The Lie-Trotter integrator in the dynamics of the symmetric free rigid body

TL;DR

This paper demonstrates that the Lie-Trotter integrator acts as a Poisson integrator when applied to the Euler equations of a symmetric free rigid body, highlighting its importance in mechanical system simulations.

Contribution

It establishes that the Lie-Trotter integrator is a Poisson integrator specifically for symmetric free rigid body dynamics, providing new insights into numerical methods for mechanical systems.

Findings

Lie-Trotter integrator is a Poisson integrator for symmetric free rigid body

Highlights importance of integrators in mechanical system analysis

Provides theoretical foundation for numerical simulation accuracy

Abstract

The numerical integration plays a fundamental role in understanding the behaviour of many mechanical systems. In this paper some important aspects of the mechanical integrators on the dynamics of a mechanical system are studied. More specific, we have shown that if that the Lie-Trotter integrator is obtained, in case of Euler equations for the dynamics of symmetric free rigid body, then it is a Poisson integrator. At the end of the paper some important remarks are presented.

Equations

$${\varphi }_t:{\mathbb{R}}^n\to {\mathbb{R}}^n,$$

$$x_{n+1}={\varphi }_t\left(x_n\right).$$

$$\stackrel{\cdot }{x}=f\left(x\right),$$

$$\frac{x_{n+1}-x_n}{h}=f\left(\frac{x_{n+1}+x_n}{2}\right).$$

$$\left\{\begin{array}[]{c}\frac{x_{n+1}-x_{n}}{h}=\sum\limits_{i=1}^{s}b_{i}\ f\left(X_{i}\right),\\ X_{i}=x_{n}+\sum\limits_{j=1}^{s}a_{ij}\ f\left(X_{j}\right).\end{array}\right.$$

$$\frac{x_{n+1}-x_n}{h}=f\left(x_n\right).$$

$$\frac{x_{n+1}-x_n}{h}=f\left(x_n\right)+f\left(x_{n+1}\right).$$

$${\varphi }_t^{\ast }\omega =\omega ,\left(\forall \right)t\in \mathbb{R}.$$

$$M=T^{\ast }\mathbb{R}\simeq {\mathbb{R}}^2,\omega =dp\wedge dq,H\left(p,q\right)=\frac{1}{2}\left(q^2+p^2\right).$$

$$\left\{\begin{array}[]{c}\overset{\cdot}{q}=p,\\ \overset{\cdot}{p}=-q,\end{array}\right.$$

$$\left\{\begin{array}[]{c}x_{n+1}^{1}=x_{n}^{1}+hx_{n}^{2},\\ x_{n+1}^{2}=-hx_{n}^{1}+\left(1-h^{2}\right)x_{n}^{2}.\end{array}\right.$$

$$H\left(x_{n+1}^1,x_{n+1}^2\right)=\left(1+h^2\right)H\left(x_n^1,x_n^2\right)+h^2\left(\frac{h^2}{2}-1\right)x_n^2+h^3{x}_n^1{x}_n^2.$$

$$d{x}_{n+1}^2\wedge d{x}_{n+1}^1=d{x}_n^2\wedge d{x}_n^1.$$

$$\left({\mathbb{R}}^{2n},\omega ,H\right),$$

$$\omega =d{p}_1\wedge d{q}^1+...+d{p}_n\wedge d{q}^n.$$

$$\left\{\begin{array}[]{c}b_{i}a_{ij}+b_{j}a_{ji}-b_{i}b_{j}=0,\\ 1\leq i,\ j\leq s.\end{array}\right.$$

$$J:M\to {\mathcal{G}}^{\ast }.$$

$${\varphi }_t^{\ast }J=J,\left(\forall \right)t\in \mathbb{R}.$$

$$J:M\to {\mathcal{G}}^{\ast }.$$

$$J:M\to \mathcal{G}.$$

$$\left(i\right){\left\{{\varphi }_t\right\}}_{t\in \mathbb{R}}\text{ is an symplectic integrator,}$$

$$\left(ii\right){\left\{{\varphi }_t\right\}}_{t\in \mathbb{R}}\text{ is an energy integrator,}$$

$$\left(iii\right){\left\{{\varphi }_t\right\}}_{t\in \mathbb{R}}\text{ is an moment integrator,}$$

$$\left(iv\right)\text{dynamics on the reduced space }\left(M_0,{\omega }_0\right)\text{ not integrable.}$$

$${\varphi }_t^{\ast }\left\{f,g\right\}=\left\{{\varphi }_t^{\ast }f,{\varphi }_t^{\ast }g\right\},\left(\forall \right)f,g\in C^{\infty }\left(P,\mathbb{R}\right).$$

$$\left\{\begin{array}[]{c}\left(D\phi_{t}\left(x\right)\right)\cdot\Pi\cdot\left(D\phi_{t}\left(x\right)\right)^{t}=\Pi\left(\phi_{t}\left(x\right)\right),\\ \left(\forall\right)\ t\in\mathbb{R},\ \left(\forall\right)\ x\in\mathbb{R}^{n},\end{array}\right.$$

$$\Pi=\left[\begin{array}[]{cc}0&x_{2}\\ -x_{2}&0\end{array}\right],$$

$$H\left(x_1,x_2\right)=A{x}_1+B{x}_2+C,$$

$$\left\{\begin{array}[]{c}h=\frac{1}{2bA-aA},\\ 2bA-aA\neq 0.\end{array}\right.$$

$$A=0\text{, sau }B=0\text{.}$$

$$\left\{\begin{array}[]{c}b_{i}a_{ij}+b_{j}a_{ji}-b_{i}b_{j}=0,\\ 1\leq i,\ j\leq s.\end{array}\right..$$

Full text

**THE LIE - TROTTER INTEGRATOR IN THE DYNAMICS OF THE SYMMETRIC FREE RIGID BODY **

Ciprian Hedrea

Department of Mathematics

Polytechnic University of Timişoara

Square of Victory no. 2, 300006

Timişoara, România

ciprian.hedrea@upt.ro

Abstract. The numerical integration plays a fundamental role in understanding the behaviour of many mechanical systems. In this paper some important aspects of the mechanical integrators on the dynamics of a mechanical system are studied. More specific, we have shown that if that the Lie-Trotter integrator is obtained, in case of Euler equations for the dynamics of symmetric free rigid body, then it is a Poisson integrator. At the end of the paper some important remarks are presented.

Keywords: Integrator; rigid body; symmetrical rigid body; mechanical system; dynamics.

2000 Mathematics Subject Classification: 37M15, 70Q05

1. Introduction:

The overall situation for mechanical integrators is a complex one and it is still evolving. There are numerical integration methods that preserve some of the invariants of the mechanical system, such as: energy, momentum, or the symplectic form.

It is well known that if the energy and momentum map include all the integrals from a certain class that one cannot create integrators that are symplectic, energy preserving and momentum preserving unless they coincidentally integrate the equations exactly up to a time parametrization [6]. In mechanics the numerical integration plays a fundamental role.

Because there are many mechanical systems whose explicit integration is unknown, in our effort to understand their behavior, it is very useful the numerical integration of their dynamics.

Recently, more research is dedicated to the use of the mechanical integrators in molecular dynamics, spin systems, magnetism, etc.

2. Numerical and symplectic integrators

Definition 2.1. A numerical integrator on $\mathbb{R}^{n}$ consists in a number of applications differentiable on class $C^{\infty}$

[TABLE]

which are differentiable in relation to $t\in\mathbb{R}.$ We write

[TABLE]

We consider in $\mathbb{R}^{n}$ the system of differential equations given by

[TABLE]

where $x\in\mathbb{R}^{n}$ and $f\in C^{\infty}\left(\mathbb{R}^{n},\mathbb{R}^{n}\right).$

In the following, we will present a series of numerical integrators that enable us to approximate the solutions of the system $\left(2.1\right)$.

$\left(i\right)$ The Gauss-Legendre integrator, given by

[TABLE]

\left(ii\right)\The Runge-Kutta integrator with $s$ steps, given by

[TABLE]

\left(iii\right)\The Euler integrator, given by

[TABLE]

\left(iv\right)\The modified Euler integrator, given by

[TABLE]

(For other examples, see [3] and [9]).

Let $\left(M,\omega,H\right)$ be a mechanical Hamiltonian system.

Definition 2.2. A numerical integrator $\left\{\phi_{t}\right\}_{t\in\mathbb{R}}$ on $M$ is called symplectic integrator if

[TABLE]

Example 2.1. Considering a mechanical Hamiltonian system $\left(M,\omega,H\right)$, where we have

[TABLE]

Then the Ruth integrator associated with this dynamic

[TABLE]

will be given as

[TABLE]

The conclusions that can be drawn immediately, are

\left(i\right)\The Integrator $\left(2.6\right)$ doesn’t conserve energy, because

[TABLE]

$\left(ii\right)$ The integrator $(2.6)$ is a symplectic integrator

More exactly we have

[TABLE]

**Proposition 2.1. **([5]) Let the Hamiltonian mechanical system

[TABLE]

where

[TABLE]

Then the corresponding Gauss - Legendre integrator is a symplectic integrator.

**Proposition 2.2. **([7], [15], [17]) Let the Hamiltonian mechanical system $\left(\mathbb{R}^{2n},\omega,H\right),$ where

[TABLE]

Then the Runge - Kutta integrator with $s$ stairs and $h$ step is a symplectic integrator if and only if

[TABLE]

**Definition 2.3. **Let $\left(M,\omega,H,G\right)$ be a mechanical Hamiltonian system with symmetry having an application moment

[TABLE]

An integrator $\left\{\phi_{t}\right\}_{t\in\mathbb{R}}$ on $M$ is called integrator moment if

[TABLE]

**Proposition 2.3. **([6]) Let $\left(M,\omega,H,G\right)$ be a mechanical Hamiltonian system with symmetry and having a moment application

[TABLE]

Then any integrator \left\{\phi_{t}\right\}_{t\in\mathbb{R}}\on $M$ wich is $G$-invariant and is symplectic, is a moment integrator.

**Proposition 2.4. **([6]) Let $\left(M,\omega,H,G\right)$ a mechanical Hamiltonian system with symmetry and having a moment application

[TABLE]

Then if $\left\{\phi_{t}\right\}_{t\in\mathbb{R}}$ is an integrator on $M$ such that

[TABLE]

Then $\left\{\phi_{t}\right\}_{t\in\mathbb{R}}$ gives the exact solution for initial system dynamics (unreduced).

Observation 2.1. Higher order integrators have generally similar properties. For details see [14], [10], [4], [19] and [16].

3. Poisson integrators

Let $\left(P,\left\{\cdot,\cdot\right\}\right)$ be a dimensional finite Poisson manifold from class $C^{\infty}.$

Definition 3.1. An integrator $\left\{\phi_{t}\right\}_{t\in\mathbb{R}}$ on $P$ is called Poisson integrator if

[TABLE]

**Observation 3.1. **In the particular case $P=\mathbb{R}^{n},$ relation $(8)$ is equivalent to

[TABLE]

where $D\phi_{t}$ designate Fréchét differential of $\phi_{t}$, and $\Pi$ is the matrix associated to the Poisson structure $\left\{\cdot,\cdot\right\},$ hence $\Pi=\left[\left\{x_{i},x_{j}\right\}\right].$

**Example 3.1. **([11]) Let $\left(\mathbb{R}^{2},\Pi,H\right)$ be a Hamilton-Poisson mechanical system , where

[TABLE]

and

[TABLE]

$A,B,C\in\mathbb{R}.$ A simple calculation shows that

\left(i\right)\The Runge - Kutta integrator with $1$- step is a Poisson integrator if and only if

[TABLE]

\left(ii\right)\The Runge - Kutta integrator with $1$- step is an energy integrator if and only if

[TABLE]

\left(iii\right)\The Gauss-Legendre integrator is a Poisson integrator,

\left(iv\right)\The Gauss-Legendre integrator is an energy integrator.

**Proposition 3.1. **([8]) Let $\left(\mathbb{R}^{n},\left\{\cdot,\cdot\right\},H\right)$ be a Hamilton-Poisson mechanical system. If the matrix $\Pi$ is constant, then the Runge-Kutta integrator with $s$-steps is a Poisson integrator if and only if

[TABLE]

**Observation 3.2. **In the particular case n=2m\ \text{and }\Pi=\left[\begin{array}[]{cc}0_{m}&I_{m}\\ -I_{m}&0_{m}\end{array}\right]\the theorem [7], [15] and [17] is found (see Proposition 2.2).

**Proposition 3.2. **([1]) Let $\left(\mathbb{R}^{n},\left\{\cdot,\cdot\right\},H\right)$ be a Hamilton-Poisson mechanical system. If the matrix $\Pi$ is constant, then the Gauss-Legendre integrator is a Poisson integrator.

Observation 3.3. In the particular case n=2m\ \text{and }\Pi=\left[\begin{array}[]{cc}0_{m}&I_{m}\\ -I_{m}&0_{m}\end{array}\right]\the theorem [5] is found (see Proposition 2.1.).

**4. The Lie-Trotter Integrator and some properties in the dynamics of symmetric free rigid body **

Let $\left(\mathbb{R}^{n},\left\{\cdot,\cdot\right\},H\right)$ be a Hamilton-Poisson mechanical system. The Lie-Trotter integrator (see [18]) usually applies when the Hamiltonian $H$ can be written as

[TABLE]

and the dynamics generated by $X_{H_{1}}$ $\left[\text{resp }X_{H_{2}}\right]$, hence $\exp\left(tX_{H_{1}}\right)$ $\left[\text{resp. }\exp\left(tX_{H_{2}}\right)\right]$ can be explicitly integrated, hence $\exp\left(tX_{H_{1}}\right)$ and $\exp\left(tX_{H_{2}}\right)$ can be explicitly calculated. Then the Lie-Trotter integrator is given by

[TABLE]

Because $\phi_{t}$ is actually composing two Hamiltonian flows, we have

**Proposition 4.1. **([12], [13]) The Lie-Trotter integrator $\left(4.1\right)$ has the following properties

$\left(i\right)\ \left\{\phi_{t}\right\}_{t\in\mathbb{R}}$ is a Poisson integrator,

\left(ii\right)\The restriction of the foliation of a manifold symplectic Poisson $\left(\mathbb{R}^{n},\left\{\cdot,\cdot\right\}\right)$ gives rise to a symplectic integrator.

Obviously the integrator $(4.1)$ is a first order integrator. This order can be grown by using some ideas from [2]. Concretely, a Poisson integrator of second order is given as

[TABLE]

The 4th order Poisson integrator is given by

[TABLE]

where

[TABLE]

The process continues and, finally, it is obtained a Poisson integrator of the even order $2n+2$ defined as

[TABLE]

where

[TABLE]

Analogously explicit Poisson integrators can be build unconcerned their even order. We note that construction is difficult because in the construction of a Poisson integrator of even order $2n$ , the second-order integrator $\phi_{t}^{2}$ is used for $3^{n-1}$ times. Therefore the number of steps $k$ is $1+3^{n-1}$, hence

[TABLE]

which grows extremely fast, and the calculations become increasingly more complex.

We proof below the next theorem:

**Theorem 4.1. **If the Lie - Trotter integrator is obtained, in the case of Euler equations for the dynamics of symmetric free rigid body, then it is a Poisson integrator.

Proof

The Euler equations for the dynamics of symmetric free rigid body are written as

[TABLE]

where $a_{1}=\dfrac{1}{I_{3}}-\dfrac{1}{I_{1}}$, the $I_{1}$, $I_{3}$ inertia tensors are components of the body and we assume in all that follows that $I_{1}>I_{3}>0$.

As we well know (see some Puta’s papers) the dynamic $\left(4.2\right)$ has the following Hamilton-Poisson realization $\left(\left(so\left(3\right)\right)^{\ast}\simeq\mathbb{R}^{3},\Pi,H\right)$, where

[TABLE]

We can say more and more precisely that the function $C\in C^{\infty}\left(\mathbb{R}^{3},\mathbb{R}\right)$ given by

[TABLE]

is one Casimir of our configuration $\left(\mathbb{R}^{3},\Pi\right).$ Actually, the symplectic foliation of the Poisson variety $\left(\mathbb{R}^{3},\Pi\right)$ is given by $\left\{0\right\}$ and by the coadjuncte orbits $\left(O_{k},\omega_{k}\right),$ where

[TABLE]

and

[TABLE]

$\left(\omega_{k}\text{ is symplectic form Kirillov-Kostant-Souriau}\right).$

It can be now seen that the Hamiltonian field $X_{H}$ splits

[TABLE]

where

[TABLE]

Then the corresponding flows have the following expressions

[TABLE]

[TABLE]

[TABLE]

Then results that Lie - Trotter integrator has the following expression

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

In conclusion the Lie-Trotter integrator $\left(4.3\right)$ is a Poisson integrator.

From the above, (see [13]), we get

Proposition 4.2: The restriction of the coadjuncte orbits $\left(O_{k},\omega_{k}\right)$ involves obtaining a symplectic integrator.

Proof: It results from the above theorem and Proposition 4.1.

Remark 4.1: The Lie-Trotter integrator is not an energy integrator.

Remark 4.2: If we compare with the Runge-Kutta integrator with $4$ steps, then we obtain almost the same results, but the Lie - Trotter integrator has the advantage that it is much easier to implement.

Remark 4.3 Similar results with the ones presented in this paper are shown in [13] for the free rigid body. By analysing [13], one can see that some of the computations are identical.

Remark 4.4: It remains an open problem to find if there are characteristic roots of the matrix $M\cdot N\cdot P$ which have even values and follow a Hopf bifurcation moving from the unit circle on the real right so that their product is allways $1$ and if there is a transition in the dynamic integration $(4.2)$ by means of the Lie - Trotter integrator, from the real world in the complex world tagged with the Hopf bifurcation for some of their values coming from $M\cdot N\cdot P$.

Remark 4.5 The square matrix $\ R=M\cdot N\cdot P$ , where $\Pi_{n}=R^{n}\cdot\Pi_{0}$, \Pi_{0}=\left[\begin{array}[]{c}1\\ 0\\ 0\end{array}\right] can also give an open problem for finding if there are values of $n$ so that the Markov chain related to the matrix can be found in a maximum possible number of states.

Acknowledgement

This year we mark a decade from the death of the regretted Professor Dr. Mircea Puta, the coordinator of my PhD thesis. I dedicate this paper in his memory and for the strong collaboration with him.

References

[1] Austin, M. A., Krishnaprassad P.S., Wang L.S., Almost Poisson integration of rigid body systems, Journal of Computational Physics, vol. 107, No. 1 (1993), 105-117.

[2] Benzel, S., Ge, Z., Scovel, C., Elementary construction of higher order Lie-Poisson integrators, Phys. Letter A, 174 (1993), 229-232.

[3] Buchner, K., Craioveanu, M. and Puta, M., Recent progress in the integration of Poisson systems via de mid-point rule and Runge-Kutta algorithm, Balkan Journal of Geometry and Its Applications, Vol. 1. Nr. 20 (1996), 9-20.

[4] Channel, P. and Scovel, C., Symplectic integration of Hamiltonian systems, Nonlinearity 3 (1990), 231-259.

[5] Feng, K., Difference schemes for Hamiltonian formalism and symplectic geometry, J. Comp. Math. 4 (1985), 279-289.

[6] Ge, Z. and Marsden, J., Lie-Poisson integrators and Lie-Poisson Hamiltonian-Jacobi theory, Phys. Lett. A 133 (1988), 134-139.

[7] Lasagni, F., Canonical Runge-Kutta methods, ZAMP 39 (1988), 952-953.

[8] Mc Lachlan, R.L., Comment on Poisson schemes for Hamiltonian systems on Poisson manifolds, Computers Math. Applic. vol. 29 No. 3 $\left(1995\right).$

[9] Moş, I., An Introduction to Geometric Mechanics, Cluj University Press. $\left(2005\right).$

[10] Neri, F., Lie-algebras and canonical integration, preprint, Dept. of Physics, University of Maryland, $\left(1987\right).$

[11] Puta, M., Poisson integrators, Analele Univ. Timişoara, vol. 31 (1993), 267-272.

[12] Puta, M., An overview of some Poisson integrators, in Enumath. 97, 2nd European Conference on Numerical Mathematics and Advanced Applications, eds. H.G. Bock, G. Kanschat, Y. Kuznetsov and J. Periaux, World Scientific,(1998), 518-523.

[13] Puta, M., Lie-Trotter formula and Poisson dynamics, Int. Journ. of Biffurcation and Chaos, vol. 9 (1999), 555-559.

[14] Ruth, R.D., A canonical integration technique, IEEE Trans. Nucl. Sci., N.S. 30 (1983), 2669-2271.

[15] Sanz-Serna, J.M., Runge-Kutta schemes for Hamiltonian systems, BIT 28 (1988), 877-883.

[16] Sanz-Serna, J.M., Symplectic integrators for Hamiltonian problems an overview, Acta Numerica 1,(1992), 243-286.

[17] Suris, Y.B., Canonical transformations generated by methods of Runge-Kutta type for the numerical integration of the system $\overset{\cdot\cdot}{x}=\partial U/\partial x$ (Russian), Zh. Vychisl. Mat. i Mat. Fiz. 29 (1989), 202-211.

[18] Trotter, H.F., On the product of semigroups of operators, Proc. Amer. Math. Soc. 10 (1959), 545-551.

[19] Yoshida, H., Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262-268.