First integrals and integrating factors of second order ...

First integrals and integrating factors of second order autonomous systems Tam´as Kalm´ar-Nagy∗ Department of Fluid Mechnics, Faculty of Mechanical Engineering Budapest University of Technology and Economics Bal´azs S´andor† Department of Hydraulic and Water Resources Engineering Faculty of Civil Engineering Budapest University of Technology and Economics Water Management Research Group - Hungarian Academy of Sciences

Abstract We present a new approach to the construction of first integrals for second order autonomous systems without invoking a Lagrangian or Hamiltonian reformulation. We show and exploit the analogy between integrating factors of first order equations and their Lie point symmetry and integrating factors of second order autonomous systems and their dynamical symmetry. We connect intuitive and dynamical symmetry approaches through one-to-one correspondence in the framework proposed for first order systems. Conditional equations for first integrals are written out, as well as equations determining symmetries. The equations are applied on the simple harmonic oscillator and a class of nonlinear oscillators to yield integrating factors and first integrals.

1

Introduction

The importance of first integrals of differential equations have long been recognized. First integrals are conserved quantities of the underlying dynamical system, providing qualitative information about its behavior. A first integral can be used to reduce the order of the differential equation, and a complete set of first integrals determines the solution of the equation. First integrals can prove useful in the stability analysis of the system: a first integral may be used to construct a Lyapunov function [1] to be used with the second method of Lyapunov [2]. First integrals can many times be related to the symmetries of the problem, but symmetries are not always necessarily related to first integrals. Gonz´ alez-L´ opez [3] discusses the connection between point symmetries and the integrability by quadratures of second-order ordinary differential equations. He provides an example of a family of secondorder ordinary differential equations integrable by quadratures whose point symmetry group is trivial, refuting the widespread belief that the existence of nontrivial point symmetries is a necessary condition for the integrability by quadratures of ordinary differential equations. The knowledge symmetries is important, sometimes just as important as the conserved quantity itself. Excellent textbooks for symmetry methods include [4] and [5]. Many different techniques have been published for deriving first integrals. Ad hoc construction of first integrals is quite widespread, some less rigorous than others. Direct construction of first integrals is shown, for example, by Sarlet and Bahar [6]. Sarlet and Bahar’s method consists of trying to construct a first integral in a manner similar to the one used in obtaining the energy integral for conservative systems, namely that of multiplying the equation of motion by an appropriate integrating factor. The most systematic, group-theoretical approaches include Noether’s theorem and its various generalizations and analysis by one-parameter families of infinitesimal transformations. The theory of Lie groups can be applied directly to any differential equation. The purpose of this paper is to present a new approach to the construction of a first integral for second order autonomous systems without invoking a Lagrangian or Hamiltonian reformulation. This paper is strongly influenced by the work of Olver [7, 8]. The book by Bluman and Anco [9] describes the connection between the integrating ∗ [email protected] † [email protected]

1

factor and the Lie point symmetry of first order systems, helping us to recognize the similar relation between the integrating factor and dynamical symmetries of second order autonomous systems. In particular, the natural space for studying symmetries is the 3-dimensional jet space (t, x, x) ˙ for first order equations and (x, x, ˙ x ¨) for second order autonomous equations. In the first order case a first integral is a family of solution curves and symmetries map solution curves into solution curves. For second order systems a first integral is a family of phase space curves and dynamical symmetries map phase space curves to phase space curves. A paper by Muriel and Romero [10] introduces a new class of symmetries (λ-symmetries) that strictly include Lie symmetries, but is more general. Later, Muriel and Romero [11] investigated the relationship between integrating factors and λ-symmetries for ordinary differential equations. In a 2009 paper Muriel and Romero [12] studied first integrals, integrating factors and λsymmetries of second-order differential equations. The knowledge of a λ-symmetry permits the determination of an integrating factor or first integral via coupled first-order linear partial differential equations. These methods include and complete other methods to find integrating factors or first integrals that are based on variational derivatives or the Prelle–Singer method [13]. Cheb-Terrab and Roche [14] presented what they termed a systematic algorithm for the construction of integrating factors for second order ordinary differential equations. They showed that there were instances of ordinary differential equations without Lie point symmetries which were solvable with this algorithm. Leach and Bouquet [15] demonstrate that the existence of integrating factors is paralleled by the existence of suitable Lie symmetries which enable one to reduce the equations to quadratures thereby emphasising the fact that integrability relies upon symmetry. In this paper we consider dynamical systems of the form x ¨ = f (x, x). ˙

(1)

Hale and Kocak [16] defines a first integral ω (x (t) , x˙ (t)) as a real-valued C1 function that is not constant on any open subset of R2 if the function ω is constant along every solution of the planar differential equation (1). That is, for any solution x (t) satisfying x (0) = x0 , x˙ (0) = v0 , the composite function satisfies ω (x (t) , x˙ (t)) = ω (x0 , v0 ) for all t for which the solution is defined. In this paper, ω and f are required to be sufficiently smooth: we assume that their second derivatives exist, i.e. ω, f ∈C2 . We present two approaches to calculate integrating factors and first integrals for Eq. (1). The first method gives a simple conditional equation for the integrating factors in the form Λ = Λ(x, x), ˙

(2)

which has a straightforward connection with similar equations resulting from variational derivatives or λ-symmetries. The second approach deduces direct relation between the integrating factors and a generalised Lie-symmetry (called dynamical symmetry) of Eq. (1). Both of these approaches are similar with procedures for first order systems x˙ = f (t, x). In particular, we exploit the analogy between first order and second order autonomous systems regarding their natural “embedding” spaces (their so-called jet spaces), both of them isomorphic to R3 . Therefore we first present the methods with first order systems (3) first, then we turn our attention to the second order sytems (1).

2

Intuitive determination of integrating factors and first integrals: first order systems

Let us consider the first order system (the independent variable is placed first) x˙ = f (t, x).

(3)

ω(t, x) = const.

(4)

The family of solution curves serve as first integrals of Eq. (3). The total derivative Dt of the first integral ω (t, x) must be zero, i.e. Dt ω (t, x) =

dω (t, x) = ωt + xω ˙ x = 0, dt

(5)

where partial differentiation is denoted with a subscript. Since x˙ − f = 0, we can write the following: ωt + xω ˙ x = Λ(t, x)(x˙ − f ), 2

(6)

where Λ(t, x) is the integrating factor. Rearranging Eq. (6) we get (ωx − Λ) x˙ + ωt + f Λ = 0.

(7)

A sufficient condition for Eq. (7) to hold is ωx = Λ, ωt = −f Λ.

(8)

Equating the mixed second derivatives (i.e. ωxt = ωtx ) results in a partial differential equation for the integrating factor: Λt + (f Λ)x = 0. (9) Remark 1 The more general formalism on writing the determining equations of the integrating factors of ordinary differential equations is based on the variational derivative of the differential algebraic expression [9, 17, 18, 19] θ(t, x, x) ˙ = Λ(x˙ − f ).

(10)

This variational derivative is also called Euler-operator and has the following form for one independent and one dependent variable: ∞ X ∂ ∂ ∂ ∂ − Dt + Dt2 − ... . (11) (−Dt )j j = E= ∂x ∂x ∂xt ∂xtt j=0 Applying this operator to θ(t, x, x) ˙ truncates the infinite sum at j = 1, and we arrive directly to expression (9): E(θ) =

∂θ ∂θ − Dt = −Λt − (f Λ)x = 0. ∂x ∂ x˙

(12)

In other words, the Euler operator annihillates the total derivatives, i.e. E(θ) = E(Dt ω) ≡ 0.

2.1

(13)

Connection between integrating factors and Lie-point symmetries

This Section is based on the books of Olver [8], Bluman and Anco [9], and Cohen [20]. Let us first look at the geometrical context of first order equations. The surface x˙ = f (x, t) is embedded in the jet space J (1) = ((t, x), x) ˙ ' R3 ,

(14)

where (t, x) is the total space of the independent and dependent variables, and x˙ spans the first jet space [9, 7]. This natural embedding will provide the connection with the treatment of second order autonomous equations. A one-parameter Lie transformation group of the plane [t, x] (the plane of the independent and dependent variables) transforms points (t, x) into points (t˜, x ˜) according to (t˜, x ˜) = g · (t, x) ,

(15)


where g = g t (t, x, ε), g x (t, x, ε) is a group element and ε is the parameter. The infinitesimal generator of the corresponding Lie-algebra has the form ∂ ∂ (16) v = ξ(t, x) + η(t, x) , ∂t ∂x where dg t dg x ξ= and η = . (17) dε ε=0 dε ε=0 For the interpretation of Lie-point symmetries of Eq. (3) we need to characterise the jet space (14) with the corresponding first (because the highest derivative is first order - for a second-order system the second prolongation will be used) extension/prolongation of the group action or the infinitesimal generator (16): v(1) = ξ

∂ ∂ ∂ +η + η (1) , ∂t ∂x ∂ x˙ 3

(18)

where η (1) = Dt η − xD ˙ t ξ = ηt + (ηx − ξt )x˙ − ξx x˙ 2

(19)

expresses the transformation of the x˙ derivative in the first jet under the group action. A group g, with generator (16) is a Lie-point symmetry group of Eq. (3) if it maps solution curves into solution curves. Or equivalently, if the differential algebraic expression F(t, x, x) ˙ = x˙ − f (t, x)

(20)

v(1) (F) = 0,

(21)

F = 0.

(22)

vanishes under the extended generator (18): provided Condition (21) means that the graph of f (t, x) (or the 0 level set of F) is invariant under the extended group action, while mapping solutions into solutions on the plane [t, x]. Since the equation is first order, the first integrals (4) are also families of solution curves ω(t, x) =const., i.e. these are conform-invariants to the generator (16) v (ω) = Ω(ω),

(23)

where Ω is an arbitrary function of the first integrals. The connection between integrating factors and Lie-point symmetries arising from the pairing of condition (6) on the total derivative of the first integrals, and condition (23) on the conformal invariance of the first integrals as families of solution curves - on the [t, x] plane, with Ω ≡ 1 (without loss of generality): Dt ω = ωt + xω ˙ x = ωt + f ωx = 0,

(24)

v (ω) = ξωt + ηωx = 1.

(25)

From this system of equations we get (η 6= ξf ) ωx =

1 , η − ξf

ωt = −

f , η − ξf

(26)

and substituting Eq. (26) into Eq. (24) results in Dt ω =

1 (x˙ − f ). η − ξf

(27)

Comparing this with Eq. (6) we find the integrating factor in terms of the coefficient functions of the symmetry generator as 1 Λ(t, x) = . (28) η − ξf The Lie-point symmetries of (3) can be calculated through the coefficients ξ and η of the generators with symmetry condition (21), which can be expanded (with x˙ = f ) as v(1) (F) = η (1) − ξft − ηfx = ηt + (ηx − ξt )f − ξx f 2 − ξft − ηfx = 0.

(29)

Now we are ready to state the following: there is a one-to-one correspondance between the Lie-point symmetries and the integrating factors of (3), because substituting Eq. (28) into the determining equation of the integrating factors (9) results in Λt + (f Λ)x = −

  1 2 η + (η − ξ )f − ξ f − ξf − ηf t x t x t x (η − ξf )2 1 =− v(1) (F) = 0. (η − ξf )2

(30)

Remark 2 The choice η = ξf

4

(31)

with arbitrary ξ solves the condition equation of symmetries (29) and hence provides infinite number of symmetries. However, these are trivial symmetries with generators of the form   A = ξ ∂t + f ∂x , (32) with which A (ω) = 0,

(33)

so each solution curve is invariant with respect to these symmetries, moreover no integrating factor corresponds to these. Dealing with these types of symmetries does not help to solve the original equation. Remark 3 The vector field (32) with ξ = 1 is called as the vector field of the equation. The commutator (or Lie-bracket) of the vector field of the equation and a Lie-point symmetry generator (16) is the following vector field:

[A, v] = [∂t + f ∂x , ξ∂t + η∂x ] = A(ξ) − v(1) ∂t + A(η) − v(f ) ∂x . (34) This new vector field acts on the first integral ω as

as well as



[A, v](ω) = A(ξ) − v(1) ωt + A(η) − v(f ) ωx ,

(35)

    [A, v](ω) = A v(ω) − v A(ω) = A(1) − v(0) ≡ 0.

(36)

Since ωt and ωx are not necessarily zero, the coefficients in Eq. (34) must be, so these are proportional to the coefficients in Dt ω = ωt + f ωx = 0, (37) hence (with v(1) ≡ 0) A(ξ) =

A(η) − v(f ) = ρ(t, x). f

(38)

Considering this in the commutator (34) results in [A, v] = ρA,

(39)

hence the Lie-point symmetry generators and the vector field of the equations are conform-invariants due to the connection between the first integrals and Lie-point symmetries.

3

Analysis of second order autonomous equations

Here we try to construct integrating factors Λ(x, x) ˙ for second order systems of the form (1) x ¨ = f (x, x) ˙ .

(40)

Analogously with the first order case (6), with ω = ω(x, x) ˙ and Λ = Λ(x, x) ˙ we can write the following formula: Dt ω = xω ˙ x+x ¨ωx˙ = Λ(¨ x − f ) = 0,

(41)

(ωx˙ − Λ)¨ x + xω ˙ x + f Λ = 0.

(42)

or equivalently Sufficient conditions for this to hold are ωx˙ = Λ, xω ˙ x = −f Λ.

(43)

The conditional equation (equality of the mixed derivatives) ωxx ˙ = ωxx˙ yields x˙ 2 Λx + x(f ˙ Λ)x˙ − f Λ = 0, which is the counterpart of (9) for second order systems (1). 5

(44)

Theorem 4 The conditional equation (44) of the integrating factors is consistent the ones resulting from the more general formalism of Euler-operators. Proof. Applying the Euler-operator (11) on the differential algebraic expression θ(x, x, ˙ x ¨) = Λ(¨ x − f)

(45)

results in

∂θ ∂θ ∂θ − Dt + Dt2 = A(x, ˙ x)¨ x + B(x, ˙ x) = E(Dt ω) ≡ 0. ∂x ∂ x˙ ∂x ¨ A sufficient condition for this equality to hold is that the expressions A and B are zero, i.e. E(θ) =

(46)

A(x, ˙ x) = 2Λx + xΛ ˙ xx˙ + (f Λ)x˙ x˙ = 0,

(47)

B(x, ˙ x) = −(f Λ)x + x(f ˙ Λ)xx˙ + x˙ 2 Λxx = 0.

(48)

and It can be easily checked that expressions (47) and (48) are derivatives of the left-hand side of Eq. (44): (x˙ 2 Λx + x(f ˙ Λ)x˙ − f Λ)x˙ = xA( ˙ x, ˙ x), (x˙ 2 Λx + x(f ˙ Λ)x˙ − f Λ)x = B(x, ˙ x).

(49)

Thus expression (44) is consistent with the ones arising from Euler-operator, moreover it is easier to solve than system (47)-(48) for the integrating factor.

3.1

Connection between integrating factors and various symmetries

From the aspect of Lie-groups, the two-jet J (2) = (x, x, ˙ x ¨ ) ' R3

(50) 3

spanned by the variable and its derivatives of the equation is isomorphic to R , just like the one-jet (14) in the first order case. The main difference from the first order case is that we considered invariant families of solution curves as first integrals in the first order case and now we are considering invariant families of phase space curves. The symmetries considered are not Lie-point symmetries, but a special type of generalised symmetries, namely dynamical symmetries. 3.1.1

Consistency with a λ-symmetry

The concept of λ-symmetries is based on a special generator field, which is similar to a Lie-point symmetry generator, but its extension on the jet space is more general than the prolongation of Lie-point symmetries [11]. For example, for the Lie-point symmetry generator ∂ ∂ v = ξ(t, x) + η(t, x) (51) ∂t ∂x the first extension has the following form: v(1) = ξ

∂ ∂ ∂ +η + η (1) , ∂t ∂x ∂ x˙

(52)

with η (1) = Dt η − xD ˙ t ξ + λ(η − xξ), ˙

(53)

which, compared with Eq. (19), is more general through the λ-part. In [12] a connection had been deduced between the integrating factors and λ-symmetries, which are connected to certain Lie-point symmetries. The conditional equation is Λx + (λΛ)x˙ = 0,

(54)

Q = η − ξ x˙

(55)

Dt Q . Q

(56)

where is the characteristic of v and λ=

6

Theorem 5 The conditional equation (44) for the integrating factors is a special form of (54), corresponding to the v = ∂t Lie-point symmetry. Proof. The time translation is a Lie-point symmetry of every equation of the form (1), f , x˙

(57)


1 2 x˙ Λx + x(f ˙ Λ)x˙ − f Λ . x˙ 2

(58)

Q = −x, ˙

λ=

the conditional equation is Λx + (λΛ)x˙ =

3.1.2

One-to-one correspondence with the dynamical symmetries

Instead of the Lie-point symmetry generator (16) which we considered in the first order case, let us introduce the infinitesimal generator of dynamical symmetries of (1), which has the general form [3, 8, 9] v = ξ(t, x, x) ˙

∂ ∂ ∂ + η(t, x, x) ˙ + ζ(t, x, x) ˙ . ∂t ∂x ∂ x˙

(59)

The vector field of Eq. (1) is ∂ ∂ ∂ + x˙ +f , ∂t ∂x ∂ x˙ and the expression (59) is a dynamical symmetry, if A=

[A, v] = ρ(t, x, x)A. ˙

(60)

(61)

We consider the following vector field (this is a special case of a dynamical symmetry which exploits our analogy between jet spaces for first and second order equations): v = η(x, x) ˙ with its first extension v(1) = η

∂ , ∂x

∂ ∂ + η (1) , ∂x ∂ x˙

(62)

(63)

where η (1) = Dt η = A(η) = xη ˙ x + f ηx˙

(64)

is a dynamical symmetry generator of Eq. (1). Theorem 6 The differential equation (1) x ¨ = f (x, x) ˙ provides dynamical symmetries with generator (63) if η(x, x) ˙ satisfies the following parabolic partial differential equation: x˙ 2 ηxx + 2xf ˙ ηxx˙ + f 2 ηx˙ x˙ + xf ˙ x ηx˙ + (f − xf ˙ x˙ )ηx − fx η = 0. (65) The corresponding transformation maps phase space curves into phase space curves, i.e. the first integrals ω = ω(x, x) ˙ = const. are invariant families of phase space curves with the corresponding integrating factors with determining equation (44) x˙ 2 Λx + x(f ˙ Λ)x˙ − f Λ = 0. Further, there is a one-to-one correspondence between these symmetries and integrating factors of the following form Λ(x, x) ˙ =

x˙ . ηx˙ xf ˙ + ηx x˙ 2 − ηf

7

(66)

Proof. In analogy with the first order case and connection equations of the invariant family of curves (25) we can write the following formulas: Dt ω = xω ˙ x + f ωx˙ = 0, v

(1)

(ω) = ηωx + η

(1)

ωx˙ = 1.

(67) (68)

Equation (68) provides that the space space curves are invariant family of curves of the dynamical symmetry generator (63). Using this equation and formula (64) for η (1) we can write the following: Dt ω =

x˙ (¨ x − f ) = Λ(¨ x − f) ηx˙ xf ˙ + ηx x˙ 2 − ηf

(69)

which proves the formula (66) and the invariance of the families of space space curves. The dynamical symmetries of (1) corresponding to the generator (63) can be calculated from (second prolongation, because this is a second order system) v(2) (F) = 0, (70) where F =x ¨ − f, and v(2) = η

∂ ∂ ∂ + η (1) + η (2) . ∂x ∂ x˙ ∂x ¨

(71) (72)

Here η (2) = Dt η (1) = A(η (1) ) = ηxx x˙ 2 + 2ηxx˙ xf ˙ + ηx˙ x˙ f 2 + ηx˙ (xf ˙ x + fx˙ f ) + ηx f.

(73)

Expanding the condition of symmetry (70) results in x˙ 2 ηxx + 2xf ˙ ηxx˙ + f 2 ηx˙ x˙ + (f − xf ˙ x˙ )ηx + xf ˙ x ηx˙ − fx η = 0, which is the symmetry condition (65) stated in the Theorem. This is a parabolic equation because its discriminant 2 is (2xf ˙ ) − 4x˙ 2 f 2 = 0. The one-to-one correspondence between the dynamical symmetries and integrating factors can be proven with substituting (66) into the integrating factor determining equation (44) x˙ 2 Λx + x(f ˙ Λ)x˙ − f Λ = xΛ ˙ 2 v(2) (F) = 0.

(74)

Remark 7 The geometrical interpretations in the jet space do not necessarily follow straightforwardly when one considers dynamical symmetries. We found one-to-one correspondence betweeen dynamical symmetries and integrating factors with direct connection of the families of phase space curves. However, we had considered a special form of generators, namely Eq. (63). However, since time translation is a point symmetry of every equation of the form (1), the graph of the differential algebraic expression F = x ¨ − f (x, x) ˙ in the two-jet (50) simply consists of the same sections translated along the t axis. The transformations corresponding to (63) are maps on the (x, x) ˙ phase space and the first integrals are invariant families of phase space curves through (68) and they are also symmetries of Eq. (1) through the condition (70). The simple structure of this two-jet is also revealed by the commutator
∂
∂ + A(η (1) ) − v(1) (f ) ∂x ∂ x˙
∂ (1) (1) ∂ (2) = η −η + v (F) ≡ 0, ∂x ∂ x˙

[A, v(1) ] = A(η) − v(1) (x) ˙

(75)

if the symmetry condition (70) holds. It means that the flows of the vector fields A and v(1) commute on the surface of the graph of first integrals ω(x, x). ˙

8

4

Application for oscillators

There are numerous papers dealing with the important topic of determining elementary first integrals of twodimensional autonomous systems of differential equations. Prelle and Singer [21] proposed a procedure for solving nonlinear first-order ordinary differential equations where the right-hand side is a rational function of the dependent and independent variables. It assumes a quasi-polynomial form for the integrating factor with a specified polynomial degree. The application of this procedure for several dynamical systems is given in [22]. Duarte et al. [13] have extended the Prelle-Singer procedure to second-order ODEs where the right hand side is a rational function of the independent variable, the dependent variable and its first derivative. Chandrasekar et al. [23] undertake the study of a generalized second order nonlinear equation using a generalized extended Prelle-Singer procedure. Ibragimov [19] introduces the concept of the adjoint for nonlinear equations and constructs Lagrangians and integrating factors. Integrating factors and the associated first integrals for Lienard-type and frequency-damped oscillators were considered in [24] where first integrals are derived from the method to compute λ-symmetries and the associated reduction algorithm. The knowledge of a λ-symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first-order linear systems of partial differential equations. Here we apply our proposed approaches to second-order oscillators x ¨ = f (x, x), ˙ i.e. we solve Eq. (44) for determining the integrating factor x˙ 2 Λx + x(f ˙ Λ)x˙ − f Λ = 0, (76) and calculate the first integral ω from ωx˙ = Λ, f ωx = − Λ. x˙

(77)

We also solve the symmetry determining equation (65) x˙ 2 ηxx + 2xf ˙ ηxx˙ + f 2 ηx˙ x˙ + xf ˙ x ηx˙ + (f − xf ˙ x˙ )ηx − fx η = 0

(78)

with the formula (66) for the integrating factor Λ=

x˙ . ηx˙ xf ˙ + ηx x˙ 2 − ηf

(79)

We start out with the simple undamped harmonic oscillator x ¨ + x = 0.

(80)

The PDE for the integrating factor is (f = −x) x˙ 2 Λx − x(xΛ) ˙ x˙ + xΛ = (xΛ ˙ x − xΛx˙ ) x˙ + xΛ = x˙ 2 Λx − xxΛ ˙ x˙ + xΛ = 0. We find

 Λ = xg ˙

x2 x˙ 2 + 2 2

(81)

 (82)

where g is an arbitrary function. Taking g = 1 (so that Λ = x) ˙ the PDE system (77) determining the first integral is ωx˙ = x, ˙ ωx = x,

(83)

from which we we recover the expected first integral ω=

x2 x˙ 2 + . 2 2 9

(84)

The symmetry determining equation x˙ 2 ηxx − 2xxη ˙ xx˙ + x2 ηx˙ x˙ − xη ˙ x˙ − xηx + η = 0

(85)

is satisfied by the linear function (dilation) η = ax + bx, ˙ with which Λ=

1 x˙ = x˙ . −ηx˙ xx ˙ + ηx x˙ 2 + ηx a (x2 + x˙ 2 )

(86) (87)

This yields
ln x2 + x˙ 2 ω= + c. 2a 2

(88)

2

x˙ With the choice a = 1/2, c = − ln 2 this results ω = ln x + , a function of the simple first integral (84). 2 Next, we investigate the integrating factors and first integrals for the damped harmonic oscillator

x ¨ + 2ζ x˙ + x = 0.

(89)

f = −2ζ x˙ − x.

(90)

Here We first look at the symmetry determining equation 2

(91)

η = ax + bx, ˙

(92)

x˙ 2 ηxx − 2x˙ (2ζ x˙ + x) ηxx˙ + (2ζ x˙ + x) ηx˙ x˙ − xη ˙ x˙ − xηx + η = 0. This is again satisfied by the dilation This yields Λ=

x˙ x˙ = . 2 2 ηx˙ xf ˙ + ηx x˙ − ηf a (x + x˙ 2 + 2ζ xx) ˙

(93)

While Eq. (76) can not easily be solved for Λ, it can easily be shown that this integrating factor solves the PDE for the integrating factor x˙ 2 Λx − x((2ζ ˙ x˙ + x) Λ)x˙ + (2ζ x˙ + x) Λ = (xΛ ˙ x − (2ζ x˙ + x) Λx˙ ) x˙ + xΛ = 0.

(94)

The “first integral” can then be calculated as (with the choice a = 1/2)
x + ζ x˙ 2ζ ω = ln x2 + x˙ 2 + 2ζ xx ˙ −p atanh p . 2 ζ −1 x˙ ζ 2 − 1

(95)

We note here that this first integral is formal in the sense that it has an obvious singularity at x = x˙ = 0 and hence it does not satisfy the smoothness criterion of our definition for first integrals. The function (93) appears in Cantwell’s book ([25], p. 116). While we found this function from the original equation through the dynamical symmetry, Cantwell rewrites the oscillator equation into Hamiltonian form. He then makes this nonexact first order equation exact with an integrating factor (with the method described in 2.1). This is the crucial point: the dynamical symmetry of the second order autonomous ODE and the Lie-point symmetry of the corresponding Hamiltonian system are the same dilation symmetry in the same phase space. We now turn our attention to nonlinear oscillators. For the Duffing oscillator x ¨ + x + δx3 = 0,

f = −x − δx3 ,

we get the integrating factor by solving Eq. (76) as   2 x˙ 2 x4 x + +δ . Λ = xg ˙ 2 2 4

10

(96)

(97)

With g = 1 the PDE system (77) determining the first integral is ωx˙ = Λ = x, ˙ f ωx = − Λ = x + δx3 , x˙ from which we recover the well-known first integral

(98)

x2 x˙ 2 x4 + +δ . 2 2 4

ω=

(99)

The nonlinear oscillator x ¨ + (1 + x) ˙ x = 0,

(100)

has been studied in the context of relaxation oscillations [26] and laser oscillations [27]. Beatty and Mickens [28] and later Mickens [29] and Kalm´ ar-Nagy and Erneux [30] investigated the nonlinear oscillator
(101) x ¨ + 1 + x˙ 2 x = 0. These oscillators are characterized by a velocity-dependent stiffness coefficient and depend on only one parameter. Their general form is x ¨ + x˙ n x + x = 0, (102) and thus f = −x (1 + x˙ n ) .

(103)

The PDE for the integrating factor is (xΛ ˙ x − x (1 + x˙ n ) Λx˙ ) x˙ + (x + (1 − n) xx˙ n ) Λ = 0, which yields
! x2 +12 F 1, n2 , 1 + n2 , −x˙ n x˙ 2 , 2

x˙ Λ= g 1 + x˙ n

(104)

where 12 F denotes the hypergeometric function and 1 2F



2 2 1, , 1 + , −x˙ n n n

 =

∞ X (1)n k=0

1+

2 n n
2 n n




−x˙ kn . k!

(105)

Here ()n denotes the Pochhammer symbol  (q)n =

1 q (q + 1) .. (q + n − 1)

n=0 . n>0

(106)

Taking g = 1, the integrating factor becomes x˙ , 1 + x˙ n and the PDE system (77) determining the first integral is Λ=

ωx˙ = Λ =

(107)

x˙ , 1 + x˙ n

f ωx = − Λ = x. x˙

(108)

The corresponding first integral is x2 1 + F 2 4 For n = 1, 2, 4 we get the explicit first integrals ω=

  2 2 1, , 1 + , −x˙ n x˙ 2 . n n

x2 + x˙ − ln (1 + x) ˙ , 2
x2 1 ω= + ln 1 + x˙ 2 , 2 2 x2 1 ω= + arctan x˙ 2 . 2 2 ω=

11

(109)

(110) (111) (112)

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Conclusions

We presented an intuitive as well as a more formal approach to determine first integrals of second order systems. The geometrical analogy between first order nonautonomous systems and second order autonomous systems was elucidated by showing that in both cases the jet spaces were isomorphic to R3 . Partial differential equations for first integrals and for determining symmetries were written out. Application of these equations were shown on the simple harmonic oscillator (both in the undamped and damped case) and a class of nonlinear oscillators.

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