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**Klas MODIN ^{1,2*}**

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

^{2}SKF 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

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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.

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 should have the form

(1.1)

where ǫ is the Levi–Civita tensor over n indices, and H1, . . . ,Hn−1 are smooth real valued functions on 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. denotes a phase space manifold of dimension n, with local coordinates *x* = (*x*_{1}, . . . , xn). The algebra of smooth real valued functions on is denoted Further, denotes the linear space of vector fields onThe Lie derivative along is denoted If X, Y then the commutator supplies with an infinite dimensional Lie algebra structure. Its corresponding Lie group is the set Diff of diffeomorphisms on with composition as group operation. (See McLachlan and Quispel [16] and Schmid [25] for issues concerning infinite dimensional Lie groups.) If Φ ∈ Diff 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 is equipped with a Poisson structure, defined by a bracket operation {·1, ·2} : 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 together with a multilinear map

that fulfils:

• total skew-symmetry

{H_{1}, . . . ,H_{k}} = sgn(σ){H_{σ1} , . . . ,H_{σk}} (1.2a)

• Leibniz rule

{GH_{1}, . . . ,H_{k}} = G{H_{1}, . . . ,H_{m}} + H_{1}{G,H_{2}, . . . ,H_{k}} (1.2b)

• fundamental identity

**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

{H_{1}, . . . ,H_{k}} = η( dH_{1}, . . . , dH_{k})

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 H_{1}, . . . ,H_{k−1} ∈ . The governing equations are

(1.3a)

which may also be written

(1.3b)

where X_{H1, . . . ,Hk−1} is defined by The corresponding flow map is denoted Notice that due to skew symmetry of the bracket, all the Hamiltonians H_{1}, . . . ,H_{k−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,

or equivalently

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

**Remark 1.3.** It is important to point out that in general not every 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 such that η(*x*) ≠ 0 there exist local coordinates (*x _{1}, . . . , x_{k}, x_{k+1}, . . . , x_{n}*) such that

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

(1.5)

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

(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 is a time transformation of the flow of system (1.5). That is,*

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

**Proof.** From equation (1.6) it follows directly that is parallel with X. Thus, and define the same phase diagrams. It remains to find the relation between t and τ . Since 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) :is 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 ∈ Diff

• A vector field 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 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 is reversible with respect to R and then the vector field in equation (1.6) is reversible with respect to

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

(2.1)

It is not obvious that the bracket corresponding to 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

**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, is also decomposable about its regular points, so the assertion follows from Theorem 1.3.

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

(2.2)

**Remark 2.1.** A functions is considered to belong to by the natural extension Likewise, is considered to be a function in depending on the parameter ξ. Thus, ¯ {·, . . . , ·¯} is defined also for elements in 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 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 is a time transformation, determined by the additional first integral G, of the flow of system (1.3). That is,

where

**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

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

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 coincides with the transformation (1.6) applied to system (1.3).

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 that so the governing equations (1.3a) are equivalent to

This is equivalent to

if and only if condition (3.1) holds. The last set of equations is exactly the condition on X_{H1, . . . ,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., Then X_{H1, . . . ,Hk−1} is reversible with respect to R if and only if

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

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 if

Proof. Since ∂Hi/∂ξ = 0 we have

Since maps ξ to ξ it holds that and

Altogether we now have

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.

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

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 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 be a Lie sub-algebra of and let Diff_{A} be the corresponding Lie sub-group of Diff Assume that can be splitted into explicitly integrable sub-system, each of which is a system inThat is, X = X_{1} +. . . +X_{k} where, andcan be computed explicitly. A numerical integrator for X is obtained by It is clear that is an approximation of, and thatFurther, by the Baker–Campbell–Hausdorff (BCH) formula, it follows that is the exact flow of a modified vector field This information is crucial for the analysis of For example, ifis the Lie-algebra corresponding to a Poisson structure on then will exactly conserve a modified Hamiltonian, which isclose 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 , coming from are valid only for exponentially long times, i.e., up to time scales of order 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 If then in general we have

Thus, is not closed under composition, so it is not a sub-group of Diff However, is closed under the symmetric group operation which we write asFurther, from the symmetric BCH formula (cf. [15]) it follows that if then the vector field Z such thatbelongs to

**Remark 4.2.** For near identity maps, is defined by taking its representation and then settingIn our case, Φ will always be an exact flowin which case

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 be reversible with respect to Assume that the set of fix-points of R is non-empty and that is a solution curve of X for which there exists with t_{1} < t_{2} such that Then γ is periodic.

Proof. For simplicity assume that t_{1} = 0 and t_{2} > 0, which is not a restriction. The curve is also a solution curve due to reversibility. Further, since R restricted to U is the identity map we have the equalities and Due to uniqueness of solutions the first equality implies γ2 = γ, which in conjunction with the second equality implies that Thus γ returns to the same point twice, so it is periodic.

**Rigid body problem**

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

(4.1a)

which explicitly reads

(4.1b)

It is straightforward to check that the system is reversible with respect to the linear diffeomorphism and in symmetry, also with respect to R_{2},R_{3} defined analogously. Thus, due to Lemma 4.1, we have the following KAM–like result for the free rigid body.

**Theorem 4.1.** Let depend smoothly on h such thatAssume that for each h, is reversible with respect to R_{1}, R_{2} and R_{3}. Then, for small enough h, the solution paths of are 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 t_{e} > 0, in which case it crosses either of the planes every half period [14]. That is, it holds thatfor some k ∈ {1, 2, 3} and Further, sinceX_{M,T} ≡ 0 is not allowed, it is known that if γ is an equilibrium, then Letbe a solution curve of and let γ be the solution curve of X_{M,T} such that Assume first that γ is not an equilibrium. Then, for any it holds that a continuous path between must cross the plane For small enough h it holds thatand approximatesand well enough to also be separated by Thus, for some Likewise,for some Since is reversible with respect to R1 it follows from Lemma 4.1 that is periodic. If γ is an equilibrium and is not, then either there exists s > 0 such that in which case the solution curve of X_{M,T} such that is periodic, so we are back to the first case, or 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 induced by the total angular momentum (M is a Casimir, cf. [14], for this Poisson structure). We denote the corresponding bracket by It is clear that is a sub-group of Diff_{η}.Consider the Hamiltonian splitting where The sub-system does not affect and all the quadratic terms contain *x _{i}*. Hence, it is in essence a linear system on and 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

This integrator has the following properties:

1. It is reversible with respect to R1, R2 and R3. Thus, its modified vector field 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., This implies that its modified vector field is the Hamiltonian vector field of a modified Hamiltonian 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 integrator 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 it follows that

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

Thus, a second order integrator is obtained by

This integrator is computationally cheaper than 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 R_{1}, R_{2} and R_{3}. Thus, its modified vector field is a R_{1}, R_{2,} R_{3} reversible perturbation of X_{M,T} , so Theorem 4.1 may be used to deduce periodic orbits of the numerical solution.

2. It is an η–map, i.e., which impliesHowever, does not correspond to a modified Nambu–Poisson system (see Remark 1.3), so there are no exactly conserved modified Hamiltonians 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

(4.2)

We have the following generalisation of Theorem 4.1.

**Theorem 4.2.** Let depend smoothly on ε such thatAssume that for each ∈, is reversible with respect to and that there exists such thatThen, for small enough ∈ the solution paths of are periodic.

**Proof.** From the definition of it follows that is a hyper-plane, and that implies for all Let γ be a solution curve of X_{M,T,G}. Since it is a time transformation of a solution curve of X_{M,T} and since it follows that there exists t_{1} < t_{2} and k ∈ {1, 2, 3} such that 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 We propose the following adaptive versions of

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

and correspondingly for These integrators have the following properties:

1. They are reversible with respect to Thus, their modified vector fields are reversible perturbation of X_{M,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 aremaps. 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 since each partial flow is an ηM–map.

As an illustration, numerical simulations with and are given. The moments of inertia are and initial data are *x*_{0} = (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 so the steps become smaller as the magnitude of the vector field X_{M,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 ).

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.

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).

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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