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Journal of Generalized Lie Theory and Applications
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Time transformation and reversibility of Nambu–Poisson systems

Klas MODIN1,2*

1Centre for Mathematical Sciences, Lund University, Box 118, SE–221 00 Lund, Sweden E-mail: [email protected]

2SKF Engineering & Research Centre, MDC, RKs–2, SE–415 50 G¨oteborg, Sweden

*Corresponding Author:
Klas MODIN
Centre for Mathematical Sciences, Lund University
Box 118, SE–221 00 Lund, Sweden
E-mail: [email protected]

Received date: September 10, 2008; Revised date: October 07, 2008

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Abstract

A time transformation technique for Nambu–Poisson systems is developed, and its structural properties are examined. The approach is based on extension of the phase space P into P¯ = P×R, where the additional variable controls the time-stretching rate. It is shown that time transformation of a system on P can be realised as an extended system on P¯, with an extended Nambu–Poisson structure. In addition, reversible systems are studied in conjunction with the Nambu–Poisson structure. The application in mind is adaptive numerical integration by splitting of Nambu–Poisson Hamiltonians. As an example, a novel integration method for the rigid body problem is presented and analysed.

Introduction

In 1973 Nambu [23] suggested a generalisation of Hamiltonian mechanics, taking the Liouville condition on volume preservation in phase space as a governing principle. Nambu postulated that the governing equations for a dynamical system on image should have the form

image (1.1)

where ǫ is the Levi–Civita tensor over n indices, and H1, . . . ,Hn−1 are smooth real valued functions on image called Hamiltonian functions. Notice that the vector field in equation (1.1) is source free (its divergence is zero), which implies that the corresponding phase flow is volume preserving.

Later Takhtajan [27] formalised Nambu’s framework by introducing the concept of Nambu– Poisson brackets on general phase space manifolds. Based on Takhtajan’s work the geometry of Nambu–Poisson structures has been explored in several papers [6,4,5,21,11,22,28,29].

In this paper we study time transformation of Nambu–Poisson systems. Such transformations are important in the construction and analysis of adaptive structure preserving numerical time stepping methods [26,10,3,7,24,18,2,20,19]. The idea is to obtain time step adaptivity by equidistant discretisation in the transformed variable, which corresponds to non-equidistant discretisation in the original time variable. Although numerical integration is a main motivation, the focus in the paper is not on numerical issues, but rather on structural properties.

The current section continues with a brief review of Nambu–Poisson mechanics, and of a time transformation method by Hairer and S¨oderlind [9]. The main results are in Section 2, where time transformation for Nambu–Poisson systems is developed. In Section 3, the Nambu–Poisson structure is studied in conjunction with reversibility. As an application, we show in Section 4 how to construct fully explicit, adaptive numerical integration methods based on splitting of the Nambu–Poisson Hamiltonians. In particular, a novel method for the free rigid body. Conclusions are given in Section 5.

We adopt the following notation. image denotes a phase space manifold of dimension n, with local coordinates x = (x1, . . . , xn). The algebra of smooth real valued functions on image is denotedimage Further,image denotes the linear space of vector fields onimageThe Lie derivative alongimage is denotedimage If X, Yimage then the commutatorimage supplies image with an infinite dimensional Lie algebra structure. Its corresponding Lie group is the set Diff image of diffeomorphisms on image with composition as group operation. (See McLachlan and Quispel [16] and Schmid [25] for issues concerning infinite dimensional Lie groups.) If Φ ∈ Diff image then Φ* denotes the pull-back map and Φ* the push-forward map imposed by Φ.

Nambu–Poisson mechanics

In Hamiltonian mechanics, the phase space manifold image is equipped with a Poisson structure, defined by a bracket operation {·1, ·2} : image that is skew-symmetric, fulfils the Leibniz rule and the Jacobi identity. Nambu–Poisson mechanics is a generalisation.

Definition 1.1. A Nambu–Poisson manifold of order k consists of a smooth manifold image together with a multilinear map

image

that fulfils:

• total skew-symmetry

{H1, . . . ,Hk} = sgn(σ){Hσ1 , . . . ,Hσk} (1.2a)

• Leibniz rule

{GH1, . . . ,Hk} = G{H1, . . . ,Hm} + H1{G,H2, . . . ,Hk} (1.2b)

• fundamental identity

image

Remark 1.1. The case k = 2 coincides with ordinary Poisson manifolds.

The first two conditions, total skew-symmetry (1.2a) and Leibniz rule (1.2b), are straightforward: they imply that the bracket is of the form

{H1, . . . ,Hk} = η( dH1, . . . , dHk)

for some totally skew-symmetric contravariant k–tensor η [27]. The third condition, the fundamental identity (1.2c), is more intricate. The range of possible Poisson–Nambu brackets is heavily restricted by this condition [27].

A Nambu–Poisson system on a Nambu–Poisson manifold of order k is determined by k − 1 Hamiltonian function H1, . . . ,Hk−1image. The governing equations are

image (1.3a)

which may also be written

image(1.3b)

where XH1, . . . ,Hk−1image is defined byimage The corresponding flow map is denoted image Notice that due to skew symmetry of the bracket, all the Hamiltonians H1, . . . ,Hk−1 are first integrals, which follows from equation (1.3a).

Due to the fundamental identity (1.2c), Nambu–Poisson systems fulfil certain properties which have direct counterparts in Hamiltonian mechanics (the case k = 2).

Theorem 1.1 (Takhtajan [27]). The set of first integrals of system (1.3) is closed under the Nambu–Poisson bracket. That is, if G1, . . . ,Gk are first integrals, then {G1, . . . ,Gk} is again a first integral.

Theorem 1.2 (Takhtajan [27]). The flow of system (1.3) preserves the Nambu–Poisson structure. That is,

image

or equivalently

image

Remark 1.2. The set of vector fields that fulfils equation (1.4) is denoted image Clearly image is closed under linear combinations, so it is a sub-space of image Further, sinceimageimage it is also closed under the commutator. Thus, image is a Lie sub-algebra of image Correspondingly, Diffη image denotes the Lie sub-group of Diff image that preserves the Nambu–Poisson structure. An element Φ ∈ Diffη image is called an η–map.

Remark 1.3. It is important to point out that in general not every image corresponds to a Nambu–Poisson system, i.e., a system of the form of equation (1.3). The reason is that the set of vector fields of the form of equation (1.3) is not closed under linear combinations.

There are also fundamental differences between Hamiltonian and Nambu–Poisson mechanics, i.e., between k = 2 and k ≥ 3. In particular there is the following result, conjectured by Chatterjee and Takhtajan [4] and later proved by several authors.

Theorem 1.3 ([6,1,22,11,13]). A totally skew-symmetric contravariant tensor of order k ≥ 3 is a Nambu–Poisson tensor if and only if it is locally decomposable about any regular point. That is, about any point image such that η(x) ≠ 0 there exist local coordinates (x1, . . . , xk, xk+1, . . . , xn) such that

image

Thus, every Nambu–Poisson tensor with k ≥ 3 is in essence a determinant on a sub-manifold of dimension k. It is not so for Poisson tensors.

Time transformation of dynamical systems

In this section we review the time transformation technique developed in Hairer and S¨oderlind [9]. Consider a dynamical system

image (1.5)

Its flow map is denoted image Introduce an extended phase spaceimage with local coordinates image The projectionimage is denoted , andimage is denoted π Let image and consider the extension of system (1.5) into

image(1.6)

The flows of the two systems are related by a reparametrisation t ↔ τ .

Theorem 1.4 (Hairer and S¨oderlind [9]). The flow of the extended system (1.6) restricted to image is a time transformation of the flow of system (1.5). That is,

image

Further, Q(x)/ξ is a first integral of system (1.6).

Proof. From equation (1.6) it follows directly that image is parallel with X. Thus,image andimage define the same phase diagrams. It remains to find the relation between t and τ . Sinceimage it follows from equation (1.6) that dt/ dτ = 1/ξ. Integration of this relation gives σ(τ, ¯x). Further, straightforward calculations and utilisation of the governing equations (1.6) show that d(Q(x)/ξ)/ dτ = 0.

Remark 1.4. It is clear that the time transformation is determined by Q. Since Q is strictly positive, the map σ( · , ¯x) :imageis bijective, i.e., the reparametrisation t ↔ τ is bijective.

In Hairer and S¨oderlind [9], the motivation for the extended time transformation (1.6) is to construct explicit adaptive numerical integrators for reversible systems. The key is that under reversibility of Q, the extended time transformation (1.6) preserves reversibility. First, recall the basic definitions of reversible systems.

Definition 1.2. Let R ∈ Diffimage

• A vector field image is called reversible with respect to R if R* ◦ X = −X ◦ R, or equivalently d(R(x))/ dt = −(X ◦ R)(x).

• A diffeomorphism Φ ∈ Diff image is called reversible with respect to R if R◦Φ=Φ-1◦ R.

It is a well known result that the flow of a system is reversible if and only if its corresponding vector field is reversible [12,8]. Now, concerning time transformation of reversible systems, it is straightforward to check the following result.

Theorem 1.5 (Hairer and S¨oderlind [9]). If imageis reversible with respect to R and imagethen the vector field image in equation (1.6) is reversible with respect to image

Nambu–Poisson extensions and time transformations

In this section we develop a time transformation technique for Nambu–Poisson systems. Let imagebe a Nambu–Poisson manifold of order k and η its Nambu–Poisson tensor. Consider again the extended phase space image Our first goal is to introduce a Nambu–Poisson structure onimage The most natural extension of the Nambu–Poisson tensor η is given by

image(2.1)

It is not obvious that the bracket corresponding to image will fulfil the fundamental identity (1.2c). For example, in the canonical Poisson case, i.e., k = 2, it is not so if n ≥ 3.

Lemma 2.1. If k ≥ 3 or k = n = 2, then ¯η given by equation (2.1) defines a Nambu–Poisson structure of order k + 1 on image

Proof. If k ≥ 3 then it follows from Theorem 1.3 that η is decomposable about its regular points, and when k = n = 2 it is obviously so. Thus, image is also decomposable about its regular points, so the assertion follows from Theorem 1.3.

The bracket associated with ¯η is denoted ¯{·, . . . , ·¯}.Letimage be the Hamiltonians for a Nambu–Poisson system on image , i.e., of the form of system (1.3). Further, let image and consider the system on imagegiven by

image (2.2)

Remark 2.1. A functions image is considered to belong toimage by the natural extensionimage Likewise,image is considered to be a function inimage depending on the parameter ξ. Thus, ¯ {·, . . . , ·¯} is defined also for elements in image and vice versa.

We continue with the main result in the paper. It states that time transformation of a Nambu–Poisson system can be realised as an extended Nambu–Poisson system.

Theorem 2.1. Let image and assume the conditions in Lemma 2.1 are valid. Then:

1. The extended system (2.2) is a Nambu–Poisson system.

2. Its flow restricted to image is a time transformation, determined by the additional first integral G, of the flow of system (1.3). That is,

image

where

image

Proof. The first assertion follows directly from Lemma 2.1, since ¯η is a Nambu–Poisson tensor. Since Hi for i = 1, . . . , k − 1 are independent of ξ, it follows from the definition (2.1) of ¯η that

image

Thus, for image the governing equations (2.2) are parallel with those of system (1.3a), i.e.,image defined the same phase diagram. The relation between τ and t is given byimage which, after integration, gives the desired form ofimage

It is straightforward to check the following corollary, which shows that the technique used by Hairer and S¨oderlind [9], reviewed in Section 1.2, is a special case.

Corollary 2.1. The case image coincides with the transformation (1.6) applied to system (1.3).

Reversible Nambu–Poisson systems

Recall that the time transformation by Hairer and S¨oderlind [9] is developed with reversible systems in mind. In the previous section we developed a similar approach, but based on the Nambu–Poisson framework. One may ask under what conditions a Nambu–Poisson system is reversible, and in what sense the time transformation technique developed above preserves reversibility. These questions are studied in this section.

As a first step, we have some results on necessary and sufficient conditions for a Nambu– Poisson system to be reversible.

Proof. Since R is a diffeomorphism it holds thatimage so the governing equations (1.3a) are equivalent to

image

This is equivalent to

image

if and only if condition (3.1) holds. The last set of equations is exactly the condition on XH1, . . . ,Hk−1 for reversibility with respect to R.

If R is a Nambu–Poisson map the assertion may be stated in the following way instead.

Corollary 3.1. Let R be a Nambu–Poisson map, i.e.,image Then XH1, . . . ,Hk−1 is reversible with respect to R if and only if

image

Proof. With F set to F ◦R, it is clear that the condition (3.2) is equivalent to the condition (3.1)

image

As a generalisation of Theorem 1.5, we now show in what way reversibility of a Nambu– Poisson system is preserved by the time transformed extended system (2.2).

Theorem 3.1. Let the system (1.3) be reversible with respect to R. Then the extended time transformed Nambu–Poisson system (2.2) is reversible with respect to image ifimage

Proof. Since ∂Hi/∂ξ = 0 we have

image

Since image maps ξ to ξ it holds that imageimage and

image

Altogether we now have

image

where the stipulation that system (1.3) is reversible have been used in conjunction with Proposition 3.1. Application of Proposition 3.1 again completes the assertion.

Application: numerical integration by splitting

The main motivation for extended time transformations is to construct adaptive numerical integration algorithms. By a numerical integrator for a dynamical system image we mean a family of near identity maps image such thatimage is an approximation of the exact flowimage Numerical solution “paths” are obtained by the discrete dynamical system imageimageThe integrator image is consistent of order p if image which in particular impliesimage It is explicit ifimagecan be computed by a finite algorithm. Notice that image is not a one parameter group, i.e.,image

When constructing numerical integrators, one typically tries to preserve as much as possible of the underlying qualitative structure of the exact flow. In our case, we like image to preserve the Nambu–Poisson structure, and in presence also reversibility. In addition, time step adaptivity is crucial in order for the integration method to be computationally efficient. Indeed, we would like to vary the step size h during the integration process according to the present local character of the dynamics, without destroying the structural properties of the method. The standard approach, motivating our work, is to utilise a time transformation t ↔ τ that preserves the structure of the original system, and then construct a τ–equidistant numerical integrator for transformed system. An equivalent view point is to say that the time transformation should regularise the problem, so that it becomes easier to integrate numerically.

Splitting is a compelling technique for the construction of structure preserving integrators [17]. The basic idea is as follows. Let imagebe a Lie sub-algebra of image and let DiffAimage be the corresponding Lie sub-group of Diff imageAssume that imagecan be splitted into explicitly integrable sub-system, each of which is a system inimageThat is, X = X1 +. . . +Xk where, image andimagecan be computed explicitly. A numerical integrator for X is obtained by image It is clear that image is an approximation ofimage, and thatimageFurther, by the Baker–Campbell–Hausdorff (BCH) formula, it follows that imageis the exact flow of a modified vector fieldimage This information is crucial for the analysis of image For example, ifimageis the Lie-algebra corresponding to a Poisson structure on image then image will exactly conserve a modified Hamiltonian, which isimageclose to the Hamiltonian for the original problem [8].

Remark 4.1. Due to convergence issues, the BCH formula needs to be truncated, which implies that assertions on image, coming from image are valid only for exponentially long times, i.e., up to time scales of order image See Hairer et. al. [8] for details.

Our notion for the construction of integrators is to utilise the results in Section 2–3, and consider splitting of the individual Nambu–Poisson Hamiltonians.

Let η be a Nambu–Poisson tensor. The set of Nambu–Poisson maps which are reversible with respect to R is denoted imageIf image then in general we have

image

Thus, imageis not closed under composition, so it is not a sub-group of Diff imageHowever, imageis closed under the symmetric group operation imagewhich we write asimageFurther, from the symmetric BCH formula (cf. [15]) it follows that if image then the vector field Z such thatimagebelongs to image

Remark 4.2. For near identity maps, imageis defined by taking its representation image and then settingimageIn our case, Φ will always be an exact flowimagein which case image

We now give a result concerning reversible systems, which is of use for the analysis of periodic numerical paths of reversible splitting methods.

Lemma 4.1. Let image be reversible with respect toimage Assume that the set imageof fix-points of R is non-empty and that imageis a solution curve of X for which there exists imagewith t1 < t2 such thatimage Then γ is periodic.

Proof. For simplicity assume that t1 = 0 and t2 > 0, which is not a restriction. The curve imageis also a solution curve due to reversibility. Further, since R restricted to U is the identity map we have the equalities imageand imageDue to uniqueness of solutions the first equality implies γ2 = γ, which in conjunction with the second equality implies that imageThus γ returns to the same point twice, so it is periodic.

image

Rigid body problem

The Euler equations for the free rigid body is a Nambu–Poisson system on the phase space image, equipped with the canonical Nambu–Poisson structure image Its two Hamiltonians are total angular momentumimage and kinetic energyimage whereimageare the principal moments of inertia. Thus, the governing equations are

image (4.1a)

which explicitly reads

image (4.1b)

It is straightforward to check that the system is reversible with respect to the linear diffeomorphism imageand in symmetry, also with respect to R2,R3 defined analogously. Thus, due to Lemma 4.1, we have the following KAM–like result for the free rigid body.

Theorem 4.1. Let image depend smoothly on h such thatimageAssume that image for each h, is reversible with respect to R1, R2 and R3. Then, for small enough h, the solution paths of imageare periodic.

Proof. It is known that if γ is a solution curve of the Euler equations, then it is either an equilibrium, or it is periodic with finite period te > 0, in which case it crosses either of the planesimage every half period [14]. That is, it holds thatimagefor some k ∈ {1, 2, 3} andimage Further, sinceXM,T ≡ 0 is not allowed, it is known that if γ is an equilibrium, then imageLetimagebe a solution curve of image and let γ be the solution curve of XM,T such that image Assume first that γ is not an equilibrium. Then, for any image it holds that a continuous path betweenimage must cross the planeimage For small enough h it holds thatimageand imageapproximatesimageand imagewell enough to also be separated by image Thus,image for some image Likewise,imagefor some image Since imageis reversible with respect to R1 it follows from Lemma 4.1 thatimage is periodic. If γ is an equilibrium and image is not, then either there exists s > 0 such that image in which case the solution curve of XM,T such thatimage is periodic, so we are back to the first case, orimage in which case the assertion follows directly from Lemma 4.1.

The traditional perception in the literature is to view the rigid body equations (4.1) as a Poisson system, with the non-canonical Poisson tensor imageinduced by the total angular momentum (M is a Casimir, cf. [14], for this Poisson structure). We denote the corresponding bracket by imageIt is clear that imageis a sub-group of Diffη.Consider the Hamiltonian splitting image whereimage The sub-systemimage does not affect image and all the quadratic terms contain xi. Hence, it is in essence a linear system on imageand therefore explicitly integrable (since the exponential map is computable for any 2 × 2–matrix). A well known second order integrator is obtained by the symmetric composition

image

This integrator has the following properties:

1. It is reversible with respect to R1, R2 and R3. Thus, its modified vector field image is a R1,R2,R3–reversible perturbation of X, so Theorem 4.1 may be used to deduce periodic orbits of the numerical solution.

2. It is a Poisson map, i.e., image This implies that its modified vector field image is the Hamiltonian vector field of a modified Hamiltonianimage so T is nearly conserved. Further, since M is a Casimir of the Poisson structure it is exactly conserved.

Remark 4.3. One may also view the rigid body equations (4.1) as a Poisson system with the Poisson tensor ηT = η(·1, dT, ·2), and then construct an integratorimage by splitting of M. This integrator will exactly conserve T, and nearly conserve M.

Following our notion, we now consider Hamiltonian splitting of both M and T. To this end, let image it follows that

image

Each such vector field is integrable by linear extrapolation, for example,

image

Thus, a second order integrator is obtained by

image

This integrator is computationally cheaper than image since computation of the exponential map, which involves evaluation of sin and cos, is not necessary. Further, it has the following properties:

1. It is reversible with respect to R1, R2 and R3. Thus, its modified vector field imageis a R1, R2, R3 reversible perturbation of XM,T , so Theorem 4.1 may be used to deduce periodic orbits of the numerical solution.

2. It is an η–map, i.e.,image which impliesimageHowever, image does not correspond to a modified Nambu–Poisson system (see Remark 1.3), so there are no exactly conserved modified Hamiltonians image Nevertheless, M and T are still nearly conserved due to the periodicity of the numerical solution.

Consider now time transformation of system (4.1) into an extended Nambu–Poisson system

image (4.2)

We have the following generalisation of Theorem 4.1.

Theorem 4.2. Let imagedepend smoothly on ε such thatimageAssume that imagefor each ∈, is reversible with respect to imageand that there exists image such thatimageThen, for small enough ∈ the solution paths of image are periodic.

Proof. From the definition of image it follows thatimage is a hyper-plane, and thatimage implies image for all image Let γ be a solution curve of XM,T,G. Since it is a time transformation of a solution curve of XM,T and since image it follows that there exists t1 < t2 and k ∈ {1, 2, 3} such that image Thus, γ is periodic due to Lemma 4.1. The proof now proceeds exactly as the proof of Theorem 4.1.

Assume G takes the splitted form imageWe propose the following adaptive versions of image

image

Notice that all of the partial flows are explicitly integrable. In particular,imageimage Further, it holds that

image

and correspondingly for image These integrators have the following properties:

1. They are reversible with respect to imageThus, their modified vector fields are image reversible perturbation of XM,T,G, so Theorem 4.2 may be used to deduce periodic orbits of the numerical solution. (Assuming ∃ε > 0 such that ∂G2/∂ξ > ε.)

2. They areimagemaps. However, they do not correspond to a modified Nambu–Poisson system (see Remark 1.3). Nevertheless, M, T and G are still nearly conserved due to the periodicity of the numerical solution. In fact, M is exactly conserved by image since each partial flow is an ηM–map.

As an illustration, numerical simulations with image and image are given. The moments of inertia are image and initial data are x0 = (0, cos(θ), sin(θ)), with θ = 0.2, which correspond to rotation “nearly” about the unstable principle axis. For the adaptive integrators the additional Hamiltonian is image so the steps become smaller as the magnitude of the vector field XM,T increases. The step size h = 0.15 is used for the non-adaptive integrators, and for the adaptive integrators ǫ is chosen to yield the same mean time step (i.e. so that the mean of image).

A comparison of solutions in the t (non-adaptive) and in the τ (adaptive) variables are given in Figure 1. Notice that the time-stretching makes the solution “smoother”. Further, the numerical errors in the Hamiltonians are plotted in Figure 2. Notice that the errors are significantly smaller for the adaptive integrators.

generalized-theory-applications-non-adaptive-integrator

Figure 1. Solution curves for the non-adaptive integrator image , and for the adaptive integrator image. Notice that the curves in the lower graph, corresponding to image, are “smoother”. This is due to the time-stretching.

generalized-theory-applications-adaptive-integrators

Figure 2. Absolute errors in the Hamiltonians. Notice that the errors in T (and M) are significantly smaller for the adaptive integrators. Thus, increased efficiency due to adaptivity is obtained. (Recall that image and image conserve M up to rounding errors, whence M is not plotted for these.)

Conclusions

A time transformation technique for Nambu–Poisson systems, based on extending the phase space, have been developed (Theorem 2.1). The technique is shown to preserve reversibility under mild conditions on the additional Hamiltonian function (Theorem 3.1). A family of numerical integrators based on splitting of the Nambu–Poisson Hamiltonians is suggested. In particular, a novel approach for numerical integration of the Euler equations for the free rigid body is presented. By backward error analysis, it is shown that periodicity is preserved (Theorem 4.1 and Theorem 4.2).

Acknowledgement

The author is grateful to Claus F¨uhrer, Gustaf S¨oderlind and Sergei Silvestrov for fruitful discussions, and to Olivier Verdier for many helpful suggestions on improvement. The author would also like to thank SKF for the support given.

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