Periodic Orbits of Hamiltonian Systems

Periodic Orbits of Hamiltonian Systems Luca Sbano Mathematics Institute, University of Warwick December 2007

Contents 1 Introduction 1.1 Hamiltonian equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3

2 Periodic solutions 2.1 Stability of periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Continuation and bifurcation of equilibrium solutions . . . . . . . . . . . . . . . . . . 2.2.1 Normal form analysis near equilibrium points . . . . . . . . . . . . . . . . . .

5 5 6 8

3 Poincar´ e map and Floquet operator 3.1 Continuation of periodic orbits in Hamiltonian systems . . . . . . . . . . . . . . . . .

8 11

4 Hamiltonian systems with symmetries 16 4.1 Symmetry and reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 Relative equilibria, relative periodic orbits and their continuation . . . . . . . . . . . 19 5 The 5.1 5.2 5.3 5.4

Variational principles and periodic orbits Lagrangian view point . . . . . . . . . . . . . . . Hamiltonian view point . . . . . . . . . . . . . . Fixed energy problem, the Hill’s region . . . . . . Continuation of periodic orbits as critical points

6 Further directions

1

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

21 22 26 27 34 37

Introduction

Periodic motions and behaviours in Nature have always been of interest to mankind. All phenomena that have some cyclic nature captured our attention because they are a sign and a clue for regularity. Therefore they are indications toward the possibility of understanding the laws of Nature. Since II sec. BC, Greek philosophers and astronomers look into the possibility of describing the motions of celestial bodies through the theory of epicycles, which are combinations of circular periodic motions. Notably from a modern point of view this theory can be interpreted as a clever geometrical application of Fourier expansions of the observed motions (see [23]).

1

The development of mechanics, the discovery that laws of Nature can be written in the language of Calculus and that laws of motion can be described in terms of differential equations, opened up the study of periodic solutions of equations of motion. In particular, since Newton and then Poincar´e [47], the main interest has been the understanding of the planetary motions and the solution of the so-called N -body problem, the reader should refer to [54] in this Encyclopaedia. The interest in periodic motions has not been restricted to celestial mechanics but became a sort of paradigm in all the areas where mechanics was successfully applied. In this article we shall illustrate some general aspects of the results regarding the theory of periodic orbits in Hamiltonian systems. Such systems are the modern formulation of those mechanical systems which are described by second order differential equations and have an energy function. As an example the reader could think to Newton’s equations for a point-mass in potential field. The equations read mx ¨(t) = −∇V (x(t)) with x(t) ∈ R3 for every t, m is the mass

(1)

V (x) is the potential and ∇ = (∂/∂x1 , ∂/∂x2 , ∂/∂x3 ). The energy function E=

m 2 kx(t)k ˙ + V (x(t)), 2

is conserved along the trajectories solving (1). It is important to say that most of the systems of interest in Physics can be naturally written in Hamiltonian form. The plan of this article is as follows: First we introduce the Hamiltonian formulations of the equations of motion for a classical mechanical system. It is well known that in many applications Hamiltonian systems derive from a Lagrangian formulation therefore this is also presented. Furthermore we introduce the Poisson formulation that is essentially the first generalisation to the Hamiltonian point of view. Then we turn to the study of the properties of periodic solutions (POs). In particular we focus on their ”local properties”, persistence and stability. In this analysis the main tool will be the implicit function theorem (IFT). In order to emphasise the utility of the IFT we present some of the proofs that contain typical calculations often scattered in the literature. Then we consider the problem of periodic orbit for Hamiltonian systems with symmetries, where we introduce the notion of relative equilibrium (RE) and relative periodic orbit (RPO). Inevitably we have also a short excursus about symmetry reduction that is the natural theoretical setting to study systems with symmetries. The second part of the article is devoted to the exposition of the study of periodic orbits by variational methods. It is usually attributed to P. Maupertuis (XVIII sec) the discovery that the equations of motion of mechanical systems can be derived by a variational principle, the so-called least action principle. According to this principle the motions are critical points of a functional called the action defined in a suitable space of paths. Variational methods turned out to be one of the most effective methods to prove the existence of periodic orbits, notably is the case of the N -body problem (see [2], [54]). The valuable feature of the variational methods is the possibility to study the existence problem by looking at the topology and geometry of the space of periodic paths without further restrictions. In the presentation of the results some elements of the proofs are illustrated in order to clarify the main ideas. In the final section there are some open problems and further directions of investigation, in particular it is presented a simple example of the so-called multi-symplectic structures that extended the possibility of applying finite-dimensional Hamiltonian approach to multi-periodic problems for a large class of partial differential equations. The study of periodic orbits has been for centuries one of the main centres of mathematical investi2

gations and developments and still presents challenges and the capacity of producing new interesting mathematical ideas to understand the complexity of Nature.

1.1

Hamiltonian equations

A Hamiltonian system is given by specifying a symplectic manifold (P, ω), where P is a differentiable manifold of even dimension, ω is a closed differential two-form and a function H : P → R called Hamiltonian. In the language of the differential forms the Hamiltonian vector XH field on P is written as iXH ω = dH (2) P n In the case P = R2n , the symplectic structure is ω0 = i=1 dxi ∧ dyi and the Hamilton vector field is . XH (z) = J ∇z H(z), (3) where z = (x, y) ∈ R2n and J is the symplectic matrix   0 idn J= . −idn 0

(4)

The Hamiltonian equations will then be dz(t) = XH (z(t)). dt

(5)

In the case P = R2n equation (5) reads z(t) ˙ = (x(t), ˙ y(t)) ˙ = (∇y H(x(t), y(t)), −∇x H(x(t), y(t))) . For a mechanical systems described by (1) the Hamiltonian function is H=

kyk2 + V (x) where (x, y) ∈ R6 2m

and its Hamiltonian equations read  y(t)   x(t) ˙ = m   y(t) ˙ = −∇V (x(t)). Note that the first equation corresponds to classical definition of momentum in mechanics (here denoted with y) and the Hamiltonian function H coincides with the energy E. For more details the reader could consult [1], [5] and also [28] in this Encyclopaedia. Lagrangian formulation In many applications in physics and in particular in mechanics Hamiltonian systems arise from the Lagrangian description. In such a setting a mechanical system is described by prescribing a differentiable manifold M (the configuration space) and a Lagrangian function L defined on the tangent bundle T M. Let L : T M → R be a Lagrangian on a manifold M of dimension n. If L is 3

hyper-regular (i.e. rank(Dv2q L(vq , q)) = n), then Hamiltonian function is naturally constructed on the cotangent bundle T ∗ M by using the Legendre transform (see [1],[5]) as follows: pi =

∂L , ∂vqi

H(p, q) =

n X

(6) vqi (p) qi − L(vq (p, q), q).

i=1

The Hamiltonian system is then defined on the co-tangent bundle of that is P = T ∗ M, which PM n is endowed with the canonical symplectic form ω = dθ where θ = i=1 pi dqi . In the Lagrangian description the equations of motion are d ∂L(vq (t), q(t)) ∂L(vq (t), q(t)) − = 0 i = 1, ..., n dt ∂vqi ∂qi

(7)

where vqi (t) = q˙i (t) i = 1, ..., n. Note that equations (7) contain second order time-derivatives. For more details see [1],[5]. Poisson formulation Let F(P) be the space of differentiable functions on (P, ω). On F(P) can be introduced a product {., .} (see [1],[5],[16]). The Poisson brackets ω(Xf , Xg ) = {f, g} for f, g ∈ F(P).

(8)

In terms of the Poisson brackets the Hamiltonian equations can be written as a derivation acting on F(P) XH (f ) = {f, H} for f ∈ F(P). (9) The Poisson brackets satisfy the following properties. For all f, g ∈ F(P) {f, g} is bilinear with respect f and g, {f, g} = −{g, f } (10)

{f g, h} = f {g, h} + g{f, h} {{f, g}, h} + {{h, f }, g} + {{g, h}, f } = 0 Jacobi identity Easy consequence of (9) and (10) is Proposition 1.1. The Hamiltonian function H is a constant of motion.

A manifold P endowed with the brackets {., .} is called a Poisson manifold. Any symplectic manifold is Poisson (see [1], [27]) but the contrary is false. In fact Poisson brackets on a symplectic manifold are always non degenerate, namely the condition {k, f } = 0 for all f ∈ F(P) implies that k is identically zero. In a general Poisson manifold there might exists non-vanishing k, which are then called Casimir functions. This is related to the fact that symplectic manifold are always even dimensional, whereas Poisson manifolds can be odd dimensional. We can look at Poisson manifolds as a useful generalisation of Hamiltonian systems, in fact in order to define Poisson brackets it is sufficient to have the ring of functions F(P). For a general a complete description of the Poisson structure the reader could see [1], [5], [27] and [16]. 4

2

Periodic solutions

Given a Hamiltonian vector field XH (z) on (P, ω) one can consider the following Cauchy problem   dz(t) = X(z(t)) dt (11)  z(0) = z0 . Equation (11) has to be meant defined on a local chart in P. Definition 2.1. We call flow or itegral flow the map t 7→ φ(t, z0 ) where z(t) = φ(t, z0 ) solves problem (11). Some simple consequences follow (see [37]): Remark 2.1. If XH is complete, then the flow is defined for all t ∈ R. Remark 2.2. If the Hamiltonian vector field is autonomous that is not explicitly dependent on time, then φ(., z0 ) satisfies the composition property φ(t, φ(s, z0 )) = φ(t + s, z0 ) and φ(., .) is called Hamiltonian flow. Let us now introduce the main object of this exposition: Definition 2.2. A flow φ : R → P, z(t) = φ(t, z0 ) is said to be a T -periodic solution of (11) if there exists T > 0 such that φ(t + T, z0 ) = φ(t, z0 ) for all t. Remark 2.3. Note that if φ(., z0 ) is a T -periodic solution, then φ(., z0 ) is n T -periodic for any n ∈ N. In fact from the definition φ(T, z0 ) = z0 and using remark 2.2 one can iterate φ(t + n T, z0 ) = φ(t + (n − 1)T, φ(T, z0 )) = φ(t + (n − 1)T, z0 ) and find φ(t + nT, z0 ) = φ(t, z0 ). Definition 2.3. T is called minimal period of φ(t, z0 ) if T = minτ ∈R+ {τ : φ(t, z0 ) for all t}.

φ(t + τ, z0 ) =

Definition 2.4. A point z ∗ ∈ P such that J ∇z H(z ∗ ) = 0 is called equilibrium solution. Obviously any equilibrium solution can be seen as a periodic solution with T = 0. One can easily show Lemma 2.1. φ(t, z0 ) is periodic of period T if and only if φ(T, z0 ) = z0 .

2.1

Stability of periodic orbits

Given a periodic orbit the first natural question is to study its stability. There are three possible stability criteria (see [1], [37]): Definition 2.5 (Liapunov stable). A periodic orbit φ(., z0 ) is Liapunov stable if for all  > there is δ(z0 , ) such that kz0 − z 0 k ≤ δ(z0 , ) implies that kφ(t, z0 ) − φ(t, z 0 )k ≤  for all t ≥ 0. The previous definition is natural for non-Hamiltonian system but in Hamiltonian context is very strong. It is useful though to compare Liapunov stability with the following weaker notions. 5

Definition 2.6 (Spectrally Stable). A periodic orbit φ(t, z0 ) is spectrally stable if the eigenvalues of DXH (z0 ) lie all on the unit circle. Definition 2.7 (Linearly Stable). A periodic orbit φ(t, z0 ) is linearly stable if it is spectrally stable and DXH (z0 ) can be diagonalised. One can show that spectral stability is implied by either linear stability and Liapunov stability. A natural notion of stability can be introduced by using the Poincar´e map and will be presented in section 3. The following results describe the structure of linear Hamiltonian systems, namely ˙ = systems whose Hamiltonian function is H(z) = 21 hz, A zi and the equations of motion read z(t) J A z(t). Theorem 2.1 ([37]). Let A be time independent. The characteristic polynomial of J A is even and if λ is an eigen-value then so are −λ, λ, −λ. Consequence of the previous result is that linear stability is equivalent to spectral stability for linear autonomous Hamiltonian systems. Linear systems can depend on parameters, it is therefore interesting to have a notion of stability with respect parametric changes: Definition 2.8. A linear stable Hamiltonian system H(z) = 12 hz, A zi is said parametric stable if for every symplectic matrix B such that kA − Bk <  the system z˙ = J B z is linearly stable. We finally give a characterisation of linear Hamiltonian systems which are parametric stable Theorem 2.2 ([37]). If the Hamiltonian H is positive (or negative) definite or all the eigenvalues are simple, then A is parametrically stable.

2.2

Continuation and bifurcation of equilibrium solutions

Let H(z, ) be a Hamiltonian function depending on a parameter  ≥ 0. Let Z0 = {z ∗ : J∇H(z ∗ , 0) = 0} be the set of equilibrium positions. An interesting problem is to study how Z0 is modified when  > 0. The local properties of Z0 depends on the spectrum of J D2 H(z ∗ , 0). If z ∗ ∈ Z0 is such that JD2 H(z ∗ , 0) has non-zero eigenvalues, then by the IFT there exits a curve z ∗ () of equilibrium points for  positive and sufficiently small. If J D2 H(z ∗ , 0) has at least one zero eigenvalue then bifurcation can occur, but the interesting point is that bifurcations are possible also when J D2 H(z ∗ , 0) is not degenerate. In fact in Hamiltonian systems generically the spectrum of the linearisation of the vector-field contains couples of complex conjugated eigenvalues (see [5]). In such a case there is a theorem due to Liapunov which shows the existence of periodic orbits - so called nonlinear normal modes in a neighbourhood of z ∗ . Let us now present Liapunov’s theorem. This result describes how a non-degenerate equilibrium can be continued into a periodic orbit. The types of orbits are called non-linear normal modes. Theorem 2.3 (Lyapunov’s Centre Theorem [1], [18]). If the Hamiltonian system has a nondegenerate equilibrium at which the linearised vector-field has eigenvalues ±iω, λ3 , ..., λn with λk /ω ∈ / Z then there exists a one-parameter family of periodic orbits emanating from z ∗ . The period tends to 2π/ω when the orbit radius tends to zero and the nontrivial multipliers tend to exp(2πλk /ω) with k = 3...n 6

Proof. Without loss of generality the non-degenerate equilibrium can be fixed in the origin. In a neighbourhood of the origin the Hamiltonian vector field can be written as follows z(t) ˙ = JA z(t) + r(z) where kr(z)k = o(kzk) that is kr(z)k is infinitesimal with respect to kzk. The spectrum of JA is Spec(JA) = {±iω, λ3 , ..., λn }. Let y =  z with  ∈ [0, 1] y(t) ˙ = JA y(t) + r(y, ) with r(y, ) infinitesimal for  → 0 uniformly for kyk bounded. For  = 0 the system becomes y(t) ˙ = JA y(t)

(12)

and admits a periodic solution with period T = 2π/ω y0 (t) = exp(t JA) y0 , JA y0 = v0 y0 . Equation (12) is linear, thereby coincides with its linearisation. Floquet multipliers are therefore (1, 1, exp(2π λ3 /ω), ..., exp(2π λn /ω)) which exp(2π λk /ω) 6= 1 by hypothesis. Now we look for a solution of   d − JA y(t) = r(y(t), ). dt

(13)

d On the space C 1 ([0, T ], R2n ) with periodic boundary condition the operator dt − JA has a 2dimesional kernel. Therefore one could solve equation (13) by looking for a solution in the form

y(t, ) = exp(t JA)y0 + u(t) d where u(t) ∈ rank( dt − JA). By IFT one can show that ku(t)k = o(1) and therefore the solution can be continued for small  as

y(t, ) = exp(t JA)(1 + o(1)) y0 that is z(t, ) =  exp(t JA)(1 + o(1)) y0 with lim kz(t, )k = 0. →0

d Remark 2.4. The analysis of the operator dt − JA used to prove the previous result is known as Liapunov-Schmidt reduction. In section 5.4 we shall give an application of it in the study of critical points.

7

2.2.1

Normal form analysis near equilibrium points

In [57] Liapunov’s theorem has been generalised to cases where the condition λ/ω ∈ Z might hold. This corresponds to the so-called resonance condition Definition 2.9 (Resonance). The set of eigenvalues {ωl }kl=1 of the linearisation DXH (z0 ) are said to be resonant if 0 there exists δ(, z0 ) such that kz − z0 k < δ implies kΠn (z) − z0 k <  for all n > 0. This stability criteria is difficult to verify. A weaker criteria is obtained by considering the linearisation of Π(z) at z = z0 : Definition 3.4 ([1, 37]). A periodic orbit φ(., z0 ) is spectrally stable if the associated Poincar´e map Π, which is restricted to the manifold defined by the integrals of motion, has a linearisation Dz Π(z0 ) with spectrum on the unit circle. By a local change of coordinates one can show that the eigenvectors of Dz Π(z0 ) are equal to the eigenvectors of V (T0 ) different from XH (z0 ) = J∇H(z0 ) [37]. The Poincar´e map removes the degeneracy of the monodromy matrix V (T ). To illustrate this point let us consider x ∈ Rn and an autonomous system x(t) ˙ = f (x(t)) (16) with f : Rn 7→ Rn differentiable. Let φ(t, x0 ) be a T -periodic solution of (16) emanating from x0 . To study the trajectories near x0 one can construct a local diffeomorphism h : Rn 7→ Rn , y = h(x) such that y0 = h(x0 ) with (16) in the form y(t) ˙ = Dh(h−1 (y(t))) f (h−1 (y(t))) = f˜(y(t)) with f˜(y0 ) = (1, 0, ..., 0).

(17)

In y coordinates the periodic orbit φ(t, x0 ) reads ψ(t, y0 ) = h(φ(t, h−1 (y0 ))) and we can define a map σ as σ(t, y) = hf˜(y0 ), ψ(t, y0 ) − yi. (18) The form of f˜(y0 ) implies σ(y, t) = ψ1 (t, y) − y0,1 = h1 (φ(t, h−1 (y0 ))) − y0,1 . An easy calculation shows that n X ∂σ ∂h1 = f (x) =1 l ∂t (0,y0 ) ∂xl l=1

(0,y0 )

which implies the existence of a return time τ (y) satisfying σ(τ (y), y) = 0 for y in a sufficiently small neighbourhood of y0 . Now we define the map Π(y) = ψ(τ (y), y). By applying the IFT we can compute X ∂h1 ∂φl ∂xm ∂τ = δ1j − (19) ∂yj ∂xl ∂xm ∂yj l,m

10

and also the Jacobian of Π: X ∂yi ∂φl ∂xm ∂Πi (y) ∂τ = f˜i (y) + . ∂yj ∂yj ∂xl ∂xm ∂yj

(20)

l,m

at y = y0 has the first column equal to f˜(y0 ) = Using (19) it is easy to see that the matrix ∂Π∂yi (y) j (1, 0, ..., 0). The diffeormophism h allowed to ”isolate” the direction of the vector-field f at x0 . This implies that the map ΠSf˜(z ) , the restriction of Π on the Poincar´e section Sf˜(z0 ) = {y : 0 hf˜(y0 ), y − y0 i = 0} ' Rn−1 maps Sf˜(z0 ) into Sf˜(z0 ) and its linearisation has no eigenvectors in the direction of f˜(y0 ). The map Πf˜(z0 ) describes the stability of the periodic orbit φ(t, x0 ). A similar construction can be carried out when f is a Hamiltonian vector field. In that case it is necessary to take into account the existence of integrals of motion and construct the Poincar´e map on the manifold where the integral of motions are fixed. A more complete study of the Poincar´e map will be presented in the next section. Let us now present some properties of the Floquet operator. Let us consider a Hamiltonian system with imaginary multipliers. In the linear time-independent case the monodromy matrix is given by V (T0 ) = exp(T0 JA) where T0 = 2π/|λ| with λ imaginary eigenvalue of JA. Now let H = 21 hz, Azi + h(z) be an Hamiltonian with h(z) = o(kzk2 ) near z = 0. The equation for the monodromy matrix is (15). Let φ0 (t) be another periodic orbit, equation (15) can be written as follows dVψ (s) Tφ = DXH (ψ0 (s)) Vψ (s) ds T0 where t = T0 s/Tφ and ψ0 (s) = φ0 (T0 s/Tφ ) then one can show Proposition 3.3 ([39]). The Floquet operator Vψ (t) is C ∞ in ψ and Tφ . Corollary 3.1 ([39]). If φ0 (t) is sufficiently close to z = 0 then Vφ (Tφ ) is arbitrarily close to exp(T0 JA). Corollary 3.1 is interesting because allows us to use the linearised dynamics. Now the analysis of the nonlinear stability can be carried out by using Krein’s theory. For this typical references are [59] and [21]. Here we we recall Theorem 3.1 (Krein). Let V (T ) be a spectrally stable monodromy matrix. Let Qλ be a quadratic form Qλ (v) = hV (T ) v, J vi where v belongs to the λ eigenspace of V (T ). Then V (T ) is in an open set of spectrally stable matrices if and only if the quadratic Qλ has definite sign. By combining corollary 3.1 and theorem 3.1 we could show that a solution φ0 (t) close enough to z = 0 is spectrally stable.

3.1

Continuation of periodic orbits in Hamiltonian systems

In general the problem of proving the existence and then constructing periodic orbits is difficult. Sometimes, for certain specific values of the parameters characterising the system it is possible to find particular solutions. Typically these are the equilibria and relative equilibria. In these cases 11

the continuation method can be a useful approach. The idea is to look at how a given periodic orbit changes according to small modification of the parameters. The method reduces the research of a periodic orbits to the problem of finding fixed points of the Poincar´e map that can be continued as function of the parameters, see Figure 1. Definition 3.5 (Continuation of an orbit). Given a dynamical system and φ0 (t) one of its orbit. We say that φ0 (t) can be continued if there exists a family of orbits φ(t, α) smoothly dependent on parameters α’s and such that φ(t, 0) = φ0 (t). Let us consider a Hamiltonian vector field XH (z, α) where z ∈ P and α ∈ Rk are k parameters. We now write the equations of motion in a form where the period T appears explicitly. After t → t/T the equations read z(t) ˙ = T J ∇H(z(t), α) = T XH (z(t), α)

(21)

with t ∈ [0, 1]. Definition 3.6. Let φ(t, z, T, α) be a solution of (21) the map . R(z, T, α) = φ(1, z, T, α) − z

(22)

is called return map. The orbit φ(t, z0 , α0 ) is T0 -periodic if R(z0 , T0 , α0 ) = 0. Now we are interested to see what is the fate of the orbit when z0 , T0 and α0 are varied, therefore it is useful to determine the local behaviour of the map R. This is collected in the following lemma Proposition 3.4 ([43]). Let φ0 (t, z0 ) be a periodic orbit with period T0 and α = 0, the following relations hold • Dz R(z0 , T0 , 0) = V (1) − id, • XH (z0 , 0) ∈ ker(V (1) − id), • DT R(z0 , T0 , 0) = XH (z0 , 0), R1 • Dα R(z0 , T0 , 0) = T0 V (T ) 0 V −1 (s)Dα XH (φ0 (s, z0 ))ds, together with the differential map DR(z0 , T0 , 0)(ξ, T, α) = (V (1) − id) · ξ + T XH (z0 , 0) + Dα R(z0 , T0 , 0) · α, where (ξ, T, α) ∈ R2n+k+1 . Initially let us consider a non-Hamiltonian dynamical system in Rn x(t) ˙ = f (x(t), ),  ∈ R.

(23)

We now illustrate how the IFT is used to construct a new solution from a given one, i.e. by continuation. We present a detailed proof for reader’s convenience.

12

Proposition 3.5 ([43], [3]). Let Γ0 = {φT0 (t, x0 ), t ≥ 0} be a periodic orbit of period 1 of x(t) ˙ = T0 f (x(t), ) for  = 0. If 0 is an eigenvalue of Dx R(x0 , T0 , 0) with multiplicity 1 then orbit Γ0 can be continued. Proof. Let us consider the map G : Rn × R+ × [0, 1] → Rn+1 defined by G(x, T, ) = (R(x, T, ), hf (x0 , 0), x − x0 i),

(24)

note that hf (x0 ), x − x0 i = 0 is the equation of the Poincar´e section Sf (x0 ) . Since Γ0 is a periodic orbit with period T0 emanating from x0 , then G(x0 , T0 , 0) = 0. As already stated the strategy is to employ the IFT to derive the continuation of Γ0 . Thus it is necessary to compute Dx,T G at (x0 , T0 , 0):   Dx R(x0 , T0 , 0) f (x0 , 0) Dx,T G(x0 , T0 , 0) = . f (x0 , 0) 0 In order to show that Dx,T G(x0 , T0 , 0) is invertible one notes that the equation Dx,T G(x0 , T0 , 0)(X, a) = (0, 0) is equivalent to the system 

Dx R(x0 , T0 , 0)(X) + f (x0 , 0) a = 0 hf (x0 , 0), Xi = 0.

(25)

Assume X 6= 0. The multiplicity of the 0 eigenvalue of Dx R(x0 , T0 , 0) is 1 and Dx R(x0 , T0 , 0) f (x0 , 0) = 0. Now from (25) we derive (Dx R(x0 , T0 , 0))2 X = 0. Therefore Dx R(x0 , T0 , 0)(X) + f (x0 , 0) a = 0 would be satisfied only for a = 0 and X = cf (x0 , 0) with c ∈ R. But this would imply c hf (x0 , 0), f (x0 , 0)i = 0 which is possible only for c = 0. The kernel of Dx,T G is empty at (x0 , T0 , 0) and therefore the map is invertible and the application of the IFT provides the existence of T () and x() for  in a neighbourhood of zero such that x(0) = x0 , T (0) = T0 and G(x(), T (), ) = 0. This corresponds to the existence of a new periodic orbit close to Γ0 . Upon the assumptions of sufficient regularity for the vector field, the IFT provides also the possibility of approximating x() and T () by constructing a Taylor expansion in . Let x() = x0 + ξ() and T () = T0 + τ () then equation (25) can be evaluated along the continuation curve defined by (ξ(), τ (), )  Z 1 df (φT0 (s, x0 ), 0)   Dx R(x0 , T0 , 0)(ξ 0 ()) + f (x0 , 0) τ 0 () + T0 V (1) ds = 0 V (s) d (26) 0   0 hf (x0 , 0), ξ ()i = 0. This set of equations allow us to compute an approximation for ξ() and τ (). For a general dynamical system in Rn (23) the possibility of constructing a Poincar´e section is related to a notion of non-degenerate periodic orbit: Definition 3.7. A periodic orbit φ(t, z0 ) is called non-degenerate if rank(V (1) − id) ⊕ R f (z0 ) = Rn . 13

Now for Hamiltonian systems the time evolution is contained in the level set determined by the the Hamiltonian function (the energy) and all integrals of motions {Fi }ki=1 XH (Fi ) = {H, Fi } = 0, i = 1, .., k. This requires a modification to the notion of non-degeneracy. In fact note that the set of integrals of motions (F1 , .., Fk ) define a map F : P 7→ Rk and span W = {∇Fi , i = 1...k}. Now we have W ⊥ = ker(Dz F (z)). One can show rank(V (1) − id) ⊂ ker(Dz F (z)) and XH (F ) = 0. The last condition is equivalent to XH (z) ∈ ker(Dz F (z)). Suppose that dim W ⊥ = 2n − k, if dim(ker(V (1) − id)) = k then dim(rank(V (1)−id) = 2n−k and hence rank(V (1)−id) = W ⊥ . Therefore in the Hamiltonian case the natural notion of non-degeneracy has to involve the presence of integrals of motion. This is obtained by taking the following definition: Definition 3.8 ([43]). A periodic orbit φ(t, z0 ) is called normal if rank(V (1) − id) ⊕ R XH (z0 ) = W ⊥ ,

(27)

where all the gradients {∇H, ∇F1 , ..., ∇Fk } are linearly independent. Lemma 3.1 ([43]). If the algebraic multiplicity of the zero eigenvalue of V (1) − id is ma = k + 1 then the condition (27) is satisfied. Finally we present how to construct the continuation of non-trivial periodic orbits in Hamiltonian systems: Theorem 3.2 ([43]). Let Γ0 = {φT0 (t, z0 ) : t ∈ [0, 1)} be an orbit of the Hamiltonian system H0 on the energy level e0 . Let the algebraic multiplicity of the eigenvalue 0 of V (1) − id be 2. Let H = H0 +  H1 be a smooth perturbation of H0 , then there exists a 2-dimensional family of normal periodic orbits φ(t, z0 ; , e) where φ(t, z0 , e0 , 0) = φ(t, z0 ). Proof. Let us consider the map G : R2n × R+ × [0, 1] → R2n+1 defined by G(x, T, e, ) = (R(z, T, ), hXH (z0 , 0), z − z0 i, H (z) − e).

(28)

Since Γ0 is periodic orbit with period T0 emanating from z0 then G(z0 , T0 , e0 , 0) = 0. The strategy is always to employ the IFT to derive the continuation of Γ0 . Thus it is necessary to compute Dz,T G at (z0 , T0 , e0 , 0). A straightforward calculation gives   Dz R(z0 , T0 , 0) XH (z0 , 0) . 0 Dz,T G(z0 , T0 , 0) =  XH (z0 , 0) ∇H0 (z0 ) 0 In order to show that Dz,T G(z0 , T0 , h0 , 0) is invertible one note that the equation Dz,T G(z0 , T0 , h0 , 0)(X, a, 0) = (0, 0, 0) is equivalent to the system   Dz R(z0 , T0 , e0 , 0)(X) + XH (z0 , 0) a = 0 hXH (z0 , 0), Xi = 0  h∇H0 (z0 ), Xi = 0 14

(29)

Since XH (z0 ) ∈ ker Dz R(z0 , T0 , e0 , 0) then from (29) we derive (Dz R(z0 , T0 , e0 , 0))2 (X) = 0 and therefore a = 0. Now lemma 3.1 implies that X ∈ W ⊥ , therefore the third equation in (29) is satisfied and the first implies X = b XH (z0 , 0) for some b ∈ R. But this would contradict b hXH (x0 , 0), XH (x0 , 0)i = 0 unless b = 0. This implies that that kernel of Dx,T G is empty at (x0 , T0 , e0 , 0) and therefore the map G is invertible. The application of the IFT provides the existence of T (e, ) and z(e, ) for  in a neighbourhood of 0 and e in a neighbourhood of e0 such that z(e0 , 0) = z0 , T (e0 , 0) = T0 . The functions T (e, ) and z(e, ) satisfy G(z(e, ), T (e, ), e, ) = 0. This corresponds to the existence of a new periodic orbit close to Γ. Upon the assumptions of sufficient regularity for the vector field, the IFT provides also the possibility of approximating z(e, ) and T (e, ) by following the same line of argument seen in proposition 3.5. Numerical studies The analysis of periodic orbits is very important for its concrete struction of numerical algorithms to construct periodic orbits and do not consider explicitly this problem but the reader is invited and [32]. Moreover there is some free and open software available http://indy.cs.concordia.ca/auto/) and [19, 20].

applications, hence the contheir continuation. Here we to consult for example [33] like for example AUTO (see

Example On P = R4 with coordinates z = (p, q) and the standard symplectic form, we consider the Hamiltonian 1 λ 1 1 kpk2 + V0 (q), V0 (q) = kpk2 − kqk2 + kqk4 (30) 2 2 2 4 with λ > 0. The reader could check that V0 (q) has the shape of a ”Mexican-hat”. The Hamilton’s equations of motion read  q˙ = p (31) p˙ = (λ − kqk2 )q. H=

Let e(t) be a unit vector with e(t + 2π/ν0 ) = e(t), there is a periodic orbit of the form q0 (t) = A0 e(t), p0 (t) = A0 e(t). ˙ This orbit is a relative equilibrium (see below for a formal definition). The initial conditions determine A0 ,√ ν0 and in particular the energy E which in turn can be used √ to parameterise A0 , ν0 : A20 = 32 (λ + λ2 + 3E), ν02 = 13 (−λ + λ2 + 3E), here E ≤ 0. Note that an easy calculation shows that the admissible values of the energy are E ≥ minq {V0 (q)} = −λ2 /4. Let φ(t, z0 ) = (p0 (t), q0 (t)) be the periodic solution, the Floquet operator is V (t) =

∂φ(t, z0 ) ∂z0

the linearised equations can be written as: dV (t) = M (t)V (t) dt

 where M (t) is a 4 × 4 matrix:

15

M (t) =

0 −D2 V0 (q0 (t))

id 0

 .

Now if we perform the transformation:    R(t) 0 cos ν0 t W (t) = S(t) V (t) = X(t) where R(t) = 0 R(t) sin ν0 t then we obtain:

− sin ν0 t cos ν0 t



dW (t) ˆ W (t) =M dt

where ˆ = S(t) ˙ S −1 (t) + S(t) M (t) S −1 (t). M ˆ is not time dependent, in fact M ˆ is the following matrix: M   Ω id ˆ = M −RT (t)D2 V0 (q0 (t)R(t) Ω where  Ω=

0 ν0

−ν0 0



T



2

and − R (t)D V0 (q0 (t)R(t) =

−3A20 + λ 0

0 −A20 + λ

 .

Now at t = T0

ˆ T0 ). V (T0 ) = S −1 (T0 ) W (T0 ) = exp(M o n p ˆ has spectrum Spec(M ˆ ) = 0, 0, ±i 6A2 − 4λ and the corresponding multipliers The matrix M 0 are:    n q o  p 4 2 ˆ exp(Spec(M )) = 1, 1, exp ±i T0 6A0 − 4λ = 1, 1, exp ±i T0 λ2 + 3E where T0 = 2π ν0 . The matrix V (T0 ) − id has a zero eigenvalue with multiplicity 2, then the periodic orbit φ0 (t, z0 ) can be continued by using theorem 3.2. Remark 3.1. Note that using the expression for ν0 one can check that for E = En where   λ2 n4 En = −1 + 2 with n ∈ Z. 3 (n − 12)2 the third and the fourth mupltipliers coalesce to 1 and therefore q0 (.) loses its linear stability in the directions transversal to itself.

4

Hamiltonian systems with symmetries

In many applications there are Hamiltonian systems with symmetries, whose most known example is the N -body problem (see [1], [5], [54]). A simpler example is equation (31) which is symmetric with respect the linear action of SO(2). For such systems the Hamiltonian function is invariant under the action of a Lie group G. We shall see that symmetries can greatly simplify the study of the dynamics. A Hamiltonian system with symmetry is a quadruple (P, ω, H, G) (see [1], [5], [55]) where 16

• (P, ω) is a symplectic manifold, • H : P → R is a Hamiltonian function, • G is a Lie group that acts smoothly on P according to G × P 3 (g, z) 7→ Φg (z) ∈ P. The map Φg preserves the Hamiltonian (G-invariance) that is H(Φg (z)) = H(z). • The action Φ is semi-symplectic namely Φ∗g ω = χ(g)ω with χ(g) = ±1, χ(.) is called temporal character. In the sequel we consider symplectic actions whereby χ(g) = 1 for all g ∈ G. One can show that for a system (P, ω, H, G) the vector field XH (z) is equivariant [1], that is XH (Φg (z)) = Dz Φg (z) XH (z).

(32)

To any element ξ in the Lie algebra g of G we can associate a infinitesimal generator ξP (z) of the action defined by dΦexp(ξ t) (z) . (33) ξP (z) = dt t=0 Remark 4.1. In many applications Hamiltonian systems are constructed from Lagrangian systems, therefore a symmetry appears usually as a group action on the configuration space M. The Hamiltonian symmtery is then the lifted action to the co-tangent bundle T ∗ M. For instance if SO(2) acts linearly on M = R2 by ΨR (q) = R q with R ∈ SO(2), then its lifted action on T ∗ M ' R4 is ΦR (q, p) = (R q, RT q) where RT is the transpose of R. For more details see [1],[37] and also [55].

4.1

Symmetry and reduction

Given a symplectic action of a group G there is a map J : P → g∗ defined by hdJ(v(z)), ξi = ω(v(z), ξP (z)) for all z ∈ P, v ∈ Tz P and ξ ∈ g.

(34)

The map J is called momentum map, this always exists locally, its global existence requires conditions on G and the topology of P (see [27]). The crucial property of the momentum map is to encoding the conserved quantities associated to the G action. This is the content of the famous Noether’s theorem that reads in modern formulation as follows Theorem 4.1 (E.Noether, [1], [5]). Let H be a G-invariant Hamiltonian on P with momentum map J. Then J is conserved on the trajectories of the Hamiltonian vector field XH . For instance in a system like (30) with a SO(2) symmetry action the momentum map is the classical angular momentum J(p, q) = p ∧ q. The momentum map J has also the property of being equivariant with respect to the co-adjoint action associated to G. In the case of G semisimple or compact the result reads Theorem 4.2 (Souriau, [27]). J(Φg (z)) = Ad∗g J(z).

17

The level sets of the momentum map are invariant with respect to the Hamiltonian flow, thus it is natural to restrict the motion to J−1 (µ). The construction of the dynamics reduced to the manifold defined by the conserved quantities is the origin of the theory of symmetry reduction. The first important result is the following Theorem 4.3 (Marsden-Weinstein). Let (P, ω, H, G) be an Hamiltonian system with a symplectic action of the Lie group G, then it is called reduced Hamiltonian system the triple (Pµ , ωµ , Hµ ) where π : P → Pµ = J−1 (µ)/Gµ , π ∗ ω = ωµ , Hµ = H ◦ π and Gµ = {g ∈ G : Ad∗g (µ) = µ}. The manifold Pµ is symplectic with symplectic form ωµ . The Hamiltonian flow on Pµ is induced by the Hamiltonian vector field XHµ defined by iXHµ ωµ = dHµ . Remark 4.2. Note that the reduced vector field XHµ is now defined on a manifold whose dimension is dim Pµ = dim P − dim G − dim Gµ . For a detailed discussion of this theorem the reader should refer to [1], [42], [16]. In many cases it can be more useful to perform the reduction through a dual approach using the description of the dynamics in terms of functions on P, namely through the Poisson approach. This permits to consider also cases where Pµ is not a smooth manifold but rather a stratified space. A very simple example is in Figure 2. The theory of singular reduction can be found in reduction R

2

r θ r=0

R2 /S 1

S 1 -orbit

Figure 2: A simple example: the reduction of S 1 group action on the plane is singular. There are two strata r = 0 and r ∈ (0, ∞). [4]. A recent exposition and generalisation is in [46]. The reader could in particular consider the exposition given in [16] and [17] where it is shown through several examples that invariant theory and algebraic methods can be applied to describe reduced dynamics on spaces with singularities. The singular reduction can be summarised in the following result Theorem 4.4 (Singular reduction,[46]). Let (P, {., .}) be a Poisson manifold and let Φ : G×P → P be a smooth proper action preserving the Poisson bracket. Then the following hold: 18

(i) The pair (F(P/G), {., .}P/G ) is a Poisson algebra, where the Poisson bracket {., .}P/G is characterised by {f, g}P/G = {f ◦ π, g ◦ π}; for any f, g ∈ F(P/G); π : P → P/G denotes the canonical smooth projection. (ii) Let h be a G-invariant function on M. The Hamiltonian flow φ(t, .) of Xh commutes with the G-action, so it induces a flow φP/G (., .) on P/G which is Poisson and is characterised by φ(t, z) = φP/G (t, π(z)). (iii) The flow φP/G (., .) is the unique Hamiltonian flow defined by the function [h] ∈ F(P/G) defined by [h](π(z)) = h(z). We will call HP/G = [h] the reduced Hamiltonian.

4.2

Relative equilibria, relative periodic orbits and their continuation

In a Hamiltonian system with symmetry there are orbits that are originated just by the symmetry invariance. These are the relative equilibria (RE) Definition 4.1 (Relative equilibria). A curve ze (t) in P is a relative equilibrium (RE) of (P, ω, H, G) if ze (t) = Φg(t) (we ) where g(t) a curve in G and we is such that XH (ze ) = z˙e (t). Note that if g(t) = g(t + T ) then ze (t) becomes a T -periodic orbit. For instance in the system (30) there is ze (t) = R(t) we = (A0 e(t), A0 e(t)) ˙ with R ∈ SO(2). There are orbits that can be considered closed ”up to a G-action”, i.e. they are closed on P/Gµ . These are the relative periodic orbits (RPO). Definition 4.2 (Relative periodic orbit). A a curve z(t) in P is a relative periodic orbit (RPO) of (P, ω, H, G) with period T if z(t + T ) = Φg (z(t)) where g ∈ G and g 6= id. An illustration is in Figure 3. G-Orbit

Φg (z(0)) = z(T )

z(0) w(t)

P/G

Figure 3: A relative periodic orbit. After a relative period T the orbit z(.) closes onto the group orbit. The projection of z(.) on P/G is closed. The reduction theory allows to redefine the RE. In fact let µ ∈ g and z ∈ Pµ be an equilibrium for XHµ . Then z gives rise to a RE in P, for take a closed curve g(t) in Gµ and define w(t) = Φg(t) (we ) 19

with we ∈ πµ−1 (z). Vice versa if z(t) is a RE then π(z(t)) is an equilibrium of the reduced flow (see [1]). In general there are two possible ways to study the RE. The first approach is to construct the reduction of the Hamiltonian system and then studies XHµ (z) = 0

(35)

{HG , f }P/G (z) = 0 for all f ∈ F(P/G)

(36)

in Hamiltonian form and in the Poisson description. The second approach is to observe that if z(t) = Φg(t) (z ∗ ) is a RE, then the condition z(t) ˙ = XH (z(t)) implies dΦg(t) (z ∗ ) (37) dH(z ∗ ) − dhJ(z ∗ ), ξi = 0 with ξP (z ∗ ) = dt t=0 On the three formulations (35), (36) and (37) one can apply all the standard continuation techniques based on the IFT. The possible difficulty in the study of the continuation of RE and RPO can be seen by looking at the linearisation of (37). The symmetry contributes to the degeneracy of the linearised equations, in fact one can show Proposition 4.1 ([38]). Let X(z) be an equivariant vector field with respect to a Lie group G. Then a RE has multiplier +1 with multiplicity at least equal to dim g; and an RPO has multiplier +1 with multiplicity at least equal to dim g + 1. In [39] RPOs are defined by looking at the fixed points of the action of G × S 1 on the space of T -periodic paths. This reads (g, θ) · z(t) = Φg (z(t + θ T )). (38) Remark 4.3. Note that (38) depends on T and it is easy to verify that T /k is a minimal period if the intersection of the isotropy subgroup of the action G × S 1 with S 1 is equal to Zk . Theorem 4.5 ([39]). Let (P, ω, H, G) a symmetric Hamiltonian system. Let ze ∈ P such that 1. d2 H(ze ) is a non-degerate quadratic form 2. d2 H(ze ) is positive definite if restricted to Vλ which is the real part of the eigenspace associated to the eigenvalue i λ, λ ∈ R. Then for every isotropy subgroup Γ of the G × S 1 -action on Vλ and  sufficiently small there exist at least Fix(Γ, Vλ ) periodic trajectories with period near 2π/|λ|, symmetry group contained in Γ and |H(z(t)) − H(ze )| = 2 . This result has been proved using a combination of a variational approach and IFT which will be considered in section 5.2. We have seen that the Poincar´e section is a useful construction to analyse the local structure of a periodic orbit. A very similar approach can be introduced for RPOs. A RPO can be seen as a combination of motions in P and in the symmetry group G. It is therefore natural to look for a suitable decomposition of the motion. In [39] it is introduced the following decomposition Tz P = W ⊕ X ⊕ Y ⊕ Z, 20

(39)

where

W = ker Dz J(z) ∩ Tz (G · z), G · z = {Φg (z) : g ∈ G}, X = Tz (G · z)/W, X = ker Dz J(z)/W, Z = Tz P/(ker Dz J(z) + Tz (G · z)).

Let Gz = {g ∈: Φg (z) = z} and Gµ = {g ∈ G : Ad∗g (µ) = µ} be the isotropy subgroups, it turns out that • ker Dz J(z) and Tz (G · z) are ω-orthogonal, • ω restricted ker Dz J(z) + Tz (G · z) is singular and W is the kernel, • ω induces a Gz -invariant symplectic form ωX on X and ωY on Y , • ω defines a Gz -invariant isomorphism between W and Z ∗ the dual of Z. The splitting (39) allows us to decompose the vector field XH and to analyse the motion along the group orbit and along the transversal directions. This is a tool used in [39] to study the Floquet operator in a neighbourhood of the RPO. For applications in the study of nonlinear normal modes and stability see [40, 41]. In [61] it is shown that such construction can be used to decompose the Poincar´e section in a part which is tangent to the conserved momentum, another part tangent to shape and a part parameterising the momentum. This approach has been also applied to the study of the geometry of mechanical systems defined on the cotangent bundle of a differentiable manifold, for this see [50] and references therein.

5

The Variational principles and periodic orbits

The idea behind variational principles is to transform a problem in differential equations into a question about critical points of a certain functional called action, whose domain is formed by trajectories of interest. Now in the study of T -periodic orbits we are interested in finding trajectories φ solving the equations of motion and satisfying φ(t) = φ(t+T ). This is a periodicity condition. The variational approach is particularly useful in the study of periodic problems because the periodicity condition is included into the definition of the space where the action is defined. Furthermore the method enables us to prove results not restricted to small perturbations. Let A : Λ 7→ R be a functional on a space of loops usually modelled on a Banach or Hilbert space. We shall see that the condition of vanishing of the first derivative of A is equivalent, in an appropriate sense, to solving the equations of motion. In order to describe the properties of A let us introduce: Definition 5.1 (Critical points). Let A[.] be differentiable functional on Λ, a path q(.) is a critical point of A[.] if DA[q](v) = 0 for all v(.) ∈ T Λ. Definition 5.2 (Critical set). The set K = {q(.) ∈ Λ : DA[q] = 0} is the critical set of A[.]. . Definition 5.3 (Critical value). A real number c is called critical value if Kc = A−1 [c] ∩ K 6= ∅.

21

5.1

Lagrangian view point

A typical way to write Newton’s equations in a variational form is by using Lagrange’s equations which are formulated through the Least Action Principle. This can be achieved for all mechanical systems that have potential forces. Consider a mechanical system whose configuration space is a Riemannian manifold M of dimension n. We denote with (q, vq ) the local coordinates in T M and with L : T M → R the Lagrangian function. In particular in the case of the so-called natural mechanical system [1] the Lagrangian has the form L(q, vq ) =

1 hvq , vq i − V (q), 2

(40)

where h., .i is the metric on T M and V : M → R is the potential. Given a path q : [0, T ] → M with integrable time derivative one can define Definition 5.4 (Action functional in Lagrangian form). Z AL [q] =

T

dt L(q(t), q(t)). ˙

(41)

0

In what follows we will be mostly interested in closed paths. Let C 2 ([0, T ], Rn ) be space of closed path of period T with two continuous time derivative. One can easily show that Proposition 5.1 ([1],[5]). Let L be a smooth Lagrangian function on M of the form (40). Let AL be defined over Λ = C 2 ([0, T ], M), then DAL [q](v) = 0 for every v(.) ∈ T Λ ' C 2 ([0, T ], T M) is equivalent to Newton’s equations with q(0) = q(T ). In particular the equations of motion in local coordinates are the Euler-Lagrange equations ˙ q(t)) ∂L(q(t), ˙ q(t)) d ∂L(q(t), − = 0 i = 1, ..., n. dt ∂ q˙i ∂qi Remark 5.1. For historical reasons the condition DAL [q](v) = 0 is called least action although is only a condition for stationary points of AL [.] in Λ. In general the Lagrangian contains a quadratic form in vq , in turn the action has a time integral of a quadratic form in q(t). ˙ This essentially shows that the natural domain for AL [.] is the Sobolev space H 1 ([0, T ], Rn ). Actually the action is defined on H 1 ([0, T ], M), which is a Hilbert manifold (see [30], [6]). This is locally described by absolutely continuous functions with the time derivative in the Lebesgue space L2 (see [29]). The interest in variational methods is related to the possibility of using critical point theory to find critical points corresponding to certain type of trajectories and then to show that such trajectories are solutions of Newton’s equations (see [35], [2], [21]). In particular this approach has been very successful in studying the problem of periodic orbits. Here is a general result in the Lagrangian setting: Theorem 5.1 ([6]). Let M be compact Riemannian manifold let L : T M → R be a Lagrangian of the form (40) with V ∈ C 1 (M, R), then there exists a periodic orbit in any free homotopy class. 22

We illustrate the ideas of the proof considering the special case where M is the n-dimensional torus Tn ' Rn /Zn . Theproblem is now to show that the action AL [.] attains a critical point, a minimum, in Λ(M) = q(.) ∈ H 1 ([0, T ], M) : q(.) is not null homotopic . Let us consider the sub-level of the action Ak = {q(.) ∈ Λ(M) : AL [q] ≤ k}. Now since V is smooth there exists minM V (q) = m > −∞ and therefore 1 AL [q] ≥ 2

Z

T 2 dt kq(t)k ˙ − m.

0

Since M is compact, the norm of q(.) in Λ(M) is equivalent to kqk ˙ 2 . This allows us to show that the action A[.] is coercive namely: Definition 5.5 (Coercivity). A : Λ → R is coercive in Λ if for all qn (.) such that limn→∞ kqn kΛ = ∞ then limn→∞ A[qn ] = 0. Moreover in Ak necessarily we have kqk ˙ 22 =

Z

T 2 dt kq(t)k ˙ ≤ 2(k + m).

0

This condition guarantees that Ak is weakly compact in the topology of Λ (see ,[35],[53],[2]) and using the coercivity the existence of a minimiser q∗ (.) can be obtained. The minimiser q∗ (.) is attained in Ak and it is a weak solution of the equations of motion. Indeed A[q ∗ ] = minΛ A[q] and DA[q∗ ](v) = 0 ∀v(.) ∈ Λ(T Rn ), namely T

Z

dt 0

n  X ∂L(q˙∗ (t), q∗ (t))

∂qi

i=1

vi (t) +

 ∂L(q˙∗ (t), q∗ (t)) v˙ i (t) = 0. ∂ q˙i

(42)

Using that q∗ (.) is absolutely continuous and that L is regular it is possible to integrate by part Z

T

dt 0

n  X ∂L(q˙∗ (t), q∗ (t))

∂ q˙i

i=1

Z

t

−

ds

∂L(q˙∗ (s), q∗ (s)) ∂qi

 v˙ i (t) = 0,

(43)

and to obtain ∂L(q˙∗ (t), q∗ (t)) − ∂ q˙i

Z

t

ds

∂L(q˙∗ (s), q∗ (s)) = ci where ci is constant in L2 ([0, T ], Rn ). ∂qi

By differentiating with respect to t we obtain the Euler-Lagrange’s equations: d ∂L(q˙∗ (t), q∗ (t)) ∂L(q˙∗ (t), q∗ (t)) − = 0 a.e. ∀i. dt ∂ q˙i ∂qi

(44)

It turns out that the equality in (44) holds for all time t because L is smooth. The reader should observe that the derivation of equations (42) and (44) are part of the proof of proposition 5.1. It is necessary to note that constants path are not in Λ(M), indeed the number of the minimisers is bounded from below by the Lusternik-Schnirelman category Cat(M) = n + 1 (see [35], [15]). Thus we exclude possible minimisers which would be trivial periodic orbits. In this very simple example 23

we can therefore appreciate the role of the definition of space of paths Λ(M) and its topology. More interesting cases can be found in the study of the N -body problem and in particular in the article [54] of this Encyclopaedia. The result in [6] can be generalised to Lagragian systems with symmetries. Assume to have a group action on the configuration space G × M 7→ M - denoted by (g, m) 7→ g.m - which preserves the Lagrangian L : T M → R and consider the problem of finding relative periodic paths γ(.) as critical points of the action AL [.]. We need to study the topology of Λg (M) = {γ(.) ∈ H 1 ([0, T ], M) : γ(t + T ) = g.γ(t)}.

(45)

g

In fact if the action AL [.] is bounded from below on Λ (M), then each connected component would contain at least a minimum that is a critical point and therefore a periodic orbit. The following analysis has been presented in [36]. In what follows for notational simplicity we denote a path and its homotopy class by the same symbol and use ∗ to denote both concatenation of paths and the induced operations on homotopy classes. Assume that M is connected. Choose a base point m ∈ M and let . Λgm (M) = {γ ∈ Λg (M) : γ(0) = m}, . the space of continuous paths from m to gm. Let Λm (M) = Λid m (M) denote the space of continuous loops based at m. Note that the space of connected components of Λm (M) is the fundamental group of M: π0 (Λm (M)) = π1 (M, m). Fix a particular path ω ∈ Λgm (M). The map Φω (γ) = ω −1 ∗ γ is a bijection Φω : π0 (Λgm (M)) → π0 (Λm (M)) = π1 (M, m), where ω −1 is the path obtained by traversing ω ‘backwards’. This bijection depends (only) on the homotopy class of ω in Λgm (M). For any α ∈ Λm (M) let g.α be the loop in Λgm (M) obtained by applying the diffeomorphism g to α and define an automorphism of π1 (M, m) by: α 7→ αg = ω −1 ∗ g.α ∗ ω. Again this depends (only) on the homotopy class of ω in Λgm (M). Now define the g-twisted action of π1 (M, m) on itself by α · β = αg ∗ β ∗ α−1

α, β ∈ π1 (M, m).

(46)

The number of connected components of the relative loop space is given by the following result: Theorem 5.2. The map Φω induces a bijection g π0 (Λg (M)) ∼ = π1 (M, m) , g

where π1 (M, m) is the set of orbits of the g-twisted action of π1 (M, m) on itself. Remark 5.2. If g is homotopic to the identity then Λg (M) is homotopy equivalent to the loop . space Λ(M) = Λid (M) and the g-twisted action of π1 (M, m) on itself is just conjugation. This is therefore the well known result that the connected components of the loop space map bijectively to the conjugacy classes of the fundamental group (see [30]).

24

Remark 5.3. The g-twisted action of π1 (M, m) on itself induces an affine action of the first homology group H1 (M) on itself: < α > · < β > = g. < α > − < α > + < β > where the brackets < . > denote the homology class and g. < α > denotes the natural action of g on H1 (M). When π1 (M, m) is abelian this is the same as the action of π1 (M, m) on itself. More generally it is easier to calculate than the π1 (M, m) action and in typical applications may be used to describe relative periodic orbits in terms of winding numbers. The analysis gives explicit results for systems whose configuration space has the property that all its homotopy groups except the fundamental group are trivial. In this case M is said to be K(π, 1), for more details see [12, 60]. Examples of K(π, 1) spaces include tori, the plane R2 with N points removed, and the configuration spaces of planar N -body problems. Theorem 5.3. Assume M is a K(π, 1). Then for any γ ∈ Λgm (M) the connected component of Λg (M) containing γ, denoted Λgγ (M), is also a K(π, 1) with g π1 (Λgγ (M)) ∼ = Zπ1 (M) (Φω (γ)),

where

. Zπg1 (M) (Φω (γ)) = {α ∈ π1 (M) : αg ∗ Φω (γ) ∗ α−1 = Φω (γ)}

i.e. the isotropy subgroup (or centraliser) at Φω (γ) of the g-twisted action of π1 (M, m) on itself. We note that all K(π, 1)’s with isomorphic fundamental groups are homotopy equivalent to each other [12, 60], and so this result determines the homotopy types of connected components of relative loop spaces. The homology groups can be computed algebraically as the homology groups of the fundamental group [14]. A simple example Let M = T 1 , the circle, and consider first the loop space Λ(T 1 ). The ‘g-twisted action’ of π1 (T 1 ) on itself is just conjugation, and since π1 (T 1 ) ∼ = Z is abelian this is trivial. So π0 (Λ(M )) ∼ = Z, the homotopy classes of loops being specified precisely by their winding numbers. Since T 1 is a K(π, 1), Theorem 5.3 says that each component of relative loop space is also a K(π, 1) with fundamental group isomorphic to Z, and therefore has the homotopy type of a circle. Now consider Λg (T 1 ) where g : T 1 → T 1 is a reflection. Choose one of the two fixed points of the reflection to be the base point m. We may choose ω to be the trivial path from m to g.m. Then for each α ∈ π1 (T 1 , m) ∼ = Z we have αg = −α and so the ‘g-twisted action’ (46) is the translation α.β = β − 2α.

(47)

g This has two orbits, π1 (T 1 ) ∼ = Z2 , and the isotropy subgroups are trivial. It follows from Theorems 5.2 and 5.3 that the space of relative loops Λg (T 1 ) has two components, each of which is contractible.

25

Nemerical studies In many interesting problems, typically in celestial mechanics, the action functional is bounded from below and therefore the expected critical points are minimisers. In recent years, in connection with the discovery of the so-called choreographic periodic orbits, C. Sim´o [51] developed an algorithm to study how to perform a numerical minimisation of the action in classes of loops with a definite symmetry type. The idea is based on the description of the orbit in terms of its Fourier coefficients and define the action AL [.] as a function on the Fourier space. The action AL [.] is then minimised on the space of Fourier coefficients. The interested reader should consult [54] in this Encyclopaedia. It would be very interesting to generalise this method to different actions and to see whether one can impose not only constrains on the symmetry type but also on topology of the space of loops.

5.2

Hamiltonian view point

The Hamilton principle gives a variational characterisation to the Hamilton equation. For Hamiltonian systems in R2n the formulation of the principle is very simple. Let H : R2n → R be the Hamiltonian function then the action functional is Definition 5.6 (Action functional in Hamiltonian form). . AH [γ] =

Z

T

dt 0

n X

Z pi (t) q˙i (t) −

T

dt H(q(t), p(t))

(48)

0

i=1

where γ(t) = (q(t), p(t)). It is worth recalling that if the Lagrangian function is hyper-regular, then system can be transformed through the Legendre transform into a Hamiltonian system on the cotangent bundle of the configuration space. In any case, given AH [.] one can show (see [5], [1]) Proposition 5.2. Let H a smooth Hamiltonian function on R2n . Let AH be defined over C 2 ([0, T ], R2n ) then DAH [q, p](v) = 0 for every v(.) ∈ C 2 ([0, T ], R2n ) is equivalent to Hamiltonian equations with q(0) = q(T ), p(0) = p(T ). Recall that in the Hamiltonian context p, q are independent variables and therefore to prove the preceding proposition it is necessary to compute the variations with respect to q’s and p’s independently. From the analytical point of view the functional AH [.] is a difficult object to study since it is unbounded from below and above, namely it is indefinite (see [35]). It is not difficult to give an example, consider n = 1 and a Hamiltonian like H = p2 /2. For indefinite functionals there is a whole theory developed to apply the mountain pass theorem [48]. There is also a generalisation of the Legendre transform [21]. The new transform (see [21]) can be applied directly to the action functional AH [.] rather than to H. This allows us to work with a convex functional. We do not enter in the details but we want to recall one result which describes the conditions for the existence of at least n periodic orbits. Theorem 5.4 ([21]). Suppose that H ∈ C 1 (R2n , R) and for some β > 0, assume that C = {z ∈ R2n : H(z) ≤ β} is strictly convex, with boundary ∂C = {z ∈ R2n : H(z) = β} satisfying 26

√ hz, ∇H(z)i > 0 for all z ∈ ∂C. Suppose that for r, R > 0 with r < R < 2 r there are two balls Br (0), BR (0) centred in the origin of R2n such that Br (0) ⊂ C ⊂ BR (0), then there are at least n distinct periodic solutions on ∂C of the Hamilton system z˙ = J ∇H(z). The functional AH [.] has been defined for Hamiltonian systems on R2n , but it admits a generalisation to symplectic manifolds which are not tangent bundles: Definition 5.7. Let (P, ω) be a symplectic manifold and H : P → R a smooth Hamiltonian function, let γ : P → R be a closed path which is the boundary of a two-dimensional connected region Σ, then Z Z T . AH [γ] = ω− dt H(z(t)). (49) Σ

0

The functional (49) is a very interesting object. It is naturally defined on closed paths which bound two-dimensional regions. The functional can be defined on ”paths” but then it would become multi-valued. In fact in M one has to introduce a 1-fom α such that ω = dα. The 1-form α is not unique and depends on the cohomology of M. Although AH is multivalued the differential DAH is single valued [34]. In fact let ϕ(s) be a finite variation with dϕ(s) ds |s=0 = ξ then Z Z T dAH [ϕ(s) = Lξ (ω) − DAH [z](ξ) = dth∇H(z), ξi, ds Σ 0 s=0 where Lξ is the Lie derivative with respect ξ. Now since dω = 0 one can show that Z

T

T

Z iξ iXH ω −

DAH [z](ξ) = 0

dth∇H(z), ξi 0

which is single valued. The theory for such multi-valued functionals has been developed by many, here we would like to cite [58], [44] and [22]. Note that functionals of the form (49) can result from the symmetry reduction. In fact as shown in [34] a Lagrangian system with a non-abelian symmetry G has a reduced dynamics determined by a variational principle of the form . AR [γ] =

Z

T

Z dt R(q(t), q(t)) ˙ −

0

βµ (q(t), q(t)) ˙

(50)

Σ

where R is the so-called Routhian and βµ is a two-form dependent on the conserved momentum µ. The construction of the Routhian and of the reduced action AR [.] can be found in [34].

5.3

Fixed energy problem, the Hill’s region

Let us consider Hamiltonian systems (P, ω, H) where the phase space is the is a cotangent bundle P = T ∗ M ' R2n , the symplectic form is then given by ω = dθ where θ is the canonical form and the Hamiltonian is 1 (51) H(p, q) = hp, pi + V (q) 2 h., .i is a Riemannian metric on Tq∗ M. The Hamiltonian flow preserves the the Hamiltonian function, therefore a natural problem is to restrict the dynamics to the manifold defined by a fixed constant 27

value of the Hamiltonian that physically corresponds to fixing the energy. Letting E be the energy value, this natural constraint can be defined by constructing the following submanifold of the phase space P ΣE = {(p, q) ∈ P : H(p, q) = E}. From ΣE it is possible to construct a new manifold that corresponds to the image of the projection of ΣE onto the configuration space M. To obtain such a projection it is sufficient to look at (51) and observe that the norm of p on Tq∗ M cannot be negative. From this, the new space, the Hill’s region, turns out to be defined as follows Definition 5.8 (Hill’s region). . PE = {q ∈ M : E − V (q) ≥ 0} Remark 5.4. The manifold PE depends on the values of E and may have boundaries. For instance consider the Hamiltonian H=

λ 1 1 kpk2 − kqk2 + kqk4 with λ > 0, 2 2 4

which has an Hill’s region defined by   λ 1 . 2 4 PE = q ∈ M : E + kqk − kqk ≥ 0 . 2 4 First note that PE is a subset of R2 and that there are the following cases: • PE = ∅ for E < −λ2 /4, • PE is a circle for E = −λ2 /4, • PE is an annulus for −λ2 /4 < E < 0, • PE is a disk minus its centre for E = 0, • PE is a disk for E > 0. Note that if PE is not empty, then the boundary of ∂PE is given by   1 λ . ∂PE = q ∈ M : E + kqk2 − kqk4 = 0 2 4 and is not empty. The boundary corresponds to the set of points where all momenta p vanish. The Hill’s region has a topology which changes according to the values of the energy. Hence it is a natural problem to search periodic orbits in it. What are the possible orbits in PE ? Assuming to have a generic Hill’s region with non-empty boundary, there are two possible types of orbits: (A) orbits joining two points of the boundary, the brake orbits, (B) orbits not intersecting the boundary, the internal periodic orbits. In general given a Hamiltonian system with non-empty Hill’s region PE we define 28

Definition 5.9 (Brake orbit). A solution q(.) of the Hamilton equation is a brake orbit of period T if q(t + T ) = q(t), q(t) ∈ PE \∂PE for all t ∈ (0, T /2) and q(0) ∈ ∂PE , q(T /2) ∈ ∂PE . Definition 5.10 (Internal periodic orbit). A solution q(.) of the Hamilton equation is an internal periodic orbit of period T if q(t + T ) = q(t) and q(t) ∈ PE \∂PE for all t. Remark 5.5. In principle one could think that brake orbits could have more than two intersections with the boundary ∂PE . This is not possible in systems with Hamiltonian (51) because the velocity on ∂PE is zero and the Hamiltonian equations are ”reversible”, that is if q(t) is a solution then q(−t) is a solution. These two properties imply that any solution q(t) such that q(ti ) ∈ ∂PE then follows the trajectory q(−t) for t > ti with reversed velocity [31], [24]. An example of Hill’s region with brake orbits and internal periodic orbit is in Figure 4.

γbrake ∂PE

γinternal

γbrake

∂PE

Figure 4: Example of a compact non-simply connected Hill’s region with two brake orbits γbrake and one internal orbit γinternal that cannot be deformed into a point.

Jacobi metric and Tonelli functional The general approach to study the periodic orbits of type (A) or (B) is to use a variational principle which is naturally defined on paths in PE . Let us consider Definition 5.11 (Jacobi metric). JE [q] =

1 2

Z 0

1



dq(s) 2

ds(E − V (q(s))

ds .

(52)

. The domain of JE [.] is Λ(PE ) = {q(.) ∈ H 1 ([0, 1], Rn ) : q(s + 1) = q(s) q(s) ∈ PE }, the following result holds Proposition 5.3. Let q(.) be a path in Λ(PE ) where JE [.] is differentiable and let q(.) +  v(.) ∈ Λ(PE \∂PE ) for all  sufficiently small, then DJE [q](v) = 0 for all v(.) ∈ T Λ(PE ) is equivalent to dq(t) = −∇V (q(t)) dt 29

after a time re-scaling defined by

dq(s) 1 dt

p . = ds ds E − V (q(s)) The proof of this proposition is quite standard and can be found in [52], [31], [11], [6],[1]. Let us observe that in the regions where E − V (q) > 0 the functional JE [.] can be derived from the Riemmanian metric X d2 ` = (E − V (q)) δij dqi ⊗ dqj (53) i,j

and indeed from this its name was originated. From the Jacobi metric a length can be defined 1

Z

. `[q] =

p 2. ˙ ds (E − V (q(s))kq(s)k

0

The Jacobi metric (53) transformed the study of the classical Newton’s equations into the study of the geodesics on PE . It is crucial to note that the Jacobi metric is vanishing on ∂PE and therefore cannot be complete in PE whenever the boundary is not empty. There is another functional that can be used to study periodic orbits in PE . This is defined as follows: Definition 5.12 (Tonelli functional). 1 TE [q] = 2

Z

1

Z

1

ds(E − V (q(s)) 0

0



dq(s) 2

. ds ds

Tonelli’s functional is a product of two integrals and therefore does not have a natural geometrical interpretation like (53). One can observe that (by Schwartz inequality) Tonelli functional is bounded from below by Jacobi length: (`[q])2 ≤ 2TE [q]. Note that also TE [.] vanishes on ∂PE . The functional TE [.] is defined on Λ(PE ) and provides another possible variational description of Newton’s equations. Indeed one can easily show (see [2]) Proposition 5.4. Let q(.) be a path in Λ(PE ) where TE [.] is differentiable and let q(.) +  v(.) ∈ Λ(PE \∂PE ) for all  sufficiently small, then DTE [q](v) = 0 for all v(.) ∈ T Λ(PE ) is equivalent to dq(t) = −∇V (q(t)) dt after time re-scaling defined by 1

Z 

dt ds

2 =Z

0 1



dq(s) 2

ds

ds

ds(E − V (q(s)) 0

30

.

Remark 5.6. Observe that in both Jacobi and Tonelli formulation there is a reparameterisation of the time and therefore one can always restrict to consider trajectories with unit period. In variational methods we aim to show that the critical set is not empty and, if possible, to estimate its cardinality. This is certainly affected by the topology of the space, where the paths are defined. Let us now consider the Jacobi metric JE [.]. This functional depends on the energy E which affects the properties of PE and Λ(PE ). There are four possible cases: • PE is compact and ∂PE = ∅, • PE is compact and ∂PE 6= ∅, • PE is not compact and ∂PE = ∅, • PE is not compact and ∂PE 6= ∅. In the next section we shall illustrate some results about the preceding four cases. We shall see that the main approaches have much in common with Riemmian geometry. For more details on variational methods the reader is encouraged to consult [15], [2], [35], [29], [53]. Orbits in compact Hill’s regions without boundary Let us consider a region PE without boundary for E > supq∈M V (q). If M is compact, then the previous condition can be satisfied for some finite E. In this case PE ' M and the Jacobi metric JE [.] becomes a Riemannian metric on M, therefore the problem of periodic orbits in M translates into the problem of closed geodesics in the Riemanian manifold M. This is solved by Theorem 5.5 (Lyusternik and Fet). Each compact Riemannian manifold contains a closed geodesic. For a proof see [29],[30]. The main tool for proving the theorem is the so called curve shortening. The manifold PE is Riemannian with metric (53). A base for the topology is given by Br (q) = {q 0 ∈ M : d(q, q 0 ) < r} where d(q, q 0 ) = inf {`[γ], γ(.) is a piece-wise smooth curve such that γ(0) = q and γ(1) = q 0 } . Remark 5.7. The reader may observe that if PE has a non empty boundary then d(., .) turns into a pseudo-metric because d(q, q 0 ) = 0 for all q, q 0 ∈ ∂PE . Certainly d(., .) is still a metric restricted to the open set PE \∂PE . Let us now give a sketch of the curve-shortening method. A standard result in Riemannian geometry guarantees that given a point q0 and a neighbourhood Bδ (q0 ), there exists δ such that any point in ∂Bδ (q0 ) can be joined to q0 by a unique geodesic [29]. Since M is compact one can use a finite family of such neighbourhoods to join any two points q0 , q1 ∈ M with a piece-wise geodesic γ(.). Now consider on M the space of closed paths of class H 1 ([0, 1], M) and consider a sequence 0 ≤ (k) (k) (k) (k) t0 < t1 < ... < tn−1 < tn ≤ 1 such that ti+1 − ti < δ 2 /(2c). Take a curve γ(.) ∈ H 1 ([0, 1], M) with JE [γ] ≤ c then define ( (k) (k) is a piece-wise geodesic curve where S (k) (γ)(t) restricted to [ti , ti+1 ], S (k) (γ) = (k) (k) is a geodesic joining γ(ti ) to γ(ti+1 ) for i = 1...n.

31

Now clearly JE [S (k) (γ)] ≤ JE [γ], `[S (k) (γ)] ≤ `[γ]. (k) (k+1) (k) The map S (k+1) (.) is constructed by taking a new partition of [0, 1] such that ti < ti < ti+1 . Since one can easily verify `[S (k+1) (γ)] ≤ `[S (k) (γ)] (54) the iteration of S (k) (.) is called curve shortening. In [24], [29], it is shown that S (k) (γ) converges uniformly to a geodesic. The limit could be just a point curve. To show that this cannot be the case one uses that on a compact manifold of dimension n there is always a map h : S d 7→ M (with 1 ≤ d ≤ n) which is not homotopic to a constant map. There have been many generalisations of Lusternik-Fet theorem and the reader is invited to refer to [30]. Brake orbits in Hill’s regions with boundary Let PE be a region with boundary for inf q∈M V (q) < E < inf q∈M V (q). In this case there is the following result: Theorem 5.6 ([11]). Suppose that PE is compact and there are no equilibrium positions in ∂PE . Then the number of brake orbits in PE is at least equal to the number of generators of π1 (PE \∂PE ). Since the Jacobi metric is singular on ∂PE it is necessary to analyse the geodesic motion near the boundary. This type of analysis goes back to the earlier works [9], [10] and [52]. The main point is to construct a new region PE− ⊂ PE on which JE [.] is positive definite. The next step is to construct a geodesic joining two points on the new boundary ∂PE− . Finally the construction on PE− is used to show that the geodesic q  (.) becomes a brake orbit in the limit  → 0. In the case of non compact PE , the existence of a brake orbit was proved in [45]. This result is based on two assumptions: (i) ∂PE is not empty and it is formed by two connected components A1 and A2 such that if xn ∈ A1 and yn ∈ A2 with kxn k → ∞, kyn k → ∞ then kxn − yn k → ∞. (ii) Let Rδ = {q ∈ M : E − δ < V (q) < E} for δ > 0. If either A1 or A2 is not compact, there is a number r > 0 with the following property: if r0 > r then there exist r1 > r0 and δ > δ ∗ > 0 such that for every q ∈ Rδ ∩ {q : kqk > r1 } and (q, p) ∈ H −1 (E) implies that the Hamiltonian flow φ(t, q, p) stays in {q : kqk > r1 } for all t ≥ 0 where defined. Theorem 5.7 ([45]). Suppose that PE is connected, inf q∈∂PE k∇V (q)k > 0 and condition (i) and (ii) hold true. Then there exists a periodic solution which is a brake-orbit. In this result the lack of compactness of PE does not allow to use the curve shortening. For this reason in [45] the direct minimisation is employed. First a new region PE−δ ⊂ PE is defined for δ > 0. Such region has now a boundary with two different connected components respectively Aδ1 and Aδ2 . Then in [45] the following minimisation problem is studied min JE [q] where q(.) ∈ H 1 and boundary conditions q(0) ∈ Aδ1 , q(0) ∈ Aδ1 . Condition (i) is able to control the lack of compactness of PE and to show that the minimsation problem admits a solution q δ (.). Condition (ii) allows to take the limit δ → 0 to obtain a brake orbit in PE .

32

Orbits in closed, non-simply connected Hill’s regions with boundaries Let us now consider a general Hill’s region which is not necessarily compact, with boundary and with non-trivial homotopy group π1 (PE ). The natural problem is to determine under which condition it is possible to prove the existence of a internal periodic orbit within a specific homotopy class of PE . A partial answer is given by Theorem 5.8 ([7],[8]). Suppose that PE is closed and bounded and that ∇V (q) 6= 0 for all q ∈ ∂PE . Then there exists a periodic orbit q(s) ∈ PE for all s. The orbit q(.) can be either a brake orbit or an internal periodic orbit. As we have already seen the Jacobi metric (but also Tonelli functional) is degenerate on the boundary ∂PE , i.e. JE [γ] is idetically zero on every closed path γ(.) such that γ(s) ∈ ∂PE for every s ∈ [0, 1]. In order to avoid this problem, in [7] and [8], a modified functional is introduced . J  [q] = JE [q] +

Z

1

ds U  [q(s)].

0

The functional J  [.] is called penalised, for if qn (.) is a sequence of curves in Λ(PE ) (the closure of Λ(PE )) weakly converging to a curve q ∗ (.) intersecting the boundary, then the form of U  (.) implies that J () [qn ] → +∞ whenever  > 0. This type of penalisation is similar to the so called strong force potential used in N -body problem (see [26], [2] and also [54] in this Encyclopedia). In this case there are two kinds of difficulties: PE is no longer compact and the boundary ∂PE is not empty. The strategy is as follows: prove that the critical level sets of J  [.] are pre-compact. This is obtained by a generalisation of the Palais-Smale condition (PS). This prevents the sequences to converge on the boundary, and it is realised by taking a functional ρ : C 1 (Λ(PE )) → R+ such that if qn (.) converges to a path intersecting the boundary, then ρ(qn ) → +∞. This allows to introduce: Definition 5.13 (Weighted Palais-Smale condition, [7]). The action J  [.] satisfies the weighted PS condition if any sequence qn (.) fulfils one of the following alternatives 1. ρ(qn ) and J  [qn ] are bounded and DJ  [qn ] → 0 and qn has a convergent sub-sequence, 2. J  [qn ] is convergent and ρ(qn ) → +∞ and there exists ν > 0 such that kDJ  [qn ]k ≥ ν kDρ(qn )k for n sufficiently large. The verification the Weighted Palais-Smale condition guarantees that the set Kc ∩ {q : ρ(q) ≤ M } is compact for every M > 0. The next step in the proof is to construct a minimax structure [48] that allows to identify the critical values. This is achieved by constructing two manifolds Q and S such that: (i) S ∩ ∂Q = ∅, (ii) if u ∈ C 0 (Q, Λ(PE )) such that u(q(.)) = q(.) for every q(.) ∈ ∂Q then h(Q) ∩ S 6= ∅. The manifolds Q and S are defined such that the critical level c = inf sup J  (h(q)) u∈U q∈Q

33

is finite. Here U = {u ∈ C 0 (Q, Λ(PE )) : u(q(.)) = q(.) if J  [q] ≤ 0}. Then the weighted PS condition allows to construct a gradient flow and to prove the existence of a critical point q  (.). In [8] it is shown also that there exists α ≤ β independent from  such that α ≤ J  [q  ] ≤ β. This uniform estimates make possible to prove that for  → 0 the path q  (.) converges uniformly to . a closed path q 0 (.) such that the set I = {t ∈ [0, 1] : q 0 (t) ∈ PE \∂PE } is not empty. Since q 0 (.) is continuous then the interval I has connected components. Note that q 0 (.) has the same homotopy type as q  (.), but in general this is not longer true for the solution of the equations motion. In fact q 0 (.) is a solution only on the connected component of I. Let ∆ be one connected components of I, on ∆ the path q 0 (.) solves the equations of motion. Therefore the topological characterisation of the periodic orbit depends on ∆, if ∆ ⊂ [0, 1] then q 0 (.) is a brake orbit, if ∆ = [0, 1] then q 0 (.) is an internal periodic orbit. It is still open the problem of finding internal periodic orbits in any prescribed homotopy class.

5.4

Continuation of periodic orbits as critical points

Variational methods can also be very useful to study the problem of continuation of periodic orbits. Here we restrict ourself to few examples. Lagrangian variational principle Let us consider a dynamical system in Rn that can be written in the Lagrangian form: L(vq , q) =

1 kvq k − V (q). 2

Let us assume (i) there exists a > 0 such that V (λ q) = λ−a V (q) for any λ > 0, (ii) there exists a 1-periodic orbit q1 (.). R1 The orbit q1 (.) is a critical point of AL [q] = 0 dsL(q(s), ˙ q(s)). Now let us consider another potential term W (q) such that W (λ q) = λ−b W (q) with b > a. Let us suppose to look for periodic orbits of period T for the system whose Lagrangian is L(vq , q) =

1 kvq k − V (q) − W (q). 2

This suggests to search for critical points x(.) of the functional  Z T  2 kx(t)k ˙ ATL [x] = dt − V (x(t)) − W (x(t)) . 2 0

(55)

The scaling properties of V and W can be used to construct a perturbation argument. In fact define x(t) = T −p q(t/T ) where q(s) is defined for s ∈ [0, 1], and p = 2/(a + 2). (56) 34

The dynamical equations for x(.) are equivalent to the Lagrangian equation for  Z 1  2 kq(s)k ˙ AL [q; ] = ds − V (q(s)) −  W (q(s)) , 2 0 where

(57)

. −2(b−a)  = T a+2 .

The scaling properties of the potential allow us to transform the given problem into a perturbation problem. Note that there is the following correspondence: a small perturbations for the scaled action (57) corresponds to large periods for (55). Now AL [., 0] = A1L [.] and in general the critical point q1 (.) is not isolated and therefore D2 AL [q1 , 0] has some degeneracy. If the manifold of critical points is degenerate along normal directions (normal degeneracy), then it is possible to continue the critical point q1 (.) into a T -periodic orbit by decomposing the continuation procedure. This approach is the so-called Liapunov-Schmidt reduction, it is used to generalize the IFT to situations where cannot be applied directly. An example will be illustrated in the last paragraph of this section in the study Hamiltonian systems. For the details about normal degeneracy the reader could consult [15]. Hamiltonian variational principle Hamiltonian equations can be thought as a vector field on a space of loops. The geometrical construction has been described in [58]. Let us consider the space C 1 ([0, 1], Rn ) of differentiable loops, we can define . dz(s) − J∇H(z(s)), z(.) ∈ C 1 ([0, 1], R2n ). X (z) = ν ds

(58)

The zero set of X (z) is formed by loops z(.) such that ν

dz(s) = J∇H(z(s)), ds

this corresponds to a periodic orbit z(.) with period 2π/ν. It turns out that X (.) is a Hamiltonian vector-field whose Hamiltonian on C 1 ([0, 1], R2n ) (see [58], [25]) is given by Z 2π 1 H[z] = ds {hν J z(s), ˙ z(s)i − H(z(s))} (59) 2π 0 and with symplectic form Z 2π 1 dshJz(s), w(s)i. (60) Ω[z, w] = 2π 0 Indeed a simple calculation shows that Ω(X (z), w) = dH[z](w). Let us now suppose to have a periodic orbit z0 (.). We want to illustrate how to study the continuation and bifurcation when the Hamiltonian is perturbed. In order to use the formulation in the loop space we can introduce the Liapunov-Schmidt reduction. In fact in general Hamiltonian vector fields at a periodic orbit have a linearisation with a non-empty kernel. The construction we now present is a very well known approach in the infinite dimensional setting. 35

Liapunov-Schmidt reduction for Hamiltonian systems [3], [25] Let (H, P, Ω) be an Hamiltonian system with P = C 1 ([0, 1], R2n ). Let X : P → Y be the Hamiltonian vector field such that z0 (.) is a zero of X (.) and (i) L = DX (z0 ) is Fredholm, (ii) J(ker(L)) = ker(L). Condition (i) implies that it is possible to make the following decomposition 1. P = ker(L) ⊕ W , W closed subspace, 2. Y = rank(L) ⊕ N , N closed subspace. In fact any z can be written as z = k + w with k ∈ ker(L) and w ∈ W . This allows us to reduce X (z) = 0 to solving the following system of equations ( Π X (k + w) = 0 ∈ rank(L) (61) (I − Π) X (k + w) = 0 ∈ N where Π : Y 7→ rank(L). The first equation in (61) can be solved with respect to w by IFT. The socond equation become the so called bifurcation equation: Xg (k) = (I − Π)X (k + w(k)) = 0.

(62)

Note that since L is Fredholm dim ker(L) < ∞ hence (62) is a finite dimensional problem. Now condition (ii) implies that Xg can be thought as a map from ker(L) into itself, furthermore this allows to show that Xg is a Hamiltonian vector-field with Hamiltonian g(k) = H(k + w(k)) and symplectic form Ω(., .). If condition (ii) does not hold a further condition has to be included in order to guarantee that Xg is a Hamiltonian vector-field. The bifurcation equation becomes therefore Ω(Xg (k), u) − dH(k + w(k))(u) = 0 ∀u. (63) In [58] the loop space approach and the reduction were used to show that if an Hamiltonian system has a manifold Σ of periodic orbits whose tangent space coincides with the kernel of D2 AH (nondegeneracy condition) then small perturbations of the Hamiltonian cannot destroy all the periodic orbits: Theorem 5.9 ([58]). Let Σ be a compact, non-degenerate manifold of periodic orbits for an Hamiltonian system (P, ω, H). Given any neighbourhood U of Σ there exists 0 > 0 such that for || < 0 , the number of periodic orbits in U for (P, ω, H +  H1 ) is no less than the Lusternik-Schnirelman category Cat(Σ/S 1 ). If Σ satisfies the additional condition that the algebraic multiplicity of 1 as an eigenvalue of the Poincar´e map is uniformly equal to the dim Σ, then Cat(Σ/S 1 ) ≥ (1 + dim Σ)/2. In [25] Liapunov-Schmidt reduction is used to prove Liapunov centre theorem and also the Hamiltonian Hopf-bifurcation. 36

6

Further directions

This article is focused on periodic orbits in Hamiltonian systems. This problem can be seen as an organising centre in the history of the development of modern mathematics, in fact it is related to many theoretical aspects: bifurcation theory, symmetry reduction, variational methods, topology of closed curves on manifolds. Still open problems remain, for example one would like to prove the existence of internal periodic orbits in every homotopy class of a Hill’s region. Another interesting direction is the analytical study of the action functional that results from symmetry reduction. A further generalisation of the Hamiltonian formalism is the study of the so-called ”multi-periodic patterns”. Here is an example: let φ, p be two functions defined on the two-dimensional torus T2 and valued in R. The functions φ, p are 2-periodic because they satisfy p(x + τ1 , y + τ2 ) = p(x, y), φ(x + τ1 , y + τ2 ) = φ(x) where (x, y) + (τ1 , τ2 ) ' (x, y) in T2 . One can pose the problem to solve a couple of partial differential equations of the form ∇2 φ +

∂ V (φ, p, x, y) = 0 ∂φ

and ∇2 p +

∂ V (φ, p, x, y) = 0 . ∂p

(64)

Here V : R2 7→ R is a smooth potential function. Note that the periodicity of φ and p is ”spatial”, that is why the term multi-periodic patterns is used. It has been shown that (64) admit a description in terms of finite dimensional multi-symplectic structure and the equations of motion can be cast into a general Hamiltonian variational principle. In fact one can define Z = (φ, u1 , u2 , p) defined as functions on T2 and the Hamiltonian S(Z, x, y) =

1 2 (u + u22 ) + V (φ, p, x, y) . 2 1

Then the equations (64) turn out to be equivalent to the variational equation DL(Z) = 0, where L is the action functional Z τ1 Z L(Z) = dx 0

0

τ2

    ∂Z ∂Z 1 J1 + J2 , Z − S(Z, x, y) dy 2 ∂x ∂y

(65)

(66)

defined on 2-periodic maps Z : T2 7→ R4 . The matrices J1 and J2 are two symplectic structures and form a multi-symplectic structure. In this example J1 and J2 read     0 −1 0 0 0 0 −1 0 1 0 0 0  0 0 0 1   , J1 =  (67) 0 0 0 −1 and J2 = 1 0 0 0 0 0 1 0 0 −1 0 0 For equations (64) one can pose the problem of finding solutions as ”multi-periodic” critical points of the action L, which is a generalisation of Hamiltonian action functional (48). An interesting reference for this problem is [13].

37

Acknowledgements The author would like to thank Heinz Hanßmann for his critical and thorough reading of the manuscript and Prof Ferdinand Verhulst for his useful suggestions.

In the bibliography Books and Reviews are denoted with a ?. The primary literature has no special marker.

References [1] ? R.Abraham, J.Marsden Foundations of Mechanics Benjamin-Cummings, Reading MA, 1978 [2] ? A.Ambrosetti, V. Coti Zelati Periodic Solutions of singular Lagrangian Systems Birckhauser, 1994 [3] ? A.Ambrosetti, G.Prodi A primer on Nonlinear Analysis CUP 1993 [4] J. M. Arms , R. Cushman, M. J. Gotay A universal reduction procedure for Hamiltonian group actions, in: T. S. Ratiu (ed.), The Geometry of Hamiltonian Systems, Springer-Verlag, New York, 1991, pp. 3351. [5] ? V.I.Arnol’d Mathematical Methods of Classical Mechanics Springer Verlag [6] V.Benci Periodic solutions for Lagrangian systems on a compact manifold, J.Diff.Eq. 63 (1986) 135-161. [7] V.Benci Normal modes of a Lagrangian system constrained in a potential well Annales de l’IHP section C tome 1 5 (1984) pag. 379-400 [8] V.Benci Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems Annales de l’IHP section C tome 1 5 (1984) pag. 401-412 [9] G.D. Birkoff Dynamical Systems with two degrees of freedom Trans. Amer. Math. Soc. 18, 199-300, (1927) [10] G.D. Birkoff Dynamical Systems Colloq. Publ. 9 Amer. Math. Soc. (1927) [11] S.V. Bolotin and V.V. Kozlov Libration in systems with many degrees of freedom J.Appl. Math. Mech. 42 256-261, 1978 [12] ? R. Bott and W. Tu, Differential forms in algebraic topology, Springer Verlag, 1995. [13] T.Bridges Canonical multi-symplectic structure on the total exterior algebra bundle Proc. R. Soc. A 462, 15311551, 2006 [14] ? K.S. Brown, Cohomology of groups, Springer Verlag, 1982. [15] ? K.Chang Infinite dimensional Morse theory and multiple solution problems, Birkhauser, 1991

38

[16] ? R. Cushman, L. Bates Global aspects of classical integrable systems Birkhauser [17] ? R.Cushmann, D Sadowski, K. Efstathiou No polar coordinates in Geometric Mechanics and Symmetry: the Peyresq Lectures Cambridge University Press, 2005 Edited by J.Montaldi, T.Ratiu [18] ? G.Dell’Antonio Agomenti scelti di meccanica SISSA/ISAS Trieste, ref. ILAS/FM-19/1995 [19] E.J. Doedel, H.B. Keller, J.P. Kernvez Numerical analysis and control of bifurcation problems: (II) Bifurcation in infinite dimensions Int. J. Bifurcation & Chaos 1 4, pages 745-772, 1991 [20] E.J. Doedel, H.B. Keller, J.P. Kernvez Numerical analysis and control of bifurcation problems: (I) Bifurcation in finite dimensions Int. J. Bifurcation & Chaos 1 3 pages 493-520, 1991 [21] ? I.Ekeland Convexity methods in Hamiltonian mechanics, Springer-Verlag, 1990 [22] ? M.Farber Topology of Closed One-Forms AMS, 2004 [23] G.Gallavotti Quasi periodic motions from Hipparchus to Kolmogorov Rend. Mat. Acc. Lincei s. 9, v. 12:125-152, 2001 [24] H.Gluck, W.Ziller Existence of periodic motions of conservative systems, Seminar on Minimal Submanifolds Ed. Bombieri, Princeton University Press, 1983 [25] ? M.Golubitsky, J.E.Marsden, I. Stewart, M.Dellnitz The constrained Liapunov-Schmidt procedure and periodic orbits Fields Institute Communications 4 (1995) [26] W. Gordon, Conservative dynamical systems involving strong force, Trans. Amer. Math. Soc. 204 (1975) 113-135. [27] ? V.Guillemin, S.Sternberg Symplectic techniques in Physics Cambridge University Press, 1984 [28] ? H.Hanßmann Dynamics of Hamiltonian Systems; in Encyclopaedia of Complexity and Systems Science, Springer (2008) [29] ? J. Jost, Riemannian geometry and geometric analysis, Springer Verlag, 1995. [30] ? Klingenberg Lectures on closed geodesics Springer [31] ? V.V.Kozlov Calculus of variations in the large and classical mechanics Russian Maths. Surveys 40, pages 37-71, 1985 [32] ? Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory Springer-Verlag, 2004 [33] ? B. Leimkuhler, Sebastian Reich Simulating Hamiltonian Dynamics CUP, 2005 [34] J. Marsden, T.S. Ratiu and J. Scheurle, Reduction theory and the Lagrange-Routh equations, J. Math. Phys., 41, 3379-3429, 2000 [35] ? J. Mawhin and M. Willhelm, Critical point theory and Hamiltonian systems, Springer Verlag, 1990.

39

[36] C.McCord, J.Montaldi, M.Roberts, L.Sbano grangian Systems Proc. Equadiff2003

Relative Periodic Orbits of Symmetric La-

[37] ? K.Meyer and G.R. Hall Introduction to Hamiltonian Systems and the N-Body Problem Springer Verlag 1991 [38] ? K.Meyer Periodic solutions of the N -body problem LNM 1719, Springer 1999 [39] J.Montaldi, M.Roberts, I Stewart Periodic solutions near equilibria of symmetric Hamiltonian systems Phil. Trans. R. Soc. Lon. A 325, pages 237-293, 1988 [40] J.Montaldi, M.Roberts, I Stewart Existence of nonlinear normal modes of symmetric Hamiltonian systems Nonlinearity 3, pages 695-730, 1990 [41] J.Montaldi, M.Roberts, I Stewart Stability of nonlinear normal modes of symmetric Hamiltonian systems Nonlinearity 3 pages 731-772, 1990 [42] ? J. Montaldi, P.L. Buono, F. Laurent Poltz Symmetric Hamiltonian Bifurcations in Geometric Mechanics and Symmetry: the Peyresq Lectures Cambridge University Press, 2005 Edited by J.Montaldi, T.Ratiu [43] F.J. Munoz-Almaraz, E. Freire, J. Galan, E. Doedel and A. Vanderbauwhede Continuation of periodic orbits in conservative and Hamiltonian systems [44] S.P. Novikov The Hamiltonian formalism and a multivalued analogue of Morse theory, Russian Math. Surveys, 37, 1 - 56, 1982 [45] D.Offin A Class of Periodic Orbits in Classical Mechanics Jou.of Diff.Equations 66, 9-117 1987 [46] J.Ortega, T.Ratiu Singular Reduction of Poisson Manifolds Letters in Mathematical Physics 46: 359372, 1998 [47] ? H. Poincar´e Le M´ethodes Nouvelles de la M´ecanique C´eleste Gauthiers-Villars, Paris [48] ? P.H. Rabinowitz Minimax Methods in Critical Point Theory with Applications to Differential Equations, C.B.M.S. Reg. Conf. Ser. No. 56, Amer. Math. Soc., Providence, RI, 1986. [49] ? J. Sanders, F. Verhulst, J. Murdock Averaging Methods in Nonlinear Dynamical Systems Springer AMS 59, 2007 [50] T. Schmah, A cotangent bundle slice theorem Diff. Geom. Appl. 25, 101-124, 2007 [51] C. Sim´ o New families of solutions in N -body problems European Congress of Mathematics, Vol. I (Barcelona, 2000), pages 101–115, Progr. Math., 201, Birkhuser, Basel, 2001 [52] H.Seifert Periodische Bewegungen mechanischer Systeme, Math.Z. 51, 197, 216 (1948) [53] ? M. Struwe, Variational methods, Springer Verlag, 1990. [54] ? S.Terracini nbody problem and choreographies; in Encyclopaedia of Complexity and Systems Science, Springer (2008)

40

[55] ? T. Ratiu, E. Sousa Dias, L. Sbano, G. Terra, R. Tudora A crush course in geometric mechanics in Geometric Mechanics and Symmetry: the Peyresq Lectures Cambridge University Press, 2005 Edited by J.Montaldi, T.Ratiu [56] ? F. Verhulst Nonlinear Dynamical Equations and Dynamical Systems Springer UT, 1990 [57] A.Weinstein Normal Modes for Nonlinear Hamiltonian Systems Inventiones Math.. 20, pages 47-57, 1973 [58] A.Weinstein Bifurcations and Hamilton Principle Math.Z. 159, pages 235-248, 1978 [59] ? V.A. Yakubovich and V.M. Starzhinsky Linear Differential Equations with Periodic coefficients vol 1,2 Wiley 1975 [60] ? G.W. Whitehead, Elements of homotopy theory (3rd edn), Springer Verlag, 1995. [61] C.Wulf, M.Roberts Hamiltonian Systems Near Relative Periodic Orbits Siam J. Applied Dynamical Systems 1, No. 1, pages 143, 2002

41