G → R and iF is the interior product defined as iF ω(X ) := ω(F, X ) for any vector field X . From now on we let H :G → R denote the trivialized Hamiltonian. A simple calculation using (2.26), (2.27) and (2.28) shows that the corresponding map f for such a Hamiltonian vector field is µ ¶ ∂H ∗ ∂H f (g , µ) = (g , µ), −R g (g , µ) . ∂µ ∂g We have come up with the following family of symplectic Lie group integrators on G × g∗ (ξi , n¯ i ) = h f (G i , M i ), n i = Ad∗exp(X i ) n¯ i , i = 1, . . . s, µ ¶ s X −1 ∗ (g 1 , µ1 ) = exp Y , (dexpY ) b i n i · (g 0 , µ0 ), Y =

s X

b i ξi ,

Xi =

i =1

i =1 s X

j =1

ai j ξ j ,

i = 1, . . . , s,

G i = exp(X i ) · g 0 , M i = dexp∗−Y

54

i = 1, . . . , s, ¶ s µ X bj aji dexp∗−X j n j , µ0 + b j dexp∗−Y − bi j =1

i = 1, . . . , s.

2.5 Symplectic integrators on the cotangent bundle of a Lie group P Here, a i j and b i are coefficients where it is assumed that is=1 b i = 1 and that b i 6= 0, 1 ≤ i ≤ s. The symplecticity of these schemes is a consequence of their derivation from a variational principle, following ideas similar to that of [3] and [67]. One should be aware that order barriers for this type of schemes may apply, and that further stage corrections may be necessary to obtain high order methods.

Let us choose s = 1 with coefficients b 1 = 1 and a 11 = θ, i.e. the RK coefficients of the well known θ-method. Inserting this into our method and simplifying gives us the method Example, the θ -method for a heavy top.

¡ ¢ ¯ = h f exp(θξ) · g 0 , dexp∗−ξ µ0 + (1 − θ) dexp∗−(1−θ)ξ n¯ , (ξ, n) ¡ ¢ (g 1 , µ1 ) = exp(ξ), Ad∗exp(−(1−θ)ξ) n¯ · (g 0 , µ0 ).

In Figure 2.6 we show numerical experiments for the heavy top, where the Hamiltonian is given as H :G → R,

H (g , µ) = 12 〈µ, I−1 µ〉 + eT3 g u 0 ,

where G = SO(3), I: g → g∗ is the inertia tensor, here represented as a diagonal 3 × 3 matrix, u 0 is the initial position of the top’s centre of mass, and e3 is the canonical unit vector in the vertical direction. We have chosen I = 103 diag(1, 5, 6) and u 0 = e3 . The initial values used were g 0 = I (the identity matrix), and µ0 = 10 I (1, 1, 1)T . We compare the behaviour of the symplectic schemes presented here to the Runge–Kutta–Munthe-Kaas (RKMK) method with the same coefficients. In Figure 2.6 we have drawn the time evolution of the centre of mass, u n = g n u 0 . The characteristic band structure observed for the symplectic methods was reported in [19]. The RKMK method with θ = 12 exhibits a similar behaviour, but the bands are expanding faster than for the symplectic ones. We have also found in these experiments that none of the symplectic schemes, θ = 0 and θ = 21 have energy drift, but this is also the case for the RKMK method with θ = 12 . This may be related to the fact that both methods are symmetric for θ = 12 . For θ = 0, however, the RKMK method shows energy drift as expected. These tests were done with step size h = 0.05 over 105 steps. See Table 2.1 for a summary of the properties of the tested methods.

55


SLGI, θ = 1/2

1 0.5

0.5

0

0

−0.5 1 0

−1 −1

1

0

RKMK, θ = 1/2

1

SLGI, θ = 0

1

−0.5 1 0

−1 −1 RKMK, θ = 0

1

0.5

1

0

0

0 −0.5 1 0

−1 −1

1

0

−1 1 0

−1 −0.5

0

0.5

1

Figure 2.6: Heavy top simulations with the symplectic (SLGI) θ-methods and RKMK θ-methods with θ = 0, 12 . The curves show the time evolution of the centre of mass of the body. The simulations were run over 5000 steps with step size h = 0.01. See the text, page 55, for all other parameter values. Table 2.1: Properties of the tested methods. The energy drift was observed numerically. RKMK

56

SLGI

Property

θ=0

θ = 1/2

θ=0

θ = 1/2

Symplectic Symmetric No energy drift

no no no

no yes yes

yes no yes

yes yes yes

2.6 Discr. gradients and integral preserving methods on Lie groups

2.6 Discrete gradients and integral preserving methods on Lie groups The discrete gradient method for preserving first integrals has to a large extent been made popular through the works of Gonzalez [29] and McLachlan, Quispel and Robidoux [51]. The latter proved the result that under relatively general circumstances, a differential equation which has a first integral I (x) can be written in the form x˙ = S(x)∇I (x), for some non-unique solution-dependent skew-symmetric matrix S(x). The idea is to introduce a mapping which resembles the true gradient; a discrete gradient ∇I : Rd ×Rd → Rd which is a continuous map that satisfies the following two conditions: ∇I (x, x) = ∇I (x),

∀x,

T

I (y) − I (x) = ∇I (x, y) (y − x),

∀x 6= y.

An integrator which preserves I , that is, I (x n ) = I (x 0 ) for all n is now easily devised as x n+1 − x n ˜ n , x n+1 )∇I (x n , x n+1 ), = S(x h ˜ y) is some consistent approximation to S(x), i.e. S(x, ˜ x) = S(x). There where S(x, exist several discrete gradients, two of the most popular are Z 1 ¡ ¢ ∇I ζy + (1 − ζ)x dζ, ∇I (x, y) = 0

(2.29)

and ∇I (x, y) = ∇I

³x + y ´

2

+

I (y) − I (x) − ∇I

¡ x+y ¢T

2 ky − xk2

(y − x)

(y − x).

(2.30)

˜ y) can be constructed with the purpose of increasing the accurThe matrix S(x, acy of the resulting approximation, see e.g. [65]. We now generalize the concept of the discrete gradient to a Lie group G. We consider differential equations which can, for a given dual two-form3 ω ∈ Ω2 (G) 3 By dual two-form, we here mean a differential two-form on G such that on each fibre of the ∗ cotangent bundle we have ωx : T∗ x G × Tx G → R, a bilinear, skew-symmetric form. Such forms

are sometimes called bivectors or 2-vectors.

57

2 An introduction to Lie group integrators – basics, new devel. and appl. and a function H :G → R be written in the form x˙ = idH ω, where iα is the interior product iα ω(β) = ω(α, β) for any two one-forms α, β ∈ Ω1 (G). The function H is a first integral since ¢ ¡ ¢ d ¡ ˙ ) = ω(dH , dH ) = 0. H x(t ) = dH x(t ) x(t dt

We define the trivialized discrete differential (TDD) of the function H to be a continuous map dH :G ×G → g∗ such that H (x 0 ) − H (x) = 〈dH (x, x 0 ), log(x 0 · x −1 )〉, dH (x, x) = R x∗ dH x .

A numerical method can now be defined in terms of the discrete differential as ¢ ¡ ¯ x 0 ) · x. x 0 = exp h idH (x,x 0 ) ω(x, ¯ is a continuous map from G ×G into the space of exterior two-forms where ω on g∗ , Λ2 (g∗ ). This exterior form is some local trivialized approximation to ω, meaning that we impose the following consistency condition ¡ ¢ ¯ x) R x∗ α, R x∗ β = ωx (α, β), ω(x, for all α, β ∈ T∗x G. We easily see that this method preserves H exactly, since H (x 0 ) − H (x) = 〈dH (x, x 0 ), log(x 0 · x −1 )〉

¯ x 0 )〉 = 〈dH (x, x 0 ), h idH (x,x 0 ) ω(x, ¡ ¢ ¯ x 0 ) dH (x, x 0 ), dH (x, x 0 ) = 0. = h ω(x,

Extending (2.29) to the Lie group setting, we define the following TDD: Z 1 ¡ ¢ 0 ∗ dH (x, x ) = R `(ξ) dH`(ξ) dξ, `(ξ) = exp ξ log(x 0 · x −1 ) · x. 0

Similarly, for any given inner product on g, we may extend the discrete gradient (2.30) to ® H (x 0 ) − H (x) − R x∗¯ dH x¯ , η [ 0 ∗ dH (x, x ) = R x¯ dH x¯ + η , η = log(x 0 · x −1 ), (2.31) kηk2

58

2.6 Discr. gradients and integral preserving methods on Lie groups where x¯ ∈ G for instance could be x¯ = exp(η/2) · x, a choice which would cause dH (x, x 0 ) = dH (x 0 , x). The standard notation η[ is used for index-lowering, the inner product (·, ·) associates to any element η ∈ g the dual element η[ ∈ g∗ through 〈η[ , ζ〉 = (η, ζ), ∀ζ ∈ g. Suppose that the ODE vector field F is known as well as the invariant H . A dual two-form ω can now be defined in terms of a Riemannian metric on G. By index raising applied to dH , we obtain the Riemannian gradient vector field grad H , and we define ω=

grad H ∧ F kgrad H k2

⇒

idH ω = F.

We consider the equations for the attitude rotation of a free rigid body expressed using Euler parameters. The set S 3 = {q ∈ R4 | kqk2 = 1} with q = (q 0 , q), (q 0 ∈ R and q ∈ R3 ), is a Lie group with the quaternion product Example.

p · q = (p 0 q 0 − p T q, p 0 q + q 0 p + p × q),

with unit e = (1, 0, 0, 0) and inverse qc = (q 0 , −q). We denote by “ ˆ " the hatmap defined in (2.20). The Lie group S 3 can be mapped into SO(3) by the Euler–Rodrigues map: E (q) = I3 + 2q 0 qˆ + 2qˆ 2 , where I3 denotes the 3 × 3 identity matrix. The Lie algebra s3 of S 3 is the set of so called pure quaternions, the elements of R4 with first component equal to zero, identifiable with R3 and with so(3) via the hat-map. The equations for the attitude rotation of a free rigid body on S 3 read ˙ = f (q) · q, q

f (q) = q · v · qc ,

and v = (0, v ),

v = 12 I−1 E (qc )m 0 ,

where m 0 is the initial body angular momentum and I is the diagonal inertia tensor, and according to the notation previously used in this section F (q) = f (q) · q. The energy function is H (q) = 21 m 0T E (q)I−1 E (qc )m 0 .

59

2 An introduction to Lie group integrators – basics, new devel. and appl. We consider the R3 Euclidean inner product as metric in the Lie algebra s3 , and obtain by right translation a Riemannian metric on S 3 . The Riemannian gradient of H with respect to this metric is then grad H = (I4 − qqT )∇H , where I4 is the identity in R4×4 and ∇H is the usual gradient of H as a function from R4 to R. We identify s3 with its dual, and using grad H in (2.31) we obtain the (dual) discrete differential grad H (q, q0 ) ∈ s3 . grad H ∧F

The two-form ω = kgrad H k2 with respect to the right trivialization can be identified with the 4 × 4 skew-symmetric matrix ωR (q) =

ξ γT − γ ξT , kγk2

ξ, γ ∈ s3 ,

ξ · q = F (q),

γ · q = grad H (q),

¯ to be where ωR (q) has first row and first column equal to zero. We choose ω ¯ q, q0 ) = ωR (¯q), ω(

¯ = exp(η/2) · q, q

η = log(q0 · qc ),

i.e. ωR frozen at the mid-point q¯ . The energy-preserving Lie group method of second order is ¡ ¢ ¯ q, q0 )grad H (q, q0 ) · q, q0 = exp h ω( and exp is the exponential map from s3 to S 3 with log: S 3 → s3 as its inverse, defined locally around the identity. In Figure 2.7 we plot the body angular momentum vector m = E (qc )m 0 on a time interval [0, T ], T = 1000, for four different methods: the Lie group energy-preserving integrator just described (top left), the built-in M AT L A B routine ode45 with absolute and relative tolerance 10−6 (top right); the ode45 routine with tolerances 10−14 (bottom left); and the explicit Heun RKMK Lie group method (bottom right). The two Lie group methods both have order 2. The energy preserving method is both symmetric, energy preserving and it preserves the constraint kqk2 = 1. The Lie group integrators use a stepsize h = 1/64. The solution of the built-in M AT L A B routine at high precision is qualitatively similar to the highly accurate solution produced by M AT L A B with tolerances 10−14 . The energy error is also comparable for these two experiments. The performance of other M AT L A B built-in routines we tried was worse than for ode45. We remark that the equations are formulated as differential

60

2.6 Discr. gradients and integral preserving methods on Lie groups

−59.9

−59.9

−59.95

−59.95

−60

−60

−60.05

5 0 −5 −2

0

−60.05 5 2

−5 −2

2

0

100

−59.9 −59.95

0

−60 −60.05

0

5 0 −5 −2

0

2

−100 100

0 −100

−100

−50

0

50

Figure 2.7: Free rigid body angular momentum, time interval [0, 1000], moments of inertia I 1 = 1, I 2 = 5, I 3 = 60, initial angular velocity I m 0 = (1, 1/2, −1)T . (Top left) energy-preserving Lie group method, h = 1/64; (top right) ode45 with tolerances 10−6 ; (bottom left) ode45 with tolerances 10−14 ; (bottom right) Heun RKMK, h = 1/64.

61

2 An introduction to Lie group integrators – basics, new devel. and appl. equations on S 3 , a formulation of the problem in form of a differential algebraic equation would possibly have improved the performance of the M AT L A B built-in routines. However it seems that the preservation of the constraint alone can not guarantee the good performance of the method. In fact the explicit (non-symmetric) Lie group integrator preserves the constraint kqk2 = 1, but performs poorly on this problem (see Figure 2.7 bottom right). The cost per step of the explicit Lie group integrator is much lower than for the energy-preserving symmetric Lie group integrator. We have given an introduction to Lie group integrators for differential equations on manifolds using the notions of frames and Lie group actions. A few application areas have been discussed. An interesting observation is that when the Lie group featuring in the method can be chosen to be the Euclidean group, the resulting integrator always reduce to some well-known numerical scheme like for instance a Runge-Kutta method. In this way, one may think of the Lie group integrators as a superset of the traditional integrators and a natural question to ask is whether the Euclidean choice will always be superior to any other Lie group and group action. Lie group integrators that are symplectic for Hamiltonian problems in the general setting presented here are, as far as we know, not known. However, we have shown that such methods exist in the important case of Hamiltonian problems on the cotangent bundle of a Lie group. There are however still many open questions regarding this type of schemes, like for instance how to obtain high order methods. The preservation of first integrals in Lie group integrators has been achieved in the literature by imposing a group action in which the orbit, i.e. the reachable set of points, is contained in the level set of the invariant. But it is not always convenient to impose such group actions, and we have here suggested a type of Lie group integrator which can preserve any prescribed invariant for the case where the manifold is a Lie group acting on itself by left or right multiplication. An interesting idea to pursue is the generalization of this approach to arbitrary smooth manifolds with a group action.

Acknowledgements This research was supported by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework

62

2.6 Discr. gradients and integral preserving methods on Lie groups Programme. The authors would like to acknowledge the support from the GeNuIn Applications and SpadeAce projects funded by the Research Council of Norway, and most of the work was carried out while the authors were visiting Massey University, Palmerston North, New Zealand and La Trobe University, Melbourne, Australia.

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Paper 2 High order symplectic partitioned Lie group methods

Geir Bogfjellmo and Håkon Marthinsen Submitted to Foundations of Computational Mathematics

3 High order symplectic partitioned Lie group methods In this article, a unified approach to obtain symplectic integrators on T∗G from Lie group integrators on a Lie group G is presented. The approach is worked out in detail for symplectic integrators based on Runge–Kutta–Munthe-Kaas methods and Crouch–Grossman methods. These methods can be interpreted as symplectic partitioned Runge–Kutta methods extended to the Lie group setting in two different ways. In both cases, we show that it is possible to obtain symplectic integrators of arbitrarily high order by this approach. Abstract.

3.1 Introduction 3.1.1 Motivation and background In general, an ordinary differential equation (ODE) can be described by a vector field on a smooth manifold where solutions of the ODE are integral curves of the vector field. Numerical approximation of solutions of ODEs is an old field of study, and a plethora of methods for obtaining numerical solutions exist. However, most of these methods assume that the manifold is Euclidean space. If the manifold is not Euclidean space, it is possible to embed the manifold in Euclidean space, and extend the vector field on the manifold to a vector field in Euclidean space such that the integral curves are ensured to remain in the image of the embedding. A standard numerical algorithm (e.g. a Runge–Kutta method) will in general result in discrete points which do not lie in the image of the embedding. An improvement of this approach is to use projection methods to obtain solutions on the manifold. These approaches, though simple, suffer from the problem that the numerical solutions depend on the particular choice of embedding, and on the particular extension of the vector field.

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3 High order symplectic partitioned Lie group methods One aspect of geometric numerical integration is to exploit structure on the manifold to define numerical methods that are intrinsic to the manifold (i.e. do not depend on a particular embedding). This structure can for instance be that of a Lie group acting on the manifold. The action of a Lie group G on a manifold M is a smooth mapping Ψ:G × M → M which respects the group structure on G. If the action is transitive, then the derivative with respect to the first component of Ψ at the group identity e is a surjective vector bundle morphism g × M → T M . Any vector field X on M can then be lifted (possibly in a non-unique manner) to a section of the vector bundle g×M . The combination of this lifting and standard charts g → G, form the basis of several classes of Lie group methods. Among them are the Crouch–Grossman (CG) methods [5] and the Runge–Kutta–Munthe-Kaas (RKMK) methods [16]. For a more detailed discussion of Lie group methods, we refer to the survey article by Iserles et al. [10] and the references therein. Another aspect of geometric numerical integration is symplecticity of numerical integrators. Many important problems from physics can be formulated as Hamiltonian ODEs on cotangent bundles over manifolds. The flow maps of these ODEs are symplectic, that is, they preserve the canonical two-form on the cotangent bundle. For Hamiltonian ODEs, it is beneficial to use symplectic integrators, due to the near-preservation of energy and excellent long-term behaviour of the numerical solutions [8, Chapter VI]. Hamilton’s principle states that the solution of a Lagrangian (in many cases also Hamiltonian) system ¢ RT ¡ ˙ ) dt moves along a path which extremizes the action integral S = 0 L q(t ), q(t among all paths q with fixed end points. One technique for deriving symplectic methods is based on the notion of discretizing Hamilton’s principle, that is, replacing the action integral with a discrete action sum, and extremizing over all discrete paths or sequences of points q 0 , q 1 , . . . , q N with fixed end points. These methods are known as variational methods or variational integrators. Variational methods are guaranteed to be symplectic since the terms L h (q k−1 , q k ) of the discrete action sum can be interpreted as generating functions (of the first type) for the numerical flow map. Variational methods have been studied by numerous authors, we refer to the review article by Marsden and West [15] or the more recent encyclopedia article by Leok [12] and the references therein for more information about variational methods. Standard Lie group methods, like RKMK methods or CG methods, give numerical solutions that evolve on the same manifolds as the exact solutions. The

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3.1 Introduction question of the existence of symplectic methods of formats similar to the ones considered by Crouch and Grossman or by Munthe-Kaas has been a topic of interest for several years. On Rn there is a unified way to extend Runge–Kutta (RK) methods on Rn to symplectic methods on T∗Rn [8, Section VI.6.3], i.e. symplectic partitioned RK (SPRK) methods. Our goal with this article is to construct and study symplectic methods of arbitrarily high order that are extended from Lie group methods, i.e. high-order symplectic Lie group integrators. We have focused on the case where M = G and the action is simply multiplication in the Lie group. In the case where M 6= G, isotropy complicates matters. The technique of extremizing a discrete action sum still yields symplectic mappings in the case M 6= G, but the presence of isotropy complicates the analysis of these integrators. The details of the isotropy case will hopefully be addressed in a later article. The idea of constructing variational methods from Lie group methods has previously been considered by several authors. Bou-Rabee and Marsden proposed in 2009 to base variational methods on RKMK methods [2], and present a class of methods of first and second order. Methods of a similar type are also considered in the survey article by Celledoni, Marthinsen and Owren [4]. In the present article, this idea is pursued further to obtain methods of arbitrarily high order. It is known to the authors that a different, but related approach to variational Lie group methods has been studied by Leok and collaborators. Their approach is based on approximating the curve q in a finite-dimensional function space, resulting in Galerkin Lie group variational integrators. The idea appears already in Leok’s doctoral thesis [11, Section 5.3], and also in other articles by Leok and co-authors. A more detailed study of this approach can be found in an article by Hall and Leok [9]. The rest of Section 1 is an introduction to ODEs on a Lie group G, the Hamilton–Pontryagin (HP) principle and the equivalent HP equations, and variational integrators in general. Section 2 begins by introducing a group structure on T∗G, or equivalently, on G × g∗ and a function f :G × g∗ → g × g∗ which together fully describe ODEs on T∗G. Next, we introduce the general format for our integrators. In Section 3, we first show that a subclass of our integrators that have been studied before [2, 4] can not obtain higher than second order on general Hamiltonian problems. We then show that our integrators can not obtain higher order than the underlying Lie group integrators. In Section 4, we present two classes of higher order integrators which are based on RKMK in-

75

3 High order symplectic partitioned Lie group methods tegrators and CG methods respectively. In Section 5, we show that both classes of methods from Section 4 can obtain arbitrarily high order, and we present general order conditions for the methods based on RKMK integrators, and conditions for order 1–3 for the CG-based integrators. We test the two classes of methods numerically in Section 6, and show that they both achieve the correct order and that they both have small energy errors over long time. Finally, in Section 7 we conclude and mention some possible topics for further work.

3.1.2 ODEs on Lie groups Let G be a finite-dimensional Lie group and g its associated Lie algebra. We denote right-multiplication with g ∈ G as R g and left-multiplication with g as L g . We use dot notation to denote translation in the tangent bundle, i.e. g · v = T L g v,

v · g = T R g v,

v ∈ T G,

and in the cotangent bundle g · p = T∗L g −1 p,

p · g = T∗R g −1 p,

p ∈ T∗G.

We also need the notation Adg := T L g ◦ T R g −1 . All autonomous ODEs on G can be written as g˙ = f (g ) · g , g (0) = g 0 , (3.1)

where g is a curve in G and the map f :G → g is determined uniquely by the vector field. We can solve this kind of equation numerically using Lie group methods [10]. Here we have chosen the right-trivialized form of this equation. We could also have used the left-trivialized form g˙ = g · f (g ) which would have resulted in only minor changes to the formulae presented later in the article. Since we are interested in solving Hamiltonian ODEs using Lie group methods, we need a group structure on the cotangent bundle of G, as well as the map f that corresponds to this type of ODEs.

3.1.3 Hamilton–Pontryagin mechanics Lagrangian mechanics on G is formulated in terms of a Lagrangian L: T G → R. Hamilton’s principle states that the dynamics is given by the curve q: R → G that extremizes the action integral Z T ˙ dt , SH = L(q, q) 0

76

3.1 Introduction where the endpoints q(0) and q(T ) are kept fixed. In [2, Theorem 3.4] it was shown that this is equivalent to the Hamilton–Pontryagin (HP) principle, which states that the dynamics is given by extremizing S HP =

T

Z 0

¡ ¢ L(q, v) + 〈p, q˙ − v〉 dt ,

where v ∈ T q G, p ∈ T q∗G are varied arbitrarily, and the endpoints of q are kept fixed. Here, we denote the natural pairing of covectors and vectors by 〈·, ·〉. This action integral leads to dynamics formulated on T∗G. To simplify further calculations, it is convenient to right-trivialize T∗G to G × g∗ via the map (q, p q ) 7→ (q, p q · q −1 ). Letting `(q, ξ) := L(q, ξ · q), ξ ∈ g, it is easy to show that the HP principle is equivalent to the right-trivialized HP principle, which has action integral S=

T

Z 0

¡ ¢ `(q, ξ) + 〈µ, q˙ · q −1 − ξ〉 dt ,

where ξ: R → g and µ: R → g∗ are varied arbitrarily, and the endpoints of q are kept fixed. Taking the variation of S, we arrive at the right-trivialized HP equations q˙ = ξ · q, ¢ ¡ µ˙ = − ad∗ξ µ + D1 `(q, ξ) · q −1 , (3.2) µ = D2 `(q, ξ),

where adx is the derivative of Adexp(x) with respect to x at the origin, and Dk ` denotes the partial derivative of ` with respect to the kth variable, i.e. a oneform. This is the ODE on G × g∗ that we need to solve.

3.1.4 Variational integrators Variational integrators are constructed by discretizing an action integral and then performing extremization with fixed endpoints. This procedure turns the action integral into an action sum. The discrete Lagrangian L h :G ×G → R is an approximation of the action integral over a small time step h, L h (q k−1 , q k ) ≈

Z

kh (k−1)h

˙ dt , L(q, q)

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3 High order symplectic partitioned Lie group methods where q: R → G extremizes the action integral with q(0) and q(T ) fixed. Letting N = T /h, the action sum becomes Sh =

N X

L h (q k−1 , q k ).

k=1

Extremizing S h while keeping q 0 and q N fixed gives us the discrete Euler– Lagrange equations D1 L h (q k , q k+1 ) + D2 L h (q k−1 , q k ) = 0,

1 ≤ k < N.

The discrete Legendre transforms define the discrete conjugate momenta p k := µk · q k := −D1 L h (q k , q k+1 ),

p k+1 := µk+1 · q k+1 := D2 L h (q k , q k+1 ). By demanding that these two definitions are consistent, we automatically satisfy the discrete Euler–Lagrange equations. If we can solve the first equation for q k+1 , we can use the second one to calculate µk+1 , giving us the variational integrator (q k , µk ) 7→ (q k+1 , µk+1 ).

3.2 From a Lie group method to a variational integrator on the cotangent bundle 3.2.1 Group structure and Hamiltonian ODEs on G × g∗ We want to numerically solve the right-trivialized HP equations (3.2), which can be viewed as a vector field on G × g∗ , or equivalently, as the ODE z˙ = f (z) · z, where z ∈ G × g∗ and f :G × g∗ → g × g∗ . For this ODE to make sense, we must choose a group product on G × g∗ . We choose the magnetic extension of G, as described by Arnold and Khesin [1, Section I.10.B]. As we will see, this group product makes the right-trivialized HP equations easily expressible as z˙ = f (z) · z.1 The magnetic extension assigns the following group product to T∗G: (g , p g )(h, p h ) := (g h, p g · h + g · p h ).

(3.3)

1 This group structure was used by Engø in [6] to construct partitioned Runge–Kutta–Munthe-

Kaas methods on T∗G, without any special regard to symplecticity.

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3.2 From a Lie group method to a var. integrator on the cotangent bundle This group structure is an extension of the group structure on G in the sense that the canonical projection T∗G → G is a homomorphism of Lie groups. We note that (g , p g )(h, p h ) = (g h, p g · h + g · p h ) ³ ´ ¡ ¢ = g h, p g · g −1 + Ad∗g −1 (p h · h −1 ) · g h . Thus, letting µ = p g · g −1 and ν = p h · h −1 , the right-trivialized version of (3.3) is the product on G × g∗ defined by ¡ ¢ (g , µ)(h, ν) := g h, µ + Ad∗g −1 ν . It can be shown that the Lie algebra associated to the Lie group G × g∗ is g × g∗ equipped with the Lie bracket [(ξ, µ), (η, ν)] = (adξ η, ad∗η µ − ad∗ξ ν). We will also need an expression for T R z ζ for z = (q, µ) ∈ G × g∗ and ζ = (η, ν) ∈ g × g∗ : ¯ ¯ ¢ d¡ T Rz ζ = exp(²η), ²ν (q, µ)¯¯ d² ²=0 ¯ ¢¯ d¡ ∗ = exp(²η)q, ²ν + Adexp(−²η) µ ¯¯ d² ²=0 ¡ ¢ = η · q, ν − ad∗η µ , We would now like to use this to write the right-trivialized HP equations (3.2) in the form of (3.1), z˙ = f (z) · z = T R z ◦ f (z). If the map f :G × g∗ → g × g∗ satisfies ´ ¡ ¢ ³ ¡ ¢ f q, D2 `(q, ξ) = ξ, D1 `(q, ξ) · q −1 (3.4) for all (q, ξ) ∈ G × g, we see that z˙ = f (z) · z, which is exactly what we need. ¡ ¢ In many cases, the map (q, ξ) 7→ q, D2 `(q, ξ) is a diffeomorphism of manifolds. If this holds, we say that the Lagrangian ` is regular. If ` is regular, the Lagrangian problem has an equivalent formulation as a Hamiltonian ODE on T∗G, where ¡ ¢ H q, D2 `(q, ξ) = 〈D2 `(q, ξ), ξ〉 − `(q, ξ) and

³ ´ ¡ ¢ f (q, µ) = D2 H (q, µ), − D1 H (q, µ) · q −1 .

(3.5) ¢ For Hamiltonians which arise in this manner, the map (q, µ) 7→ q, D2 H (q, µ) is also a diffeomorphism of manifolds. In fact the map is the inverse of the one above. Hamiltonians for which this hold are also called regular. ¡

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3 High order symplectic partitioned Lie group methods

3.2.2 General format for our integrators It is natural to consider discrete Lagrangians based on approximation of the action integral by quadrature. The procedure adopted in the present article is inspired by the approach in [8, Section VI.6.3], originally found in [21]. In this reference, the symplectic partitioned Runge–Kutta methods are derived by considering the discrete Lagrangian L h (q 0 , q 1 ) = h

s X

b i L(Q i , Q˙ i )

(3.6)

i =1

where Q i = q0 + h

s X

a i j Q˙ j ,

j =1

and b i , a i j are the coefficients of a Runge–Kutta method. The Q˙ i are chosen to extremize the sum above under the constraint q1 = q0 + h

s X

b i Q˙ i .

i =1

As shown in [8, Section VI.6.3], the resulting integrator is exactly the partitioned Runge–Kutta integrator where the position is integrated using the original coefficients b i , a i j , while the momentum is integrated by using the coefficients bˆ i = b i , aî j = b j − b j a j i /b i . In the following, we will generalize the approach used in [8, Section VI.6.3] to Lie groups. Consider the discrete Lagrangian L h (q 0 , q 1 ) = Lˆ h (Q 1 , . . . ,Q s , ξ1 , . . . , ξs ) = h

s X

b i `(Q i , ξi ),

i =1

where b i are non-zero quadrature weights, and the auxiliary variables Q 1 , . . . ,Q s , ξ1 , . . . , ξs are chosen to extremize Lˆ h under the constraints ¡ ¢ Y (Q 1 , . . . ,Q s , ξ1 , . . . , ξs , q 0 ) − log q 1 q 0−1 = 0, (3.7) ¢ ¡ X i (Q 1 , . . . ,Q s , ξ1 , . . . , ξs , q 0 ) − log Q i q 0−1 = 0, i = 1, . . . , s. The functions Y and X i will typically arise from Lie group integrators, as we will see later on. The formulation of the discrete Lagrangian is that of a constrained optimization problem. As done in [8, Section VI.6.3], we solve this by introducing Lagrange multipliers. Let Λ be the Lagrange multiplier corresponding

80

3.2 From a Lie group method to a var. integrator on the cotangent bundle to the constraint containing Y , and let λi be the Lagrange multiplier corresponding to the equation containing X i for i = 1, . . . , s. To obtain a variational integrator, we extremize

s ¡ ¢® X ¡ ¢® Lˆ h − Λ, Y − log q 1 q 0−1 − λi , X i − log Q i q 0−1 , i =1

while keeping q 0 and q 1 fixed. Varying this with respect to Λ, λi , ξi and Q i , we obtain the set of equations

q 1 = exp(Y )q 0 ,

Q i = exp(X i )q 0 , µ ¶ µ ¶ X ∂X j ∗ ∂Y ∗ ∂Lˆ h = Λ+ λj , ∂ξi ∂ξi ∂ξi j µ ¶ ¶ µ ³¡ X ∂X j ∗ ¢∗ ´ ∂Y ∗ ∂Lˆ h = Λ+ λ j − dexp−1 λi · Q i , Xi ∂Q i ∂Q i ∂Q i j

(3.8)

for all i = 1, . . . , s. To find the integrator based on the discrete Lagrangian L h , we need to evaluate the partial derivatives of L h with respect to q 0 and q 1 . In doing so, we consider Q 1 , . . . ,Q s , ξ1 , . . . , ξs as functions of q 0 and q 1 defined implicitly by (3.7) and (3.8). The partial derivatives of L h are then µ ¶ ∂L h X ∂Lˆ h ∂Q j ∂Lˆ h ∂ξ j = + , ◦ ◦ ∂q 0 ∂q 0 ∂ξ j ∂q 0 j ∂Q j µ ¶ ∂L h X ∂Lˆ h ∂Q j ∂Lˆ h ∂ξ j = ◦ + ◦ . ∂q 1 ∂q 1 ∂ξ j ∂q 1 j ∂Q j

(3.9)

The functions Q 1 , . . . ,Q s , ξ1 , . . . , ξs satisfy the constraints (3.7) for all q 0 , q 1 . By

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3 High order symplectic partitioned Lie group methods differentiating the constraints we see that the identities µ ¶ ∂Y X ∂Y ∂Q j ∂Y ∂ξ j + ◦ + ◦ + dexp−1 0= −Y ◦T R q 0−1 , ∂q 0 ∂Q ∂q ∂ξ ∂q 0 0 j j j ¶ µ X ∂Y ∂Q j ∂Y ∂ξ j ◦ + ◦ − dexp−1 0= Y ◦T R q 1−1 , ∂Q ∂q ∂ξ ∂q 1 1 j j j µ ¶ ∂X i X ∂X i ∂Q j ∂X i ∂ξ j + + ◦ ◦ + dexp−1 0= −X i ◦T R q 0−1 ∂q 0 ∂q 0 ∂ξ j ∂q 0 j ∂Q j ∂Q i , ∂q 0 µ ¶ X ∂X i ∂Q j ∂X i ∂ξ j ∂Q i 0= ◦ ◦ + − dexp−1 , X i ◦T RQ i−1 ◦ ∂q 1 ∂ξ j ∂q 1 ∂q 1 j ∂Q j − dexp−1 X i ◦T RQ −1 ◦ i

i = 1, . . . , s, (3.10)

all hold. We combine the discrete Legendre transforms −µ0 · q 0 =

∂L h , ∂q 0

µ1 · q 1 =

∂L h , ∂q 1

with (3.8) and (3.9), and simplify using (3.10) to obtain the equations ! Ãµ µ ¶ ¶ X ∂X j ∗ X¡ ¡ ¢∗ ¢∗ ∂Y ∗ Λ+ Λ+ dexp−1 µ0 = λ j · q 0−1 + dexp−1 −Y −X j λ j , ∂q 0 ∂q 0 j j ¡ ¢ ∗ −1 µ1 = dexpY Λ. Using the identity (3.4), we get ´ ¡ ¢ ³ ¡ ¢ f Q i , D2 `(Q i , ξi ) = ξi , D1 `(Q i , ξi ) · Q i−1 , and defining n i , M i ∈ g∗ by

∂Lˆ h = hb i D1 `(Q i , ξi ) = hb i n i · Q i , ∂Q i

for i = 1, . . . , s, we get Ãµ

∂Lˆ h = hb i D2 `(Q i , ξi ) = hb i M i , ∂ξi

! ¶ ¶ µ X ∂X j ∗ ¡ ¢∗ ∂Y ∗ hb i n i = Λ+ λ j · Q i−1 − dexp−1 λi , Xi ∂Q i ∂Q i j µ ¶ µ ¶ X ∂X j ∗ ∂Y ∗ hb i M i = Λ+ λj . ∂ξi ∂ξi j

82

3.3 First and second order integrators Combining everything above, the variational integrator is defined by the set of equations Ãµ ! ¶ µ ¶ X ∂X j ∗ X¡ ¡ ¢∗ ¢∗ ∂Y ∗ µ0 = Λ+ λ j · q 0−1 + dexp−1 Λ+ dexp−1 −Y −X j λ j , ∂q 0 ∂q 0 j j Ãµ ¶∗ µ ¶∗ ! X ∂X j ¡ ¢∗ ∂Y Λ+ λ j · Q i−1 − dexp−1 λi , hb i n i = Xi ∂Q i ∂Q i j µ ¶ µ ¶ X ∂X j ∗ ∂Y ∗ Λ+ λj , hb i M i = ∂ξi ∂ξi j (ξi , n i ) = f (Q i , M i ),

Q i = exp(X i )q 0 ,

q 1 = exp(Y )q 0 , ¡ ¢∗ Λ. µ1 = dexp−1 Y

i = 1, . . . , s,

(3.11) Notice that we no longer involve the Lagrangian. We only need to evaluate the vector field through the map f . This opens up the possibility of applying the method to degenerate Hamiltonian systems (or indeed to any ODE on T∗G).2 It should be noted that since the integrator can be formulated as a variational integrator on G, the group structure chosen for T∗G in (3.3) is not consequential. Indeed, the integrator is uniquely defined by (3.7), (3.8), and (3.9), which do not depend on the introduced group structure on T∗G. For any choice of group structure on T∗G such that the canonical projection T∗G → G is a homomorphism of Lie groups, there is an equivalent formulation of the integrator in (3.11). Note that f is defined via the group structure, and a change of group structure would lead to f being changed as well.

3.3 First and second order integrators In the article by Celledoni et al. [4], a special case of variational integrators of the form introduced in the previous section was considered. These integrators serve as an example of application of the formulae above. In these methods, let a i j and b i be the coefficients of an s-stage Runge–Kutta method which satisfies 2 Variational methods for degenerate Hamiltonian systems using Type II generating functions

have been proposed by Leok and Zhang [13].

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3 High order symplectic partitioned Lie group methods b i 6= 0 for all i . Let the discrete Lagrangian be given by

L h (q 0 , q 1 ) = h

s X

b i `(Q i , ξi ),

i =1

and the constraints by (3.7) and

Y =h

s X

b i ξi ,

i =1

Xi = h

s X

ai j ξ j ,

i = 1, . . . , s.

j =1

We can see that for i , j = 1, . . . , s, ∂Y = 0, ∂q 0 ∂Y = 0, ∂Q i ∂Y = hb i , ∂ξi

∂X j ∂q 0 ∂X j ∂Q i ∂X j ∂ξi

= 0, = 0, = ha j i .

By inserting these into (3.11), we get the set of equations X¡ ¡ ¢∗ ¢∗ µ0 = dexp−1 Λ+ dexp−1 −Y −X j λ j , j

¡ ¢∗ hb i n i = − dexp−1 λi , Xi X hb i M i = hb i Λ + ha j i λ j , j

(ξi , n i ) = f (Q i , M i ),


q 1 = exp(Y )q 0 , ¡ ¢∗ µ1 = dexp−1 Λ. Y

84

i = 1, . . . , s,

3.3 First and second order integrators In these equations, Λ and λ j can be eliminated, giving the integrator ³ ´ X X b i M i = b i dexp∗−Y µ0 + h b j Ad∗exp(X j ) n j − h b j a j i dexp∗X j n j , j

Xi = h

X

j

ai j ξ j ,

j


(ξi , n i ) = f (Q i , M i ), X Y = h b j ξj ,

i = 1, . . . , s,

(3.12)

j

q 1 = exp(Y )q 0 , ³ ´ X µ1 = Ad∗exp(−Y ) µ0 + h b j Ad∗exp(X j ) n j . j

Here we have used the identity dexpx ◦ dexp−1 −x = Adexp(x) . Equation (3.12) is equivalent to the method presented in [4, Section 5]. Methods of this form suffer from an order barrier. They can not obtain higher accuracy than second order. The proof, presented below, is closely related to a similar order barrier for commutator-free Lie algebra methods [3]. Proposition 3.3.1. The integrators of the format (3.12) can not achieve higher

than second order on general Hamiltonian differential equations. Proof. The proof proceeds by applying the variational method (3.12) to a partic-

ular class of regular Hamiltonian problems and a particular choice of starting values. We show that in this case, the Lie group part of the solution has an error of at most second order, thus the variational method is at most of second order as well. Let a Hamiltonian on G × g∗ be given by H (q, µ) = 〈µ, v(q)〉 + T (µ), where v:G → g is smooth, but otherwise arbitrary, and T : g∗ → R is a nondegenerate quadratic function of µ. Using (3.5), we find that the corresponding Hamiltonian vector field is Ãµ ¶ ! ! dT ∂v ∗ −1 f (q, µ) = v(q) + ,− µ ·q , dµ ∂q Ã

(3.13)

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3 High order symplectic partitioned Lie group methods and the differential equation is ¶ µ dT · q, q˙ = v(q) + dµ Ãµ ¶ ! ∂v ∗ µ˙ = − µ. µ · q −1 − ad∗ v(q)+ dT ∂q dµ

We note that ¯ dT ¯ = 0.

dT dµ

³¡ ¢ ´ ∂v ∗ µ · q −1 are both linear in µ, in particular and U (q)µ := − ∂q

dµ µ=0

A particular class of solutions to this ODE consists of those solutions which satisfy µ(t ) = 0 for all t . For these solutions q(t ) solves the ODE q˙ = v(q) · q. We want to show that the numerical solution from (3.12), when applied to this problem, preserves the invariant µ = 0, and that the method reduces to a conventional Lie group method in this case. If we apply (3.12) to the Hamiltonian vector field (3.13) and set µ0 = 0, we get, among others, the equation n i = U (Q i )M i . Inserting this into the first equation of (3.12), we get b i M i = hb i dexp∗−Y

X j

b j Ad∗exp(X j ) U (Q j )M j − h

X j

b j a j i dexp∗X j U (Q j )M j ,

where i = 1, . . . , s. Clearly, this system of equations has M i = 0 for i = 1, . . . , s as a solution. Additionally, if we assume that Y and X j go to zero as h goes to zero, M i = 0 is the only solution for small enough step-length h. Therefore, n i = U (Q i )M i = 0, and µ1 = 0. The remaining equations of (3.12) are Xi = h

X

ai j ξ j ,

j


¯ dT ¯¯ ξi = v(Q i ) + = v(Q i ), dµ ¯µ=Mi X Y = h b j ξj ,

i = 1, . . . , s,

j

q 1 = exp(Y )q 0 . We recognize these equations as a commutator-free Lie group method with one exponential, or equivalently, an RKMK method with cut-off parameter 0, applied to the ODE q˙ = v(q)· q. As explained in [18], commutator-free methods

86

3.4 Higher order integrators with one exponential cannot satisfy the third order conditions, and the solution is at most second order accurate.

By repeating the argument with any variational integrator of the form described in (3.11), we get a generalization. Proposition 3.3.2. A variational integrator on T∗G of the form (3.11) based on a

Lie group integrator can not achieve higher order than the underlying Lie group integrator. We should note that first and second order methods of the format described P in (3.12) do exist. Specifically, a method of that format is first order if is=1 b i = 1 Ps and second order if, in addition, i , j =1 b i a i j = 1/2. The proof is a special case of Theorem 3.5.3 with cut-off parameter r = 0. Example 3.3.3 (The midpoint method). Let us choose s = 1. The method is of

second order if and only if we choose b 1 = 1 and a 11 = 1/2. This method is also symmetric, since the substitutions h → −h, q 0 ↔ q 1 , µ0 ↔ µ1 , Y → −Y in (3.12) yields the same method after some manipulation of the equations. This property is utilized in Section 3.6.1 to achieve high order by composition.

3.4 Higher order integrators The methods discussed in the previous section were limited to at most second order. To obtain higher order integrators, we consider two approaches, based on two well-known classes of Lie group integrators. The first approach is based on the Runge–Kutta–Munthe-Kaas (RKMK) methods. This approach was already considered by Bou-Rabee and Marsden in 2009 [2]. The work in the present article builds on the work by Bou-Rabee and Marsden, and examines in detail the case when the cut-off parameter r (q in [2, 16]) in the RKMK method is larger than 0, and provides a complete order theory for variational methods based on RKMK methods. The second approach is based on Crouch–Grossman (CG) methods. This approach has to our knowledge not been explored before. We show that these methods can achieve arbitrarily high order, but the complete order theory of these methods remains unresolved.

87


3.4.1 Variational Runge–Kutta–Munthe-Kaas integrators A popular class of Lie group integrators is the class of RKMK integrators. For our purposes, these integrators can be written xi = h

s X j =1

a i j dexp−1 (r ),x j ξ j ,

Q i = exp(x i )q 0 ,

ξi = f (Q i ), i = 1, . . . , s, s X Y =h b i dexp−1 (r ),x i ξi ,

(3.14)

i =1

q 1 = exp(Y )q 0 , where dexp−1 (r ),x = id −

r B X 1 k adx + (adx )k 2 k! k=2

is the Taylor series approximation to dexp−1 x , and a i j , b i are the coefficients of a Runge–Kutta method. If the RK method is of order p and r ≥ p − 2, the resulting Lie group integrator is of order p as well [8, Theorem IV.8.4]. Variational methods based on RKMK methods were considered by BouRabee and Marsden in [2], though the methods they present in detail are at most second order, since they only consider the case r = 0. The methods in this case are essentially the methods considered in Section 3.3. For r > 0, some complications arise for the variational integrator, since the x i are not explicitly given by ξ1 , . . . , ξs . Our solution is to treat both x i and ξi as unknowns and the equations for x i in (3.14) as restrictions. The Lagrange multipliers λi corresponding to the equations for x i cannot be eliminated from the equations in a general manner, so the dimension of the non-linear equation to be solved at each step is larger than that of the corresponding symplectic method applied to T ∗ Rn , or the simpler integrator with r = 0. In the variational integrator we set the discrete Lagrangian to Lh = h

88

s X j =1

b j `(Q j , ξ j ),

3.4 Higher order integrators and let the constraints be given by (3.7), that is q 1 = exp(Y )q 0 ,


and

Y =h Xi = h

s X j =1 s X

j =1

b j dexp−1 (r ),x j ξ j , a i j dexp−1 (r ),x j ξ j ,

i = 1, . . . , s,

¡ ¢ where x i = log Q i q 0−1 . Note that on the solution set of the constraints, X i = x i . Applying the variational equations from (3.11), the integrator is given by ³¡ ´ X ¡ ¢∗ ¢ −1 ∗ ∗ µ0 = dexp−1 − h b dexp ◦ P (X , ξ ) i i i Λ −Y −X i (r ) i

´ X ³¡ X ¢∗ ¡ ¢ −1 ∗ ∗ + dexp−1 − h a dexp ◦ P (X , ξ ) λj , i i j i −X j −X i (r ) j

i

¡ ¢∗ ¡ ¢∗ ∗ hb i n i = − dexp−1 λi + hb i dexp−1 ◦ P (r Xi Xi ) (X i , ξi )Λ X ¡ ¢ −1 ∗ ∗ + h a j i dexp X i ◦ P (r ) (X i , ξi )λ j , j

´ X ¡ ¢∗ ³ hb i M i = h dexp−1 bi Λ + a j i λ j , (r ),X i

(3.15)

j


(ξi , n i ) = f (Q i , M i ),

q 1 = exp(Y )q 0 , ¡ ¢∗ µ1 = dexp−1 Λ, Y

i = 1, . . . , s,

∗ ∗ ∗ where P (r ) (x, ξ) is a polynomial in adx and adξ of degree r , defined as the

adjoint of the partial derivative of dexp−1 (r ),x ξ with respect to x. Specifically, ∗ P (0) (x, ξ) = 0, 1 ∗ P (1) (x, ξ) = ad∗ξ , 2 1 1 1 ∗ P (2) (x, ξ) = ad∗ξ − ad∗ξ ad∗x + ad∗x ad∗ξ , 2 6 12 r B k−1 X X ¡ ¢k−i −1 1 k ∗ ∗ P (r ad∗ i ad∗x . ) (x, ξ) = adξ − ad ξ x 2 k=2 k! i =0

89

3 High order symplectic partitioned Lie group methods By applying Ad∗exp(X i ) to both sides of the second equation in (3.15), we see that the first equation can be simplified. Using this and rearranging the rest of the equations while assuming that b i 6= 0, we arrive at the set of equations ³ ´ X Λ = dexp∗−Y µ0 + h b i Ad∗exp(X i ) n i , i

λi = −hb i dexp∗X i

³ ´ X ∗ n i + hP (r (X , ξ ) b Λ + a λ , i i i j i j ) j

1¡ dexp−1 bi Λ + (r ),X i bi X X i = h a i j dexp−1 (r ),X j ξ j , ¢∗ ³

Mi =

X

´

a ji λj ,

j

(3.16)

j

¡ ¢ (ξi , n i ) = f exp(X i )q 0 , M i , X Y = h b i dexp−1 (r ),X i ξi ,

i = 1, . . . , s,

i

q 1 = exp(Y )q 0 , X ¡ ¢ µ1 = Ad∗exp(−Y ) µ0 + h b i Ad∗exp(X i ) n i , i

which define a symplectic integrator on T∗G. The identity dexpx ◦ dexp−1 −x = Adexp(x) was used to obtain the last line of the equations above. We call the integrators defined by (3.16) variational Runge–Kutta–Munthe-Kaas methods or VRKMK methods for short. One can easily check that if the Lie group is abelian, a VRKMK method simplifies to a symplectic partitioned Runge–Kutta method. In implementations of these methods, it is required that exp: g → G and dexp∗ : g × g∗ → g∗ are calculated to machine precision to obtain symplecticity. In the numerical tests of Section 3.6, we choose G = SO(3) and use Rodrigues’ formula [14, Section 9.2] to calculate these expressions. In a general setting, calculating these expressions usually involves analytic functions of matrices. The equations defining the integrator can be solved as a set of non-linear equations in the unknowns X i , M i and λi , i = 1, . . . , s, as the other quantities in (3.16) are given explicitly in terms of the aforementioned variables as well as q 0 and µ0 . If G is an n-dimensional Lie group, this is a total of 3ns scalar unknowns. Other choices of independent and dependent unknowns are possible. One could reduce the number of unknowns by starting with coefficients of an explicit RK method and thereby obtain explicit expressions for x i and be able to eliminate

90

3.4 Higher order integrators the λi in the integrator. The variational method based on an explicit RK method would still be implicit, however, and due to the order conditions presented later in this article, the increased number of stages required for a particular order would offset the reduction in number of unknowns per stage obtained by using an explicit method as the underlying method, so it is unclear if this simplification is useful in practice. Alternative approach

The authors have discovered an alternative approach which reduces the number of unknowns in the equations (3.16). The alternative approach requires a modification of the variational principle described in Section 3.2, and full details and analysis goes beyond the scope of this article. A modification of the RKMK methods can be obtained by replacing the Taylor approximation dexp−1 (r ) in (3.14) with a Padé approximation. After trivial manipulations, the modified RKMK method is xi = h

X

a i j ξ˜ j ,

j

Q i = exp(x i )q 0 , dexp(r ),xi ξ˜i = f (Q i ), X Y = h b i ξ˜i ,

i = 1, . . . , s,

i

q 1 = exp(Y )q 0 , 1 where dexp(r ),ξ = 1 + 21 adξ + · · · (r +1)! adrξ . In this formulation, the x j are explicit in the ξ˜ j , so there is no need to introduce restrictions for the equations x i = P h j a i j ξ˜ j .

The discrete Lagrangian in this formulation is L h (q 0 , q 1 ) = h

X j

b j `(Q j , dexp(r ),x j ξ˜ j ),

which is not of the format (3.6) discussed in Section 3.2. Therefore the general formulae for integrators (3.11) derived earlier do not apply. However, the general idea can still be pursued, and the resulting integrator can be formulated on the Hamiltonian side.

91


3.4.2 Variational Crouch–Grossman integrators The methods of Crouch and Grossman form another important class of Lie group methods. Crouch and Grossman formulated their integrators in terms of rigid frames, i.e. finite collections of vector fields on a manifold. On a Lie group, a suitable rigid frame is a basis for the right-invariant vector fields on G corresponding to a basis of g. In this setting, the Crouch–Grossman methods can be defined as follows. Let b i , a i j be the coefficients of an s-stage RK method. The Crouch–Grossman method [8, Section IV.8.1] with the same coefficients is defined by the equations Q i = exp(ha i s ξs ) · · · exp(ha i 1 ξ1 )q 0 , ξi = f (Q i ),

q 1 = exp(hb s ξs ) · · · exp(hb 1 ξ1 )q 0 . The order of a CG method is determined by the order conditions developed in [19]. Using the general format (3.11), we set the discrete Lagrangian to

L h (q 0 , q 1 ) = h

s X

b i `(Q i , ξi ),

(3.17)

i =1

with b i 6= 0 and constraints given by (3.7), that is q 1 = exp(Y )q 0 ,

Q i = exp(X i )q 0 , and ¡ ¢ Y = log exp(hb s ξs ) · · · exp(hb 1 ξ1 ) , ¡ ¢ X i = log exp(ha i s ξs ) · · · exp(ha i 1 ξ1 ) .

Inserting this into the equations defining a variational integrator (3.11), we

92

3.4 Higher order integrators obtain X¡ ¡ ¢∗ ¢∗ µ0 = dexp−1 Λ+ dexp−1 −Y −X j λ j , j

¡ ¢∗ hb i n i = − dexp−1 λi , Xi

¡ ¢∗ hb i M i = hb i dexp∗hbi ξi ◦ Ad∗exp(hb s ξs )··· exp(hbi +1 ξi +1 ) ◦ dexp−1 Λ Y X ¡ ¢∗ + h a j i dexp∗ha j i ξi ◦ Ad∗exp(ha j s ξs )··· exp(ha j ,i +1 ξi +1 ) ◦ dexp−1 X j λj , j

(ξi , n i ) = f (Q i , M i ),


q 1 = exp(Y )q 0 , ¡ ¢∗ µ1 = dexp−1 Λ. Y

Eliminating Λ and λ j , and rearranging, we get the integrator q1 = q s ,

Qi = Qi s ,

q j = exp(hb j ξ j )q j −1 ,

Q i j = exp(ha i j ξ j )Q i , j −1 ,

µ¯ 0 = Ad∗q0 µ0 ,

(ξi , n i ) = f (Q i , M i ), µ¯ 1 = Ad∗q1 µ1 ,

µ¯ 1 = µ¯ 0 + h M i = dexp∗hbi ξi ◦ Ad∗(q i )−1 µ¯ 1 − h

s X

q 0 = q0 ,

Q i 0 = q0 ,

∗ n¯ i = AdQ ni , i

(3.18)

b j n¯ j ,

j =1

s b a X j ji j =1

bi

∗ ¯j, dexp∗ha j i ξi ◦ AdQ −1 n ji

for all i = 1, . . . , s. We call the integrators defined by (3.18) variational Crouch– Grossman methods or VCG methods for short. The last equation in (3.18) can also be written as M i = dexp∗hbi ξi ◦ Ad∗(q i )−1 µ¯ 0 µ ¶ s X aji ∗ ∗ ∗ ∗ +h b j dexphbi ξi ◦ Ad(q i )−1 − dexpha j i ξi ◦ AdQ −1 n¯ j , ji bi j =1

(3.19)

which we will need in the order analysis of Proposition 3.5.6. In the case that the Lie group is abelian, the integrator simplifies to the same symplectic, partitioned RK method as in the abelian case for the VRKMK integrator.

93


3.5 Order analysis In analyzing the order of variational methods we use variational error analysis as described by Marsden and West [15, Section 2.3]. We recite two definitions from this reference which are useful in the following sections. The exact discrete Lagrangian is given by L Eh (q 0 , q 1 ) =

h

Z 0

¡ ¢ ˙ ) dt , L q(t ), q(t

where q(t ) is the solution to the Euler–Lagrange equations with q(0) = q 0 , q(h) = q 1 . A discrete Lagrangian L h is said to be of order p if ¡ ¢ ¡ ¢ L h q(0), q(h) = L Eh q(0), q(h) + O (h p+1 ),

for all solutions q(t ) of the Euler–Lagrange equations. The following theorem is a special case of [15, Theorem 2.3.1].3 Theorem 3.5.1. Given a regular Lagrangian L and a discrete Lagrangian L h of

order p, then the symplectic integrator defined by L h is of order p. Both classes of methods presented in this article depend on Butcher coefficients a i j and b i . Furthermore, when applied to an abelian Lie group (for instance Rn ), both classes become symplectic, partitioned RK methods where the position is integrated with the RK method with coefficients a i j and b i , while the momentum is integrated with the RK method with coefficients aî j = b j − b j a j i /b i and bˆ i = b i . The order conditions for SPRK methods have been explored in detail by Murua [17]. Since an abelian Lie group is a special case, the order of the SPRK method is an upper bound for the order of the variational Lie group method with the same coefficients. The order of the underlying Lie group method is also an upper bound according to Proposition 3.3.2.

3.5.1 Order of VRKMK integrators The VRKMK methods described in Section 3.4.1 are fully described by the Butcher coefficients a i j and b i , and the cut-off parameter r . The cut-off parameter r limits the order of the RKMK method (3.14) on G. The order of the 3 Patrick and Cuell [20] demonstrate an inaccuracy in the proof in [15]. However, they also show

that the relevant result still holds.

94

3.5 Order analysis RKMK method is the minimum of the order of the RK method based on the same coefficients and r + 2. [8, Section IV.8.2] As explained above, the order of the VRKMK method is bounded from above by the order of the SPRK method based on the same Butcher coefficients, and by the order of the RKMK method. Since the order conditions for RK methods for a particular order form a subset of the order conditions for the SPRK method, we can a priori say that the order of the VRKMK method is bounded from above by the order of the SPRK method and r + 2. Theorem 3.5.3 states that in the case of a regular Lagrangian, the order of the VRKMK method is in fact the minimum of these two bounds. The proof of this theorem relies on the following lemma. Lemma 3.5.2. Assume that the continuous Lagrangian is regular, and that the

SPRK method based on the coefficients a i j and b i is of order d . Then the discrete Lagrangian of the SPRK method (3.6) is also of order d . ˙ = 〈p, q〉 ˙ − H (q, p) the corProof. Let H (q, p) be a regular Hamiltonian, L(q, q) ¡ ¢ responding Lagrangian, and q(t ), p(t ) an exact solution to the Hamiltonian system. The resulting integrator is order d accurate if and only if the original coefficients b i and a i j together with bˆ i = b i and aî j = b j − b j a j i /b i fulfil the order conditions up to order d for a partitioned Runge–Kutta method. We will apply the partitioned Runge–Kutta method to the system · ¸ " ∂H # q˙ = ¡ ∂p ¢ , S˙ L q, q˙

p˙ = −

∂H , ∂q

where the (q, S)-component is integrated using the coefficients b i , a i j , and the p-component is integrated using the coefficients bˆ i , aî j .4 These are simply the Hamiltonian equations augmented with the differential equation for the action integral S. As starting values, we use q 0 = q(0), p 0 = p(0) and S 0 = S(0) = 0. The exact solution of the system at t = h is given by the solution to the Hamiltonian equation, q(h), p(h), and Z h ¡ ¢ ˙ dt = L Eh q 0 , q(h) . S(h) = L(q, q) 0

4 Since b ˆ i = b i and the right hand side is independent of S, we could instead have grouped S

with p without any change.

95

3 High order symplectic partitioned Lie group methods The numerical solution obtained with one step of the partitioned Runge–Kutta method is, using the notation of Section 3.2.2, q 1 = q(h) + O (h d +1 ), p 1 = p(h) + O (h d +1 ) s X ¡ ¢ S1 = h b i L(Q i , Q˙ i ) = L h (q 0 , q 1 ) = L Eh q 0 , q(h) + O (h d +1 ), i =1

since the method is order d . Using Taylor series expansion and that p 1 = ¡ ¢ ¡ D2 L h (q 0 , q 1 ), we see that L h (q 0 , q 1 ) − L h q 0 , q(h) = 〈p 1 , q 1 − q(h)〉 + O q 1 − ¢2 q(h) = O (h d +1 ), which completes the proof. Theorem 3.5.3. If the symplectic, partitioned Runge–Kutta method based on the

coefficients a i j and b i is of order at least p, and the cut-off parameter r satisfies r ≥ p − 2, then the variational Runge–Kutta–Munthe-Kaas method with the same coefficients is at least of order p for regular Hamiltonians. Proof. The proof consists of two steps. We introduce the limit case where the

cut-off parameter r goes to infinity, that is, the method where dexp−1 (r ),x is replaced by dexp−1 . To distinguish between the two RKMK methods, we will x denote the “full” RKMK method by RKMK(∞), and the RKMK method with cut-off parameter r by RKMK(r ). The variational integrators based on the two methods are denoted VRKMK(∞) and VRKMK(r ), respectively. In the first step, we show that the discrete Lagrangian which defines the VRKMK(∞) method is of order p. The proof relies on two facts. Firstly, that the discrete Lagrangian of the SPRK method is of order p. Secondly, that the discrete Lagrangians of the VRKMK(∞) and of a special case of the SPRK method are obtained as extremal values of the same object function and under the same constraints. In the second part, we show that if we apply the VRKMK(∞) and VRKMK(r ) methods to the same initial values, their difference after one step goes to zero as O (h r +3 ). Let q: [0, a] → G be a solution to the Euler–Lagrange equation with q(0) = q 0 , ¡ ¢ and assume that a > 0 is sufficiently small such that σ(t ) = log q(t )q 0−1 is uniquely defined for all t ∈ [0, a]. The exact discrete Lagrangian is given by L Eh

96

¡

¢ q 0 , q(h) =

h

Z 0

¡ ¢ ` q(t ), ξ(t ) dt ,

(3.20)

3.5 Order analysis ˜ T g → R as ˙ ) · q(t )−1 . If we define `: where ξ(t ) = q(t ¡ ¢ ˜ σ) ˙ = ` exp(σ)q 0 , dexpσ σ ˙ , `(σ, ¡ E

we can rewrite (3.20) as L h

(3.21)

¡ ¢ q 0 , q(h) = L˜ Eh 0, σ(h) where ¢

¡ ¢ L˜ Eh 0, σ(h) =

h

Z 0

¡ ¢ ˙ ) dt . `˜ σ(t ), σ(t

This is an exact discrete Lagrangian on the vector space g, which we approximate by the action sum arising from the underlying RK method, X ¡ ¢ ˜ i , η i ), L˜ RK b i `(y (3.22) h 0, σ(h) = h i

where y i = h j a i j η j , i = 1, . . . , s and the sum is extremized under the conP straint σ(h) = h i b i η i . Under the assumptions of the theorem, the order of the SPRK method is at least p, so by Lemma 3.5.2, the discrete Lagrangian of the SPRK method is order p accurate, P

¡ ¢ ¡ ¢ p+1 ˜E L˜ RK ). h 0, σ(h) = L h 0, σ(h) + O (h

Inserting (3.21) into (3.22) gives X ¡ ¢ ¡ ¢ ¡ ¢ ˜ RK L RK b i ` exp(y i )q 0 , dexp y i η i , h q 0 , q(h) = L h 0, σ(h) = h

(3.23)

i

¡ P ¢ where the sum is extremized under the constraint q(h) = exp h i b i η i q 0 . Now, the discrete action sum arising from RKMK(∞) is X ¡ ¢ ¡ ¢ L RKMK(∞) q 0 , q(h) = h b i ` exp(X i )q 0 , ξi , (3.24) h i

which is extremized under the constraints X X i = h a i j dexp−1 X j ξj , j

i = 1, . . . , s,

³ X ´ q(h) = exp h b i dexp−1 ξ X i i q0 . i

We see that under the identifications yi = X i ,

η i = dexp−1 X i ξi ,

97

3 High order symplectic partitioned Lie group methods the objective functions (3.23) and (3.24) are identical and are extremized under the same constraints. Thus their extremal values are identical and we have proved ¢ ¡ ¢ ¡ ¢ ¡ ˜ RK L RKMK(∞) q 0 , q(h) = L RK h q 0 , q(h) = L h 0, σ(h) h ¡ ¢ ¡ ¢ = L˜ Eh 0, σ(h) + O (h p+1 ) = L Eh q 0 , q(h) + O (h p+1 ), concluding the first part of the proof. For the second part of the proof, we consider the integrator in (3.16) and the variational integrator based on RKMK(∞) with the same initial data (q 0 , µ0 ). Let ξi , n i , X i , M i , λi , Λ and Y be as in (3.16), and ξ(∞) , etc. be the corresponding i quantities in VRKMK(∞). We define δξi = ξi − ξi(∞) and so on, and consider the difference between ¡ ¢∗ VRKMK(∞) and VRKMK(r ). Since q 1 = exp(Y )q 0 and µ1 = dexp−1 Λ, the Y leading order of the difference between the two integrators is given by the leading orders of δY and δΛ. It is clear from the equations in (3.16) defining the integrator that as h → 0, λi , λ(∞) , X i , X i(∞) , Y and Y (∞) all go to zero as O(h). i Furthermore, δξi , δn i , δM i and δΛ must also go to zero as h → 0. By using the expressions in (3.16), the equations ξi = ξ0 +O(h), n i = n 0 +O(h), Λ = µ0 +O(h), and X i = hc i ξ0 +O(h 2 ), and the series expansions of dexp∗ and Ad∗exp(·) , we find that X ¡ ¢ δΛ = − 12 ad∗δY µ0 + h b i ad∗δX i n 0 + δn i + higher order terms, i

X aji B r +1 r +1 r +1 ¡ ∗ ¢r +1 µ0 + δΛ + δλ j h ci adξ0 δM i = − 12 ad∗δX i µ0 − (r + 1)! j bi

+ O(h r +2 ) + h.o.t., ¡ ¢ B r +1 r r ¡ ∗ ¢r +1 δλi = −hb i 21 ad∗δX i n 0 + δn i + hb i h c i adξ0 µ0 (r + 1)! ³¡ ´ ¢ 1 + hb i 12 ad∗δξi − 16 ad∗ξ0 ad∗δX i + 12 ad∗δX i ad∗ξ0 µ0 + 12 ad∗ξ0 δΛ

+ O(h r +2 ) + h.o.t., X δX i = h a i j δξ j + O(h r +3 ) + h.o.t., j

(δξi , δn i ) = T(q0 ,µ0 ) f (δX i · q 0 , δM i ) + h.o.t., X ¡ ¢ δY = h b i δξi − 21 adδX i ξ0 + O(h r +3 ) + h.o.t. i

98

3.5 Order analysis In the equations above, “higher order terms (h.o.t.)” denote terms that are dominated by at least one of the preceding terms. We continue by combining the equations and dropping terms of higher order. Consider the equation for δξi , and insert the expression for δX i . We obtain ∂f 1 ∂f 1 (δX i · q 0 ) + (δM i ) + h.o.t. ∂q ∂µ ´ ∂f ∂f 1 ³ X 1 (δM i ) + O(h r +3 ) + h.o.t. h a i j δξ j · q 0 + = ∂q ∂µ j

δξi =

=

∂f 1 (δM i ) + O(h r +3 ) + h.o.t., ∂µ

and δX i = h

X j

ai j

∂f 1 (δM i ) + O(h r +3 ) + h.o.t. ∂µ

Similarly, successively we get ∂f 2 ∂f 2 (δX i · q 0 ) + (δM i ) + h.o.t. ∂q ∂µ ∂f 2 = (δM i ) + O(h r +3 ) + h.o.t., ∂µ X δY = h b i δξi + O(h r +3 ) + h.o.t.

δn i =

i

´ ∂f 1 ³X =h b i δM i + O(h r +3 ) + h.o.t., ∂µ i X 1 δΛ = − 2 ad∗δY µ0 + h b i δn i + O(h r +3 ) + h.o.t., i

¶ B r +1 r r ¡ ∗ ¢r +1 h c i adξ0 µ0 + 12 ad∗δξi µ0 − δn i + O(h r +2 ) + h.o.t., δλi = hb i (r + 1)! µ ¶ X aji ¡ ¢r +1 B r +1 r +1 δM i = h −c ir +1 + b j c rj ad∗ξ0 µ0 + O(h r +2 ). (r + 1)! b i j µ

From the last equation, we see that X i

b i δM i = O(h r +2 ),

99

3 High order symplectic partitioned Lie group methods which yields, successively,

X i

δY = O(h r +3 ),

b i δn i = O(h r +2 ), δΛ = O(h r +3 ),

concluding the proof.

An immediate consequence of the proof is that there exist methods in this class of arbitrarily high order. For instance, the Gauss methods [8, Section II.1.3] form a class of Runge–Kutta methods which achieve arbitrarily high order. Since these methods themselves are symplectic, aî j = b j − b j a j i /b i = a i j , and the variational method based on a Gauss method is a partitioned Runge–Kutta method with the same coefficients for both position and momentum. The variational method is equivalent to the Gauss method itself applied to the Hamiltonian ODE, and has therefore the same order as the Gauss method itself. When r is large enough, the VRKMK method based on the coefficients of a Gauss method achieves the same order as the Gauss method.

3.5.2 Order of VCG integrators In this section, we will prove that there exist VCG integrators of any order. To show this, we will need the following lemma.

Lemma 3.5.4 (Composition of VCG integrators). Let (A (1) , b (1) ) and (A (2) , b (2) )

be the Butcher tableaux of Runge–Kutta methods with s (1) and s (2) stages, and γ a real number. The composition method formed by first applying the VCG method based on (A (1) , b (1) ) with step length γh and then the VCG method based on (A (2) , b (2) ) with step length (1 − γ)h, is a VCG method with Butcher tableau

100

3.5 Order analysis

γA (1) γb 1(1) .. .

···

γb 1(1)

···

γb 1(1)

···

0 γb s(1) (1) .. .

(1 − γ)A (2)

γb s(1) (1) γb s(1) (1)

(1 − γ)b 1(2)

···

(1 − γ)b s(2) (2)

Proof. Consider the discrete Lagrangians corresponding to the two VCG integrators that are to be composed,

L (1) (q 0 , q 1 ) = h h with constraints

s (1) X i =1

¡ ¢ b i(1) ` Q i(1) , ξ(1) , i

¡ ¢ ¡ ¢ Q i(1) = exp ha i(1) ξ(1) · · · exp ha i(1) ξ(1) q 0 , 1 1 s (1) s (1) ¡ ¡ (1) (1) ¢ (1) ¢ q 1 = exp hb s(1) q0 , (1) ξ s (1) · · · exp hb 1 ξ1

and L (2) (q 0 , q 1 ) = h h with constraints

s (2) X i =1

¡ ¢ b i(2) ` Q i(2) , ξ(2) , i

¡ ¡ (2) (2) ¢ (2) ¢ ξ · · · exp ha i 1 ξ1 q 0 , Q i(2) = exp ha i(2) (2) (2) s s ¡ (2) (2) ¢ ¡ (2) (2) ¢ q 1 = exp hb s (2) ξs (2) · · · exp hb 1 ξ1 q 0 ,

as well as the composition discrete Lagrangian ¯ + L (2) ¯ q 1 ), L (c) (q 0 , q 1 ) = L (1) (q , q) (q, h γh 0 (1−γ)h where q¯ is chosen so that L (c) is extremized. It was proved in [15, Theorem 2.5.1] h

that the integrator corresponding to L (c) is the composition method that results h

101

3 High order symplectic partitioned Lie group methods from composing the integrator corresponding to L (1) with the integrator corresγh ponding to L (2) . Denote by (A (c) , b (c) ) the Butcher tableau with s (c) = s (1) +s (2) (1−γ)h stages given above. Then Ã (1) ! s s (2) X X (c) (1) ¡ (1) (1) ¢ (2) ¡ (2) (2) ¢ L h (q 0 , q 1 ) = h γ b i ` Q i , ξi + (1 − γ) b i ` Q i , ξi i =1

=h with constraints

s (c) X

i =1

i =1

¡ ¢ b i(c) ` Q i(c) , ξ(c) , i

¡ ¢ ¡ ¢ Q i(c) = exp ha i(c) ξ(c) · · · exp ha i(c) ξ(c) q 0 , 1 1 s (c) s (c) ¡ ¡ (c) (c) ¢ (c) ¢ q 1 = exp hb s(c) q0 . (c) ξ s (c) · · · exp hb 1 ξ1

Proposition 3.5.5. There exist methods of any order among the VCG integrators. Proof. From [8, Section II.4], we know that if we compose a one-step method with itself using different step sizes, we can obtain arbitrarily high order, provided we choose the number of steps and the step sizes appropriately. Thus, we obtain VCG methods of any order by composition. Proposition 3.5.6. For VCG integrators applied to regular Lagrangian problems,

the order conditions for first and second order are the same as for the underlying Runge–Kutta method, i.e. s X i =1

where c i =

P

j

b i = 1,

and

s X

1 bi ci = , 2 i =1

ai j .

Proof. We use variational order analysis, as presented in [15, Section 2.3]. Let

the exact discrete Lagrangian be denoted Z h ¡ ¢ ¡ ¢ E ˙ ) dt , L h q 0 , q(h) = L q(t ), q(t 0

where q 0 = q(0).

The exact discrete Lagrangian can be expanded in powers of h: ¯ ¶ ∞ h k µ dk ¡ ¢¯ ¡ ¢ X E E L h q 0 , q(h) = L q , q(h) ¯¯ . k h 0 k! dh h=0 k=0

102

3.5 Order analysis From the right-trivialized HP equations (3.2) and (3.4), it is straight-forward to show that ¿ À ¢ d d ¡ d ˙ ˙ ) = 〈µ, ξ〉 = 〈µ, ˙ ξ〉 + 〈µ, ξ〉 = 〈 f 2 (z), f 1 (z)〉 + µ, L q(t ), q(t f 1 (z) , dt dt dt ¡ ¢ where z = (q, µ) and f (z) = f 1 (z), f 2 (z) . Thus, using (ξ0 , n 0 ) = f (z 0 ) = f (q 0 , µ0 ), ˙ we get and `(q, ξ) = L(q, q), ¡ ¢ L Eh q 0 , q(h) = h`(q 0 , ξ0 ) µ ¿ À¶ ¢ h2 ∂ f1 ∂ f1 ¡ + 〈n 0 , ξ0 〉 + µ0 , (ξ0 · q 0 ) + n 0 − ad∗ξ0 µ0 + O (h 3 ). 2 ∂q ∂µ

Similarly, we can expand the discrete Lagrangian in powers of h by using (3.17) together with Q i |h=0 = q 0 and ξi |h=0 = ξ0 : Lh

¡

¯ ¶ ∞ h k µ dk ¢ X ¡ ¢¯ q 0 , q(h) = L q , q(h) ¯¯ k h 0 h=0 k=0 k! dh ¯ ¶ µ k k ∞ s Xh X ¯ d = h b i `(Q i , ξi )¯¯ k h=0 i =1 k=0 k! dh ¯ ¶ ³X ´ 2µ X ¯ h d =h b i `(q 0 , ξ0 ) + + O (h 3 ). 2 bi `(Q i , ξi )¯¯ 2 dh h=0 i i

By comparing equal powers of the two expansions, we see that the first order P condition is i b i = 1, as in RK methods. The second term needs more work. We apply (3.4) together with n i |h=0 = n 0 and M i |h=0 = µ0 and get ¯ µ¿ À ¿ À¶¯ X X ¯ dQ i dξi ¯¯ d = 2 bi ni · Q i , 2 bi `(Q i , ξi )¯¯ + Mi , dh dh dh ¯h=0 h=0 i i ¯ À ¯ À ¿ ¿ X X dξi ¯ dQ i ¯¯ ¯ = 2 bi n0 · q0 , + 2 µ0 , b i . dh ¯ dh ¯ i

h=0

i

h=0

We calculate the derivatives of Q i and ξi with respect to h using (3.18): ¯ X dQ i ¯¯ = a i j ξ0 · q 0 = c i ξ0 · q 0 , dh ¯h=0 j ¯ ¯ ¯ ¶ µ X dξi ¯ X ∂ f 1 dQ i ¯¯ ∂ f 1 dM i ¯¯ ¯ bi = bi ◦ + ◦ dh ¯h=0 ∂q dh ¯h=0 ∂µ dh ¯h=0 i i

103

3 High order symplectic partitioned Lie group methods We also need the derivative of M i . In this expression, we apply (3.19) and simplify using the first order condition: ¯ ³ ´ X dM i ¯ X 1 ∗ ¯ bi = 1 − b c i i n 0 − adξ0 µ0 . ¯ dh h=0 2 i i Putting these equations together, we obtain ¯ X ¯ d = 2 b i c i 〈n 0 , ξ0 〉 `(Q i , ξi )¯¯ dh h=0 i i ¿ µ ¶À ´ X X ∂ f1 ∂ f1 ³ ∗ + µ0 , 2 b i c i (ξ0 · q 0 ) + 2 1 − b i c i n 0 − adξ0 µ0 . ∂q ∂µ i i P Thus, to get second order, we need the second order RK condition i b i c i = 1/2.

2

X

bi

The computation for third order is similar, but much more complicated. We give the third order conditions here, without proof. Proposition 3.5.7. For VCG methods applied to regular Lagrangian problems,

using b i aî j + b j a j i = b i b j and cî =

Ps

ˆ

j =1 a i j , the conditions for third order are s X

s X s X

1 b i c i2 = , 3 i =1

1 bi ai j c j = , 6 i =1 j =1 ¶ µ s iX −1 X 1 bi = , bi ci bj + 2 3 i =1 j =1 s X

1 b i cî2 = , 3 i =1 ¶ µ s iX −1 X bi 1 b i cî bj + = , 2 3 i =1 j =1 s X

i =1

b i3 = 0.

The first two conditions come from standard RK methods, the third condition comes from CG methods and the fourth comes from SPRK methods, while

104

3.6 Numerical tests

−q

q β

Figure 3.1: Dipole on a stick the final two conditions are new. The last condition also appear in the order conditions for compositions of one-stage RK methods. It is noteworthy that the final condition forces at least one of the weights b i to be negative. The general order theory for VCG integrators is not complete and needs further study.

3.6 Numerical tests To test our methods, we constructed a Hamiltonian test problem which we call “dipole on a stick” (see Figure 3.1). The problem models a pendulum consisting of a long straight rod of length 1, with one end fixed (but freely rotating) at the origin, and a shorter rod of length 2α with its centre attached perpendicularly to the long rod at the other end. At each of the endpoints of the shorter rod there are charged particles with masses m/2 and electric charges ±q. The rods are assumed to be massless. The pendulum is affected by gravity in the negative e3 -direction and the electric field generated by a charged particle at position z = (0, 0, −3/2)T of charge β. The physical constants for specific gravity and electric force are set equal to 1. We chose this test problem so that it would have chaotic behaviour and conserved energy, with SO(3) as configuration space. If we let y + (t ), y − (t ) denote the positions of the positive and negative charge, respectively, the position of the pendulum can be described uniquely by the matrix g (t ) ∈ SO(3) such that y ± (t ) = g (t )y ±0 , where y ±0 = (0, ±α, −1)T is the

105


Table 3.1: Butcher-tableaux of the RKMK methods tested

1 2

1 2

r =0

+

p 3 6 p 3 6

1 4 1 4

+

p 3 6

1 2

r =2

0

0

1 2

1 2

0

0

1

−1

2

0

2 3

1 6

r =1

(a) Second order Gauss method

−

0

1 6

1

1 2 1 2

0

1 4

−

p 3 6

1 4 1 2

(c) Fourth order Gauss method

(b) Kutta’s third order method p 15 10 1 2 p 15 1 + 2 10 1 2

−

5 36 5 36

5 36 p + 2415 p + 3015 5 18

p 15 15 2 9 p 15 2 + 9 15 4 9

2 9

−

5 36 5 36

− −

p 15 30 p 15 24

5 36 5 18

r =4

(d) Sixth order Gauss method

position of the two charged particles in reference or body coordinates. Using the standard identification of so(3) with R3 and of so(3)∗ with R3 via the standard inner product, the state of the system (g , µ), can be represented with g ∈ SO(3) as a 3 × 3 real matrix, and µ ∈ so(3)∗ as a vector in R3 . The right-trivialized Hamiltonian of this system is °−1 ° °−1 ¢ ¡° 1 H (g , µ) = µT g I−1 g T µ + meT3 g e3 + qβ °g y +0 − z ° − °g y −0 − z ° , 2

where I = m diag(1 + α2 , 1, α2 ) is the inertia tensor of the pendulum.

3.6.1 Order tests The VRKMK methods that were tested are based on the 1-, 2- and 3-stage Gauss methods, and Kutta’s third order method. These methods are defined by the

106

3.6 Numerical tests

Table 3.2: Butcher-tableaux of the VCG methods tested

1 2

1 2 γ1 1 2 1 − 12 γ1

1 2

1 γ1 =

1 − 12 γ1

2−2

0

0

γ1

1 2 γ2

0

γ1

γ2

1 2 γ1

γ1

γ2

γ1

, 1/3

γ2 =

−21/3

2 − 21/3

(b) Fourth order DIRK method based on triple jump

(a) Second order midpoint method

1 2 γ1 γ1 + 21 γ2 γ1 + γ2 + 12 γ3 1 2 1 − (γ1 + γ2 + 21 γ3 ) 1 − (γ1 + 12 γ2 )

1

1 2 γ1

1 2 γ1

0

0

0

0

0

0

γ1

1 2 γ2

0

0

0

0

0

γ1

γ2

1 2 γ3

0

0

0

0

γ1

γ2

γ3

1 2 γ4

0

0

0

0

0

γ1

γ2

γ3

γ4

1 2 γ3

γ1

γ2

γ3

γ4

γ3

1 2 γ2

0

γ1

γ2

γ3

γ4

γ3

γ2

1 2 γ1

γ1

γ2

γ3

γ4

γ3

γ2

γ1

γ1 = 0.78451361047755726381949763, γ2 = 0.23557321335935813368479318,

γ3 = −1.17767998417887100694641568, γ4 = 1.31518632068391121888424973 (c) Sixth order DIRK method

107

3 High order symplectic partitioned Lie group methods Butcher tableaux and cut-off parameters in Table 3.1. These methods can be shown to satisfy the extra order conditions for variational integrators to their respective orders. The order of the VCG methods were also tested. To obtain higher order, symmetric composition of the midpoint method (which is symmetric, see Example 3.3.3) as described in [8, Section V.3.2] was used. The Butcher tableaux of the resulting methods are the same as those of the fourth and sixth order diagonally implicit Runge–Kutta methods (DIRK) shown in Table 3.2. The parameters γ1 , . . . , γ4 were derived by Yoshida [22]. The methods were implemented in M AT L A B , using a modified version of the DiffMan package [7] for defining Lie algebra and Lie group classes and functions on these spaces. The sets of non-linear equations (3.16) and (3.18) were solved by fixed-point iteration. The iteration was terminated when the norm of the residual became less than 10−11 . In these tests we have used the data m = q = β = 1,

α = 0.1,   1 0 0 g (0) = 0 0 −1 , 0 1 0 µ(0) = g (0)Ig (0)T e2 . ¡ ¢ The initial data µ(0) is chosen so that the first component of f g (0), µ(0) is e2 . The errors in µ(0.5) and g (0.5) with respect to a reference solution are shown in Figure 3.2. The errors plotted are kµ − µref k2 + kg − g ref k2 , where the first k·k2 is the Euclidean vector norm, and the second is the subordinate matrix norm. The reference solution was calculated using the sixth order VRKMK method with step size h = 10−3 . The dashed lines are reference lines for the appropriate orders and are the same lines in the two plots. As is evident from the plots, errors from fixed point iteration dominates the errors for the 6th order methods when h is small, and the methods appear to obtain their theoretical order. Analytically, the second order VRKMK and VCG methods are actually identical. The implementations of the two methods are different, as the non-linear equations are set up in slightly different manners. The result of this is that the numerical solutions differ slightly. For this numerical test, the error constants of the VRKMK methods are smaller than those of the VCG methods.

108

10−1

10−1

10−6

10−6 Error

Error

3.7 Future work

10−11 2nd order 3rd order 4th order 6th order

10−16 10−21

10−3

10−2 Step size

10−1

(a) Absolute error of VRKMK methods

10−11 10−16 10−21

2nd order 4th order 6th order 10−3

10−2 Step size

10−1

(b) Absolute error of VCG methods

Figure 3.2: Order plot. Dashed lines are reference lines for the appropriate orders

3.6.2 Long time behaviour The long term behaviour of the methods was also investigated. In Figure 3.3, the energy error of the numerical solution is plotted over the time span (0, 1000). We have used step size h = 0.01 (105 integration steps). Only the second and fourth order methods were tested on this time span. As can be seen from the plots, the energy error is small, approximately 10−3 for both second order methods, and approximately 10−7 for the fourth order VRKMK method and about 10−5 for the fourth order VCG method.

3.7 Future work The reformulation of RKMK methods with a Padé approximation of dexp−1 is briefly discussed in Section 3.4.1. This reformulation makes it possible to eliminate the Lagrange multipliers λi , which is beneficial for computational efficiency. We expect that the proof of the order of VRKMK methods, Theorem 3.5.3, will carry over to these methods without complications. Implementation and study of this approach would make the variational RKMK methods more competitive in terms of computational cost. Another class of Lie group methods is formed by the commutator-free Lie group methods. The approach described in this article can easily be used to

109

Energy error


1 0.5 0

Energy error

0

Energy error

100

200

300

400

500

600

800

900 1,000

800

900 1,000

800

900 1,000

800

900 1,000

1 0.5 0 100

200

300

400

500

600

700

Time Second order VCG method

·10−3

1 0.5 0 0

100

200

1

300

400

500

600

700

Time Fourth order VCG method

·10−5

0.5 0 0

100

200

300

400

500

600

Time Figure 3.3: Energy error

110

700

Time Fourth order VRKMK method

·10−7

0

Energy error

Second order VRKMK method

·10−3

700

3.7 Future work formulate symplectic Lie group methods based on commutator-free methods. Formulation, implementation and study of variational commutator-free methods are aspects that can be pursued in the future. A desirable result would be generalization of these integrators to homogeneous spaces. This has proven to be more difficult than one could hope. In general, the problem arises due to isotropy. If M is a homogeneous G-space with dim(M ) < dim(G), then the infinitesimal action at a point z ∈ M ,

g 3 ξ 7→

∂ exp(t ξ) · z ∈ T z M , ∂t

is not injective. Therefore, to identify a vector in T z M with some element in g, a choice has to be made. The main idea of variational integration is to minimize the action. Inspired by this, one could attempt the following approach, sketched out for a variational method based on the one-stage θ-method for 0 ≤ θ ≤ 1. Assume the action is from the left, and denote the action as g · q, and the infinitesimal action as ξ · q. Let `(q, ξ) = L(q, ξ · q) be the “trivialized” Lagrangian, and use the discrete Lagrangian ¡ ¢ L h (q 0 , q 1 ) = min h` exp(hθξ) · q 0 , ξ (3.25) ξ

where the minimum is taken over all ξ such that exp(hξ) · q 0 = q 1 . If the minimizing equation can be solved, this discrete Lagrangian can be used to construct symplectic integrators. However, the following example shows that in some cases, the minimizing equation has no solution. Let M = R and the group action that of affine functions R → R, i.e., for (a, b) ∈ (R à {0}) × R = G, (a, b) · q = aq + b. ˙ = 12 q˙ 2 . In this case it turns Let the Lagrangian be that of a free particle, L(q, q) out that the minimizing problem (3.25) can be expressed as an unconstrained one-dimensional problem, µ ¶ h xehθx 2 L h (q 0 , q 1 ) = min (q 1 − q 0 )2 . x∈R 2 ehx − 1

However, this has no solution if q 1 6= q 0 , so L h (q 0 , q 1 ) is not defined. Furthermore, for 0 < θ < 1, the expression has a maximizer, so a naive solution to the extremization problem would return the maximizing solution. The symplectic method based on such a solution is not even consistent.

111


3.8 Conclusion In this article, a set of equations defining symplectic integrators for ODEs on T∗G were presented, as well as two classes of integrators using these equations. The integrators obtained are formulated intrinsically on T∗G, and any drift away from the manifold in numerical solutions is due to round-off errors. The integrators were developed as variational methods for Lagrangian problems, and are therefore symplectic when applied to Hamiltonian differential equations. Both classes that were studied, were shown to contain methods of arbitrarily high order, although the computational cost per time step increases with the order. Effective implementation of the methods has not been a major goal in this article, we have instead focused on the properties of these methods. The two classes of symplectic methods are based on, respectively, the Runge– Kutta–Munthe-Kaas methods, and the Crouch–Grossman methods. The methods have a partitioned structure where the position on the Lie group is integrated by the Lie group method while the momentum is integrated by formulae which involve various functions on g∗ . We can therefore say that these methods are partitioned Lie group methods and are Lie group methods in a wide understanding of that term. To the knowledge of the authors, this is the first time that symplectic partitioned Lie group methods have been presented and studied in the level of detail done in this article.

Acknowledgements We would like to thank our supervisor, Brynjulf Owren, for encouragement and many helpful discussions. We would also like to thank Klas Modin for fruitful discussions leading to the writing of this paper. Finally, we would like to thank the two anonymous referees for helpful comments and suggestions. The research was supported by the Research Council of Norway, and by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme.

112

3.8 Conclusion

References [1]

V. I. Arnol’d and B. A. Khesin. Topological methods in hydrodynamics. Vol. 125. Applied Mathematical Sciences. Springer-Verlag, New York, 1998. DOI: 10.1007/b97593.

[2]


[3]


[4]

E. Celledoni, R. I. McLachlan, B. Owren and G. R. W. Quispel. ‘Geometric properties of Kahan’s method’. In: J. Phys. A 46.2 (2013), pp. 025201, 12. DOI: 10.1088/1751-8113/46/2/025201.

[5]


[6]

K. Engø. ‘Partitioned Runge–Kutta methods in Lie-group setting’. In: BIT 43.1 (2003), pp. 21–39. DOI: 10.1023/A:1023668015087.

[7]

K. Engø, A. Marthinsen and H. Z. Munthe-Kaas. ‘DiffMan: an objectoriented MATLAB toolbox for solving differential equations on manifolds’. In: Appl. Numer. Math. 39.3–4 (2001). Special issue: Themes in geometric integration, pp. 323–347. DOI: 10.1016/S0168-9274(00)000428.

[8]


[9]

J. Hall and M. Leok. Lie Group Spectral Variational Integrators. 2014. arXiv: 1402.3327 [math.NA]. Submitted.

[10]


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3 High order symplectic partitioned Lie group methods [11]

M. Leok. ‘Foundations of computational geometric mechanics’. PhD thesis. California Institute of Technology, 2004. URL: http://resolver. caltech.edu/CaltechETD:etd-03022004-000251.

[12]

M. Leok. Variational Integrators. In: Encyclopedia of Applied and Computational Mathematics. Ed. by B. Engquist. Springer-Verlag Berlin Heidelberg, 2013. URL: http://www.springerreference.com/.

[13]

M. Leok and J. Zhang. ‘Discrete Hamiltonian variational integrators’. In: IMA J. Numer. Anal. 31.4 (2011), pp. 1497–1532. DOI: 10.1093/imanum/ drq027.

[14]


[15]


[16]


[17]

A. Murua. ‘On order conditions for partitioned symplectic methods’. In: SIAM J. Numer. Anal. 34.6 (1997), pp. 2204–2211. DOI: 10 . 1137 / S0036142995285162.

[18]

B. Owren. ‘Order conditions for commutator-free Lie group methods’. In: J. Phys. A 39.19 (2006), pp. 5585–5599. DOI: 10.1088/0305-4470/39/ 19/S15.

[19]


[20]

G. W. Patrick and C. Cuell. ‘Error analysis of variational integrators of unconstrained Lagrangian systems’. In: Numer. Math. 113.2 (2009), pp. 243– 264. DOI: 10.1007/s00211-009-0245-3.

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3.8 Conclusion [21]

Y. B. Suris. ‘Hamiltonian methods of Runge–Kutta type and their variational interpretation’. Russian. In: Mat. Model. 2.4 (1990), pp. 78–87. URL: http://mi.mathnet.ru/eng/mm2356.

[22]

H. Yoshida. ‘Construction of higher order symplectic integrators’. In: Phys. Lett. A 150.5–7 (1990), pp. 262–268. DOI: 10.1016/0375-9601(90) 90092-3.

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Paper 3 Geometric integration of non-autonomous Hamiltonian problems

Håkon Marthinsen and Brynjulf Owren To be submitted

4 Geometric integration of non-autonomous Hamiltonian problems Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of dimensions. We show that one possible extension of symplectic methods in the autonomous setting to the non-autonomous setting is obtained by using canonical transformations. Many existing methods fit into this framework. We also perform experiments which indicate that for exponential integrators, the canonical and symmetric properties are important for good long time behaviour. In particular, the theoretical and numerical results support the well documented fact from the literature that exponential integrators for non-autonomous linear problems have superior accuracy compared to general ODE schemes. Abstract.

4.1 Introduction An important property of a Hamiltonian system is that its flow is a symplectic map. The idea of devising numerical methods which are themselves symplectic maps goes back into the previous century, some early references are [7, 25]. The monographs by Hairer, Lubich and Wanner [12] and Leimkuhler and Reich [20] may be consulted for an extensive treatment. Such numerical methods are called symplectic integrators and their success is often explained through the well known fact that any symplectic map can be identified as the exact flow of a local, perturbed Hamiltonian problem. This ensures good long time behaviour in the sense that the exact Hamiltonian is approximately conserved over

119

4 Geometric integration of non-autonomous Hamiltonian problems exponentially long times and that the numerical approximation also nearly preserves invariant tori of the exact flow. Methods which do not possess this symplectic property will often exhibit a drift in the energy and even their global accuracy will typically deteriorate faster over long times than symplectic schemes. As discussed in [3], a particularly attractive feature of the Hamiltonian formulation of mechanics compared to its Lagrangian counterpart is that the former distinguishes between the geometry of the problem represented by a symplectic structure and the dynamical aspects which are represented by the Hamiltonian function. In the Lagrangian formulation this feature is absent since the symplectic structure is partly encoded in the Lagrangian function. Turning now to time-dependent systems, we assume that the dependent variables belong to some cotangent bundle T∗Q. The usual Hamiltonian description introduces a contact structure on the space T∗Q × R. By definition, this structure depends on the time-dependent Hamiltonian H (q, p, t ), and thus the separation between the geometry and dynamics is again lost. The geometric meaning of a canonical transformation is therefore no longer clear as in the autonomous case. A common approach is to extend the system by adding an extra position variable. This variable can be interpreted as a new time variable. Then one may consider the extended phase space T∗(Q × R) which can be furnished with a symplectic form, see for instance [26]. In the late 1990s a renewed interest in the numerical solution of linear nonautonomous differential equations was sparked, in particular through some pioneering papers by Iserles and Nørsett, see e.g. [18], where they developed numerical methods based on the Magnus expansion [21]. There are also other similar ways of representing the exact flow of such problems, for instance the Fer expansion [8], see also [17] and [4]. This activity resulted in several new contributions to the numerical solution of linear and quasi-linear nonautonomous PDEs, see e.g. [9, 10, 15]. Another application branch of such methods is highly oscillatory linear non-autonomous ODEs. Asymptotic analysis can be used to show excellent behaviour of the global error when the dominating frequencies of the problem tend to infinity, see for instance [11] and [16]. In this paper, we attempt to present a more geometric view on integrators for non-autonomous systems, and we give particular attention to methods which have an exponential character, such as Magnus integrators. We use the definition of canonical transformations introduced by Asorey, Cariñena

120

4.2 Four classes of problems and Ibort [3]. Their framework is relatively general and we shall consider the question of which numerical integrators can be characterized as canonical transformations. In particular we shall see that the most common exponential integrators for non-autonomous linear problems can be furnished with such a property. Finally, we provide numerical evidence showing that canonicity in this sense together with symmetry of the scheme appear to be important for the long term behaviour of integrators. It is well known from the literature that if such methods are also exponential, they can have excellent properties, as is for instance the case for Magnus integrators. However, being exponential without any of these two additional properties will typically not yield a good approximation of the Hamiltonian over long times.

4.2 Four classes of problems In this section we consider the four possible combinations of autonomous and non-autonomous, linear and non-linear differential equations. Autonomous, linear (AL) problems.

The AL case can be written as

y˙ = Ay,

y(0) = y 0 ,

where A ∈ Rd ×d is constant. The solution to AL problems can be represented exactly by means of the matrix exponential, y(t ) = exp(t A)y 0 , thus, numerical methods for this class amount to considering methods of computing or approximating the matrix exponential, see e.g. [24]. We will not consider AL problems in this paper. Autonomous, non-linear (AN) problems.

y˙ = f (y),

The AN case can be written as

y(0) = y 0 ,

where f : Rd → Rd . Most numerical schemes for ordinary differential equations are conveniently applied to problems written in this format and are treated in several monographs and textbooks such as [13].

121

4 Geometric integration of non-autonomous Hamiltonian problems

Non-autonomous, linear (NL) problems.

y˙ = A(t )y,

The NL case can be written as y(0) = y 0 ,

where A: R → Rd ×d . Since this problem class constitutes a subset of the nonlinear problems, most general numerical schemes for ODEs can be applied also to this class. However, there exist several classes of integrators which are tailored for this problem type, two of which are the Magnus methods [18] and methods based on the Fer expansion [8]. In particular, such methods have found applications to non-autonomous linear PDEs such as the timedependent Schrödinger equations [15] and to highly oscillatory problems, see [16, 19] and the references therein. We can turn NL problems into AN problems by substituting t with a new variable y d +1 and appending the ODE y˙d +1 = 1. This process is called autonomization. By doing this, we are replacing a linear problem by a non-linear problem, which may be more difficult to solve numerically. NL problems are the main focus of this paper. Non-autonomous, non-linear (NN) problems.

y˙ = f (y, t ),

The NN case can be written as

y(0) = y 0 ,

where f : Rd × R → Rd . NN problems can be turned into AN problems by autonomization. This class of problems is not the main focus in this paper.

4.3 Autonomous and non-autonomous Hamiltonian mechanics In this section, we discuss the dynamics of autonomous (i.e. time-independent) and non-autonomous (i.e. time-dependent) Hamiltonian systems.

4.3.1 Autonomous Hamiltonian systems We will first review the basics of autonomous Hamiltonian systems [2, 22]. Let Q be a smooth n-dimensional manifold, and denote its cotangent bundle as T∗Q. The manifold Q is called the configuration space, and T∗Q is called the phase space. We will often use (q, p) as an element of T∗Q, where q ∈ Q and

122

4.3 Autonomous and non-autonomous Hamiltonian mechanics p ∈ T∗q Q. A Hamiltonian H : T∗Q → R, together with a symplectic 2-form ω0 on T∗Q, determine the Hamiltonian vector field X H via the equation i X H ω0 = −dH ,

(4.1)

where d and i are the exterior derivative and the interior product, respectively. In canonical (also called Darboux) coordinates (q i , p i ), we can write ω0 = dp i ∧ dq i (with implicit summation over repeated indices), and (4.1) turns into Hamilton’s equations, q˙ i =

∂H , ∂p i

p˙i = −

∂H ∂q i

,

for all 1 ≤ i ≤ n.

It can be easily proved [22, Section 5.4] that the autonomous Hamiltonian H and the symplectic form ω0 are conserved along the integral curves of X H . Hamiltonian AN problems can be solved numerically by standard symplectic integrators [12, Chapter VI], e.g. using symplectic, partitioned Runge–Kutta (SPRK) methods.

4.3.2 Non-autonomous Hamiltonian systems We will now consider then non-autonomous case, i.e. when H depends on time as well as phase space, so H : T∗Q × R → R. The characterization of Hamiltonian vector fields using the symplectic 2-form (4.1) is no longer appropriate, since T∗Q × R is an odd-dimensional space, while the symplectic 2-form requires an even-dimensional phase space. Hamilton’s equations still apply unchanged, but H is no longer conserved along the integral curves of X H . Let · ¸ 0 In Jn := , −In 0

where In is the n × n identity matrix. Writing y = (q, p) as a column vector, Hamilton’s equations in canonical coordinates become h i ∂H ∂H ∂H ∂H y˙ = Jn H yT , H y = ∂q 1 · · · ∂q n ∂p 1 · · · ∂p n . For the case of Hamiltonian NL problems, we need y˙ = A(t )y. Consider a generic Hamiltonian which is quadratic in the phase space variables, H = − 12 y T Jn A(t )y.

We may assume without loss of generality that A(t ) ∈ sp(2n). Since Jn A + A T Jn = 0, it follows that Jn A is symmetric, and we get y˙ = Jn H yT = A(t )y.

123


Contact structure

The usual way to tackle this problem is to apply contact structure [1, Chapter 5], [2, Appendix 4]. We use notation similar to Asorey, Cariñena and Ibort [3]. Let τ: T∗Q ×R → T∗Q be the projection (q, p, t ) 7→ (q, p), and let ω0 = dp i ∧dq i ˜ 0 := τ∗ ω0 . The be the canonical symplectic form on T∗Q, as before. Define ω contact structure on T∗Q × R is then given by the contact form ˜ 0 − dH ∧ dt . ωH := ω This enables us to define the (time-dependent) vector field X H via i X H ωH = 0,

i X H dt = 1,

or in canonical coordinates q˙ i =

∂H , ∂p i

p˙i = −

∂H ∂q i

,

t˙ = 1,

for all 1 ≤ i ≤ n.

(4.2)

The contact form ωH is preserved along the flow of X H , but H is not. Extended phase space

An alternative to using contact structure is to append one more dimension to T∗Q × R, thus obtaining an even-dimensional extended phase space T∗(Q × R). Because of this, we may now mimic the autonomous case and define the Hamiltonian system using symplectic forms. We denote the new variable as u. Let (q, p, t , u) ∈ T∗(Q × R), and let µ: T∗(Q × R) → T∗Q × R be the projection (q, p, t , u) 7→ (q, p, t ). We define the extended Hamiltonian K : T∗(Q × R) → R as K := H ◦ µ + u, and the symplectic form on the extended phase space as ˜ 0 + du ∧ dt , Ω0 := µ∗ ω

(4.3)

or in canonical coordinates, Ω0 = dp i ∧ dq i + du ∧ dt . The vector field X K is then defined the same way as in the autonomous case by i X K Ω0 = −dK ,

124

4.3 Autonomous and non-autonomous Hamiltonian mechanics which in canonical coordinates can be written as ∂K ∂H =− , ∂t ∂t (4.4) for all 1 ≤ i ≤ n. Note that H does not depend on u, so we can consider the ˙ the equations equation for u˙ as superfluous. If we disregard the equation for u, are the same as for the contact structure approach (4.2). Thus, the integral curves in extended phase space project (via µ) onto the integral curves defined by the contact structure in T∗Q × R. However, if we want to retain the usual notion of symplecticity of the flow of the vector field, we need to retain the ˙ equation for u. Analogous to the autonomous case, both Ω0 and K are conserved along the flow of X K . If we choose the initial values q 0 , p 0 , t 0 , and u 0 = −H (q 0 , p 0 , t 0 ), we get that K = 0 along the flow. This allows us to interpret −u as the energy of the Hamiltonian system. q˙ i =

∂H ∂K = , ∂p i ∂p i

p˙i = −

∂K

∂q i

=−

∂H

∂q i

,

t˙ =

∂K = 1, ∂u

u˙ = −

4.3.3 Canonical transformations It is a well known fact that symplectic integrators for autonomous problems have excellent long-time properties, however it is not clear whether the same is true for non-autonomous problems. An enticing thought is to use the constructions from the previous section so that we get a well-defined concept replacing symplecticity for the non-autonomous case. The solution employed by Asorey, Cariñena and Ibort [3] is to extend symplectic maps to canonical transformations, as defined below. Definition 4.3.1. A canonical transformation of a time-dependent system (T∗Q ×

R, ωH ) is a pair (ψ, ϕ) of diffeomorphisms, ψ on T∗(Q × R) and ϕ on T∗Q × R such that

1. µ ◦ ψ = ϕ ◦ µ, and 2. ψ∗ Ω0 = Ω0 (i.e. ψ is a symplectomorphism). The condition µ ◦ ψ = ϕ ◦ µ means that the diagram T∗(Q × R)

µ

ϕ

ψ

T∗(Q × R)

T∗Q × R

µ

T∗Q × R

125

4 Geometric integration of non-autonomous Hamiltonian problems commutes. A consequence of Definition 4.3.1 is that ψ must be a symplecto¡ ¢ ¯ ¯ T∗(Q × morphism of the form ψ(q, p, t , u) = ϕ(q, p, t ), ϕ(q, p, t , u) , where ϕ: R) → R. We will sometimes refer to ψ as a canonical transformation when there exists a ϕ such that (ψ, ϕ) is a canonical transformation. Many integrators already exist in T∗Q × R, e.g. Magnus integrators for linear problems. Given such an integrator ϕ, we seek a matching ψ such that (ψ, ϕ) is a canonical transformation. Using ideas similar to those of Asorey, Cariñena and Ibort [3], we have the following theorem which characterizes canonical transformations where time is advanced by a constant h. Theorem 4.3.2. Let ϕ be a diffeomorphism of T∗Q × R where the t -component is

advanced by a constant time-step h. Then the following are equivalent. (i) (ψ, ϕ) is a canonical transformation. (ii) There exists a function W : T∗Q × R → R such that ˜0 = ω ˜ 0 − dW ∧ dt , ϕ∗ ω

(4.5)

and ψ = (ϕ ◦ µ, u + W ◦ µ). Proof. Assume that (i) is true. We know that ψ∗ Ω0 = Ω0 . Inserting (4.3), apply-

ing µ ◦ ψ = ϕ ◦ µ and ψ∗ dt = dt , and rearranging, we get

˜0 −ω ˜ 0 ) = −d(ψ∗ u − u) ∧ dt . µ∗ (ϕ∗ ω

(4.6)

Let ν: T∗Q × R → T∗(Q × R) be any map such that µ ◦ ν = id. Our candidate function is W = ν∗ (ψ∗ u − u). We apply ν∗ to both sides of (4.6), insert the candidate function, and get ˜0 −ω ˜ 0 = −dW ∧ dν∗ t . ϕ∗ ω In the following, we will use the same symbol for the coordinate function for time t in both T∗(Q × R) and T∗Q × R. Since µ∗ t = t , we also have that ν∗ t = t and we end up with (4.5). Inserting (4.5) into (4.6), applying µ∗ t = t , and rearranging, we obtain d(µ∗W − ψ∗ u + u) ∧ dt = 0.

(4.7)

Let f := µ∗W − ψ∗ u + u. From (4.7), we see that f can only depend on t . If we apply ν∗ to f and insert the candidate function, we see that f ◦ ν = 0. Since f only depends on t , this implies that f = 0, proving that u ◦ ψ = u + W ◦ µ.

126

4.3 Autonomous and non-autonomous Hamiltonian mechanics Conversely, assume now that (ii) is true. The map ψ is given by µ◦ψ = ϕ◦µ

together with

u ◦ ψ = u + W ◦ µ.

(4.8)

We apply µ∗ and (4.8) to (4.5) and get ˜ 0 = µ∗ ω ˜ 0 − µ∗ (dW ∧ dt ). ψ∗ µ∗ ω Since µ∗ t = t , we get ˜ 0 = µ∗ ω ˜ 0 − d(ψ∗ u − u) ∧ dt . ψ∗ µ∗ ω Using (4.3), we obtain ψ∗ (Ω0 − du ∧ dt ) = Ω0 − du ∧ dt − d(ψ∗ u − u) ∧ dt , or, by applying the fact that ψ∗ dt = dt , ¡ ¢ ψ∗ Ω0 − Ω0 = d ψ∗ u − u − (ψ∗ u − u) ∧ dt = 0.

From this point, we will work in canonical coordinates. This will make the connection with existing numerical methods clearer, as well as provide formulas that can be used directly in numerical calculations. We will regard q = (q i )ni=1 , Q = (Q i )ni=1 , p = (p i )ni=1 , and P = (P i )ni=1 as column vectors. Let z = (q, t , p, u) and Z = (Q, t + h, P,U ) = ψ(z) be column vectors in R2n+2 , with U = u + W (q, p, t ). Since Q and P are independent of u, the Jacobian matrix Z z := ψ0 (z) can be written as 

Qq  0  Zz =   Pq Wq

Qt 1 Pt Wt

Qp 0 Pp Wp

 0 0  , 0 1

∂Q 1  ∂q 1

···

∂Q 1 ∂q n 

∂Q ∂q 1

···

∂Q ∂q n



where

 Q q :=  ...  n



..  , .   n

and similarly for the other submatrices. Let Y y :=

· Qq Pq

¸ Qp , Pp

and

Y t :=

· ¸ Qt . Pt

127

4 Geometric integration of non-autonomous Hamiltonian problems Proposition 4.3.3. In canonical coordinates, condition (4.5) in Theorem 4.3.2 is

equivalent to Y y ∈ Sp(2n) together with £ W y := Wq

¤ Wp = −Y tT Jn Y y .

(4.9)

Proof. Assume that (4.5) is satisfied. In canonical coordinates, (4.5) is

dP i ∧ dQ i = dp i ∧ dq i −

µ

∂W ∂q

dq i + i

¶ ∂W dp i ∧ dt . ∂p i

It is straight-forward to show that this is equivalent to the five equations In = P pTQ q −Q pT P q ,

0 = P qTQ q −Q qT P q , 0 = P pTQ p −Q pT P p ,

Wq = P tTQ q −Q tT P q , Wp = P tTQ p −Q tT P p .

(4.10) (4.11) (4.12) (4.13) (4.14)

Equations (4.10)–(4.12) may be written as Y yT Jn Y y = Jn , which implies Y y ∈ Sp(2n). Using U = u + W , we can write conditions (4.13)– (4.14) as W y = −Y tT Jn Y y . To prove the converse, simply reverse the proof. In the autonomous setting, canonical transformations are equivalent to symplectic maps. To see this, assume that we are in the autonomous setting, and are given a symplectic map Y = ϕ(y). Then Y y ∈ Sp(2n), Y t = 0, and (4.9) is satisfied by, say, W = 0, giving U = u. Thus, ϕ can be turned into a canonical transformation simply by appending the trivial update equation U = u. This is compatible with the earlier observation that −u may be regarded as the energy of the system.

128

4.4 Canonical transf. and integrators for non-auton. Hamiltonian systems

4.4 Canonical transformations and integrators for non-autonomous Hamiltonian systems In this section we take a look at how existing methods with constant timestep fit into the framework of canonical transformations. We consider the two situations where we are given either ϕ: T∗Q × R → T∗Q × R or ψ: T∗(Q × R) → T∗(Q × R), and would like to find the complementing map such that (ψ, ϕ) is a canonical transformation.

4.4.1 Constructing a canonical transformation from a given map ϕ In general, this situation is already covered by Theorem 4.3.2. Corollary 4.4.1. Let (Y , T ) = ϕ(y, t ) = M (t )y, t + h , where M (t ) ∈ Sp(2n). Then

¡

¢

(ψ, ϕ) is a canonical transformation, with ψ = (ϕ ◦ µ, u + W ◦ µ) and W = 12 y T M (t )T Jn M 0 (t )y.

(4.15)

Proof. From Proposition 4.3.3, we need that Y y = M (t ) ∈ Sp(2n), which is clearly

satisfied. Condition (4.9) says that we must find a W such that W y = −Y tT Jn Y y = −y T M 0 (t )T Jn M (t ).

Integrating with respect to y and transposing the result, we obtain (4.15). Thus, by Theorem 4.3.2, (ψ, ϕ) is a canonical transformation. The class of methods ϕ in Corollary 4.4.1 contains among others, Magnus ¡ ¢ methods [18] M (t ) = exp hX (t ) , Fer methods [17] and commutator-free meth¡ ¢ ¡ ¢ Q ods [5] M (t ) = i exp hX i (t ) , and Cayley methods [23] M (t ) = cay hX (t ) , where X (t ) and X i (t ) are elements of sp(2n). All of these can be applied to non-autonomous linear Hamiltonian problems y˙ = A(t )y (i.e. NL problems). To get consistent methods, we must choose M carefully. In fact, by considering the modified vector field of the methods, we get that the methods of this class are consistent if M (t )|h=0 = I2n , and ¯ dM (t ) ¯¯ = A(t ). dh ¯h=0

129

4 Geometric integration of non-autonomous Hamiltonian problems Example 4.4.2. Magnus integrators fit into this format by choosing M (t ) =

¡ ¢ exp hX (t ) for X : R → sp(2n). Consistency requires

X (t )|h=0 = A(t ). ¡ ¢ Since M 0 (t ) = h dexphX (t ) X 0 (t ) M (t ), and M (t )T Jn M (t ) = Jn , we can apply Corollary 4.4.1 and express U = u + W ◦ µ as U =u+

¢ h T ¡ y Jn dexp−hX (t ) X 0 (t ) y, 2

U =u+

¢ h T ¡ Y Jn dexphX (t ) X 0 (t ) Y . 2

or alternatively as

4.4.2 Constructing a canonical transformation from a given map ψ Assume that we are given a symplectomorphism ψ: T∗(Q × R) → T∗(Q × R), where the t -component is advanced by a constant time-step h, i.e. ψ is the map (q, t , p, u) = z 7→ Z = (Q, t + h, P,U ). We seek a map ϕ such that (ψ, ϕ) is a canonical transformation. This is only possible if Q and P are independent of u, since we need µ ◦ ψ = ϕ ◦ µ. In the following proposition, we will use local coordinates and write x = (q, p, t ) ∈ T∗Q × R, and κ: (q, p, u) 7→ (q, p, h, u). Proposition 4.4.3. Let H : T∗Q × R → R be a Hamiltonian. Any symplectomorph-

ism Z = ψ(z) which can be expressed in coordinates as Z = z + κ ◦ F (H x ) ◦ µ(z),

where F : Diff(R2n+1 ) → Diff(R2n+1 ), is a canonical transformation. Proof. The only way we can satisfy µ ◦ ψ = ϕ ◦ µ is if both Q and P are independent of u. The vector H x consists of the partial derivatives of H , which is independent of u, so F (H x ) also has to be independent of u. Thus, the only component of Z that can depend on u is U , proving that there exists a ϕ satisfying µ ◦ ψ = ϕ ◦ µ. Corollary 4.4.4. Symplectic partitioned Runge–Kutta (SPRK) methods applied

to the Hamiltonian problem with extended Hamiltonian K = u + H ◦ µ are canonical transformations.

130

4.4 Canonical transf. and integrators for non-auton. Hamiltonian systems Proof. This follows immediately from Proposition 4.4.3. Example 4.4.5. Let us check that SPRK methods actually are canonical trans-

formations by calculating W and ϕ. An SPRK method is given by a Butcher tableau with coefficients a i , j and b i 6= 0. The second Butcher tableau (marked by a hat) in the partitioned method is given by the first one via the formulas aî , j = b j − a j ,i b j /b i and bˆ i = b i . The method will then be ∂K k¯ i = (Q¯ i , P¯i ), ∂p¯ s X Q¯ i = q¯ + h a i , j k¯ j ,

∂K l¯i = − (Q¯ i , P¯i ), ∂q¯ s X P¯i = p¯ + h aî , j l¯j ,

j =1

Q¯ = q¯ + h

s X

j =1

b i k¯ i ,

P¯ = p¯ + h

i =1

s X

b i l¯i .

i =1

We can rewrite this using µ ¶ ∂H ¯ ˆ := (Q i , P i , Ti ), 1 , k i (k i , k i ) = ∂p ¶ µ ¯l i := (l i , lî ) = − ∂H (Q i , P i , Ti ), − ∂H (Q i , P i , Ti ) , ∂q ∂t

and we obtain Qi = q + h

s X

ai , j k j ,

j =1

Q = q +h

Pi = p + h s X

bi ki ,

where c i =

aî , j l j ,

j =1

P = p +h

i =1

T = t + h,

s X

U = u +h

s X

s X

Ti = t + c i h, bi l i ,

i =1

b i lî ,

i =1

Ps

j =1 a i , j . From these formulas, we see that

W =h

s X

b i lî ,

i =1

131

4 Geometric integration of non-autonomous Hamiltonian problems and Q, P , T and W are indeed independent of u. Thus, we have found ϕ. For non-autonomous, linear problems y˙ = A(t )y, y = (q, p), we have the Hamiltonian H = − 12 y T Jn A(t )y, so writing Yi = (Q i , P i ), we obtain · ¸ ki = A(Ti )Yi , li

lî = 21 YiT Jn A 0 (Ti )Yi .

We note in passing that backward error analysis can be trivially adapted to the situation of non-autonomous Hamiltonian systems. Since we are applying a symplectic method ψ to a Hamiltonian ODE z˙ = f (z) in extended phase space with Hamiltonian K , we can apply the result from [12, Theorem IX.3.1], showing that the modified equation is also Hamiltonian, and has Hamiltonian K˜ (z) = K (z) + hK 2 (z) + h 2 K 3 (z) + · · · . The differential equation t˙ = 1 is integrated exactly by the numerical method, so we can write K˜ = u + H˜ ◦ µ. Thus, ( H˜ − H ) ◦ µ = hK 2 + h 2 K 3 + · · · , which shows that the energy error for the numerical method is of order no less than the order of the symplectic method in extended phase space.

4.5 Numerical experiments In the numerical experiments, we would like to consider the situation where the non-autonomous problem can be viewed as a small perturbation of an autonomous problem with bounded energy. Other more challenging problems, such as the Airy equation (which has unbounded energy), will not be considered here. Consider the time-dependent harmonic oscillator with Hamiltonian ´ ¢ 1 ³¡ H (q, p, t ) = 1 + ² sin(αt ) q T q + p T p , (4.16) 2 where q, p ∈ Rn , 0 < ² ¿ 1, and 0 < α ¿ 1. As we saw in Section 4.3.2, this Hamiltonian corresponds to the linear ODE y˙ = A(t )y,

132

A(t ) =

0 ¡ ¢ − 1 + ² sin(αt ) In

·

¸ In . 0

4.5 Numerical experiments We can think of this oscillator as a slowly varying perturbation of the usual harmonic oscillator. The time-dependent perturbation ensures that the energy H and the symplectic 2-form ω0 of the system are no longer conserved, but since the perturbation is small and periodic, we expect that the energy is bounded as long as there is no resonance.

4.5.1 Long-time performance We believe that canonical methods may be well suited for non-autonomous Hamiltonian problems where we seek a long-time numerical solution with qualitatively good results. We will investigate this by considering the symmetric, 4th order Magnus method based on two-stage Gauss–Legendre quadrature [12, Example IV.7.4]. We will call this method the Lie–Gauss method. The update map in T∗Q × R is ! p 2 µ ¶ 3h h (Y , T ) = ϕ(y, t ) = exp (A 1 + A 2 )+ [A 2 , A 1 ] y, t +h , 2 12 Ã

1 2

p 3 6

1 2

A i = A(t +c i h),

(4.17) p 3 6 . To obtain a canonical method, we follow the

where c 1 = − and c 2 = + construction from Corollary 4.4.1. This gives us the auxiliary update map u 7→ U as given by Example 4.4.2. We use initial values q 0 = (1, 2, 3, 4), p 0 = (4, 1, 2, 3), t 0 = 0, u 0 = −H (q 0 , p 0 , t 0 ), parameters α = 0.1 and ² = 0.3, and step-length h = 0.3. In order to evaluate the accuracy of the method, we generate a reference solution using the same method, but with a step-length h = 0.03. Since the method is fourth-order, the reference solution is a much more accurate solution than the other one. Denote the Hamiltonian evaluated in the reference solution and the approximate solution by Hex and Hk , respectively. Furthermore, let u k be the u-component of the approximate solution, and let K k = u k + Hk be the extended Hamiltonian evaluated in the approximate solution. In Figure 4.1a, we display Hk − Hex and −u k − Hex as functions of time t k . We see that both Hk and −u k stay close to Hex over long time. If we had plotted K k , we would have observed that this is well preserved over long time, as expected of canonical methods. In Figure 4.1b, we plot Hk and −u k together with Hex to get a better understanding of our simulation. In order to get a visible separation of the three curves, we had to switch to a step-length h = 0.6 together with a lower-order method, namely the 1st order Magnus method based on the

133


5

·10−5

0

−5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 ·104

(a) Hk − Hex (blue) and −u k − Hex (red)

34

32

30

28

26

230

235

240

245

250

255

260

265

270

275

280

285

290

295

300

(b) Close-up of Hk (blue), −u k (red), and Hex (brown)

Figure 4.1: Long-time behaviour. Time t k is plotted along the x-axis.

134

4.5 Numerical experiments ¡ ¢ update y 7→ exp h A(t ) y (i.e. the Lie–Euler method). We observe that the two approximates to the energy, Hk and −u k , oscillate near the reference solution.

4.5.2 Symmetric methods and canonical transformations In the previous subsection, we applied a symmetric and canonical method to a non-autonomous Hamiltonian problem, and observed good long-time behaviour. In this subsection, we test the four different combinations of symmetric and canonical methods on the same problem. Three of the methods, namely methods (a), (b), and (d) below are 2nd order Runge–Kutta methods applied to the Hamiltonian equations (4.4) in extended phase space. Method (c) is different and is explained in detail below. For each Runge–Kutta method, we indicate whether the method is symmetric and/or canonical. We choose (b) and (d) so that neither of them are conjugate to symplectic in order to rule out this potential source of unwanted good long-time behaviour [6]. We choose the following methods: (a) The midpoint method (both symmetric and canonical) (b) Kahan’s method (only symmetric) (c) A projection-based method (only canonical) (d) Lobatto IIIC (neither symmetric nor canonical) The midpoint method is canonical since it can be viewed as an SPRK method (see Example 4.4.5). The projection-based method is based on the idea of projecting the truncated Taylor series of the exact solution onto the symplectic Lie algebra so that we obtain an integrator that can be extended to a canonical transformation. This demonstrates that we can get canonical methods (and good long-time behaviour) even if we use projections. Using projection as a device for energy preservation is known to give unsatisfactory results in many cases (see [12, pp. 112–113]). ¡ ¢ The exact solution of y˙ = A(t )y is y(t + h) = I2n + h A(t ) + O(h 2 ) y(t ). Our goal is to use the Taylor series to obtain a consistent method of the format ¡ ¢ Y = M (t )y = exp hX (t ) y with X (t ) ∈ sp(2n), as discussed in Example 4.4.2. Let Π: gl(2n) → sp(2n) be the linear projection Π(X ) = 21 (X + Jn X T Jn ).

135

4 Geometric integration of non-autonomous Hamiltonian problems The projection-based method is then defined as ¡ ¢ Y = M (t )y = exp ◦Π ◦ log I2n + h A(t ) y,

(4.18)

together with the auxiliary update equation ¢ h T ¡ Y Jn dexphX (t ) X 0 (t ) Y , 2 ¡ ¢ with X (t ) = h1 Π ◦ log I2n + h A(t ) . By Taylor expansion of the logarithm, we see that X (t )|h=0 = A(t ). Thus, the method is consistent, i.e. of order one. We use initial values q 0 = (1, 2, 3, 4), p 0 = (4, 1, 2, 3), t 0 = 0, u 0 = −H (q 0 , p 0 , t 0 ), parameters α = 0.123 and ² = 0.6, and step-length h = 0.3. The time evolution of the Hamiltonian evaluated in the numerical solution, as well as minus the auxiliary variable u k are shown in Figure 4.2. We observe that all the methods perform well, except for Lobatto IIIC.

U =u+

4.5.3 Canonical, symmetric, and exponential methods In the final experiment, we compare methods with combinations of three different properties, namely canonical, symmetric, and exponential methods. We have met canonical and symmetric methods earlier, but not exponential methods. By exponential, we mean methods that solve the ODE exactly if they are applied to an autonomous, linear (AL) problem, i.e. if A(t ) is actually independent of t . Magnus methods are exponential, since all their commutators will disappear, leaving the exact solution in the AL case. The methods tested are: Lie–Gauss The fourth order Lie–Gauss method given by (4.17). Lie–midpoint The method given by

Y = exp(h A 1/2 )y,

A 1/2 = A(t + h/2).

Lie–Euler The method given by

Y = exp(h A 0 )y,

A 0 = A(t ).

Gauss–Legendre The fourth order Gauss–Legendre Runge–Kutta method.

136

4.5 Numerical experiments

32 31 30 29 28

0

100

200

300

400

500

600

700

800

900

1,000

800

900

1,000

800

900

1,000

800

900

1,000

(a) Midpoint method (both symmetric and canonical) 32 31 30 29 28

0

100

200

300

400

500

600

700

(b) Kahan’s method (only symmetric) 32 31 30 29 28

0

100

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300

400

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700

(c) Projection-based method (only canonical) 32 31 30 29 28

0

100

200

300

400

500

600

700

(d) Lobatto IIIC (neither symmetric nor canonical)

Figure 4.2: Symmetric and canonical methods. H (q k , p k , t k ) is plotted in blue and −u k in red, both with t k along the x-axis.

137

4 Geometric integration of non-autonomous Hamiltonian problems Midpoint The standard midpoint Runge–Kutta method. Kahan Kahan’s method (viewed as a Runge–Kutta method [6]). Projection The projection-based method given by (4.18). Radau IIA The Radau IIA Runge–Kutta method of order three [14, Table IV.5.5].

This method was chosen as an example of a method which is neither exponential, symmetric, nor canonical. Symplectic Euler The symplectic partitioned Runge–Kutta method [12, The-

orem VI.3.3] Q = q + hH p (P, q, t ),

P = p − hH q (P, q, t ).

ExpNonCan The method given by

¡ ¢ Y = exp h A 0 (I2n + h A 0 [A 0 , A 1 ]) y,

A i = A(t + i h).

This method has been constructed to be exponential, but not canonical, since we apply the exponential map to something which lies outside of sp(2n). The commutator ensures that we get the exact solution if we apply the method to an AL problem. ExpSymNonCan The method given by

¡ ¢ Y = exp h A 1/2 (I2n + h A 1/2 [A 0 , A 1 ]) y,

A i = A(t + i h).

This method is similar to ExpNonCan, but has been modified to ensure that it is symmetric. All the methods advance time using T = t + h. We ignore the u-component of the canonical methods, since in this experiment we are measuring the energy error |Hk − Hex |, which is independent of u. In addition to these methods, we also include some compositions of symmetric methods using the triple jump of order 4 [12, Example II.4.2]. See Table 4.1 for a summary of the properties and order of each of the methods. We test these methods on the same Hamiltonian as before, with initial values q 0 = (1, 2, 3, 4), p 0 = (4, 1, 2, 3), t 0 = 0, u 0 = −H (q 0 , p 0 , t 0 ), parameters α = 0.123 and ² = 0.1, and step-length h = 0.3. The reference solution, giving Hex , is

138

4.5 Numerical experiments 102

Radau IIA Symplectic Euler ExpNonCan ExpSymNonCan Midpoint Midpoint w/triple pt. Gauss–Legendre Lie–Euler Lie–midpoint Lie–midpoint w/triple pt. Lie–Gauss

101

|Hk − Hex |

100 10−1 10−2 10−3 10−4 10−5

0

1

2

3

tk

4

5 ·104

Figure 4.3: Smoothed energy errors. calculated using the fourth order Lie–Gauss method with h = 0.02. The time interval of the experiment is [0, 50 000]. The energy error |Hk − Hex | oscillates rapidly around zero, and therefore, plotting this quantity is not helpful. Instead, we divide the time interval into subintervals containing 500 samples each, and plot the maximum energy error within each subinterval. This procedure smooths out the oscillations, but retains the relevant information about the size of the energy error. The smoothed energy error is presented in Figure 4.3. Many of the schemes yield very similar results, and therefore we only plot some of them. In particular, the midpoint and Kahan methods give similar results for this example. In Table 4.1, we summarize the results of this experiment. The top part of the table consists of the methods with the smallest energy errors. They all have maximum energy errors of less than ≈ 0.15, which is the maximum possible error we can get if the rapidly oscillating component of Hk (see Figure 4.1b) is completely out of phase with the exact solution. We will call this maximum phase error. The middle part of the table consists of the methods which follow the slowly oscillating component of Hk fairly well, but which attain the maximum possible phase error of ≈ 0.15. The last part of the table contains the worst methods, with errors larger than the maximum possible phase error. From the table, we see that for this problem, the best methods are the ones

139


Table 4.1: Order, properties, and maximum energy errors for a selection of different methods. Properties: C (canonical), S (symmetric), E (exponential). Method

Order

Properties

Max energy error

Lie–Gauss Lie–midpoint with triple jump Lie–midpoint Lie–Euler Gauss–Legendre

4 4 2 1 4

CSE CSE CSE C E CS

3.20 · 10−5 1.50 · 10−4 4.56 · 10−3 2.50 · 10−2 7.98 · 10−2

Midpoint with triple jump Midpoint ExpSymNonCan Kahan with triple jump Projection Kahan

4 2 1 4 1 2

CS CS SE S C S

ExpNonCan Symplectic Euler Radau IIA

1 1 3

E C

1.49 · 10−1 1.49 · 10−1 1.49 · 10−1 1.50 · 10−1 1.53 · 10−1 1.68 · 10−1 2.61 · 100 6.44 · 100 3.15 · 101

which are canonical, symmetric, and exponential. The methods that perform the worst only have one or none of these properties. Even though the Gauss– Legendre method is placed in the top tier of the table, we observe in Figure 4.3 that the energy error keeps growing for the whole time interval. The other methods in this part of the table have energy errors that remain at the same level throughout the interval.

4.6 Conclusion In this paper we have taken a new look at numerical integrators for Hamiltonian problems where the energy function depends explicitly on time. Using the framework of canonical transformations defined by [3], we have characterized integrators which are canonical according to this definition. In particular we have studied methods for linear non-autonomous equations, a problem

140

4.6 Conclusion class which has attracted considerable interest from the numerical analysis community in recent decades. We have not obtained analytical results which rigorously support the hypothesis that canonical methods can be expected to have good long time behaviour. However, numerical tests for a toy problem, a smooth oscillator, seem to corroborate such an assumption. It is unclear whether the by now classical approach of backward error analysis will be a useful tool in studying error growth of canonical methods since the analysis should allow for highly oscillatory problems and linear PDEs. We believe however, that the notion of canonical transformations used in this paper may be a viable route to gain a better insight into the excellent properties of exponential integrators applied to linear non-autonomous Hamiltonian problems.

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