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Discrete Hamiltonian variational integrators. (English) Zbl 1232.65177

This paper is concerned with the study of some properties of a discrete Hamiltonian dynamics corresponding to a continuous Hamiltonian system with a Type II generating flow. Recall that an $n$-dimensional Hamiltonian system defined by $H=H\left(p,q\right)$, $q={\left({q}^{1},{q}^{2},\cdots ,{q}^{n}\right)}^{T}$, $p={\left({p}_{1},{p}_{2},\cdots ,{p}_{n}\right)}^{T}$, is said of Type II generating flow if the solutions $q=q\left(t\right)$, $p=p\left(t\right)$, $t\in \left[0,T\right]$ make stationary the functional $F\left[q\left(·\right),p\left(·\right)\right]\equiv p\left(T\right)q\left(T\right)-{\int }_{0}^{T}\left[p\left(t\right)\stackrel{˙}{q}\left(t\right)-H\left(q\left(t\right),p\left(t\right)\right)\right]dt$ in the set of smooth functions with the boundary conditions $\delta q\left(0\right)=0$, $\delta p\left(T\right)=0$. Here due to the relation between the Type II continuous and discrete Hamiltonian dynamics through a variational formulation the authors are able to derive discretizations of Hamiltonian systems that retain the essential properties of the flow. Furthermore, when the Hamiltonian is hyperregular it is possible through the Legendre transformation to translate this correspondence to the Lagrangian formulation.

The paper is organized as follows: In Section 2 the variational formulation of Type II continuous systems is revised together the generating functions. Next, in Section 3, for a given partition $0={t}_{0}<{t}_{1}<\cdots <{t}_{N}=T$ of $\left[0,T\right]$ and a discrete curve ${\left\{\left({q}_{k},{p}_{k}\right)\right\}}_{k=0}^{N}$ associated to this partition, a discrete Type II Hamilton’s variational principle associated to a discrete functional ${F}_{d}$ of $F$ is presented. The discrete right Hamilton’s equations and the discrete generating function are given. The theory is clearly summarized at the end of Section 3. In Section 4, Galerkin type discretizations are considered for the practical solution of these discrete Hamiltonians. For a given set of basis functions ${\left\{{\psi }_{i}\left(\tau \right)\right\}}_{i=1}^{s}$, $\tau \in \left[0,1\right]$ and quadrature nodes ${\left\{{c}_{i}\right\}}_{i=1}^{s}$ the authors give a systematic procedure to construct a Galerkin variational integrator, presenting the Lagrangian point of view when the Hamiltonian is hyperregular. It is shown that a special class of Galerkin variational integrators can be implemented as symplectic partitioned Runge-Kutta methods (SPRK). Some simple examples are presented to illustrate the theory. In Section 5 the preservation properties of the discrete Hamiltonian are considered. Finally, in Section 6, the results of some numerical experiments are given to illustrate the preservation properties of some simple Type II Hamiltonians.

##### MSC:
 65P10 Numerical methods for Hamiltonian systems including symplectic integrators 37M15 Symplectic integrators (dynamical systems) 65L06 Multistep, Runge-Kutta, and extrapolation methods 65L60 Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE 37J05 Relations of dynamical systems with symplectic geometry and topology 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods