Approximate first integrals for discrete Hamiltonian systems.

*(English)*Zbl 0894.65036For readers interested in numerical methods: By applying the formalism developed for continuous Hamiltonian ordinary differential equations (ODEs) derived from a fixed end-points variational problem, some criteria are proposed for choosing an appropriate integration stepsize. It is shown that the resulting ‘polygonal’ solution will be close to the continuous one.

For readers interested in the relations between continuous and discrete formulations: The argumentation is based on a well-known fact: A Hamiltonian ODE system \(L= 0\) with periodic coefficients can be reduced to an autonomous recurrence system \(L_n= 0\) by the Poincaré method of sections. Since for \(L= 0\) the (nonautonomous) Hamiltonian \(H=\text{const.}\) is not an integral of motion, the discretized Hamiltonian \(H_n=\text{const.}\) is also not one. It is equally well known that even a second-order autonomous \(L_n= 0\) can describe chaotic dynamics. The associated \(H_n=\text{const.}\) provides then no information on the structure of the phase space. The authors assume implicitly that this structure is ‘locally orderly’. The variable stepsizes are not related to ‘intermediate iterates’ (fractional values of \(n\)).

For readers interested in the relations between continuous and discrete formulations: The argumentation is based on a well-known fact: A Hamiltonian ODE system \(L= 0\) with periodic coefficients can be reduced to an autonomous recurrence system \(L_n= 0\) by the Poincaré method of sections. Since for \(L= 0\) the (nonautonomous) Hamiltonian \(H=\text{const.}\) is not an integral of motion, the discretized Hamiltonian \(H_n=\text{const.}\) is also not one. It is equally well known that even a second-order autonomous \(L_n= 0\) can describe chaotic dynamics. The associated \(H_n=\text{const.}\) provides then no information on the structure of the phase space. The authors assume implicitly that this structure is ‘locally orderly’. The variable stepsizes are not related to ‘intermediate iterates’ (fractional values of \(n\)).

Reviewer: I.Gumowski (Thoiry)

##### MSC:

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |

65L12 | Finite difference and finite volume methods for ordinary differential equations |