Symplecticity of coupled Hamiltonian systems

TL;DR

This paper establishes a specific condition under which coupled finite-dimensional Hamiltonian systems preserve their Hamiltonian nature, focusing on symplectic form conservation, which is crucial for accurate numerical discretization.

Contribution

It introduces a novel condition ensuring that the composition of two coupled Hamiltonian systems remains Hamiltonian, addressing a gap in understanding symplectic integrator applicability.

Findings

Derived a condition for coupled systems to be Hamiltonian

Showed that composition may not always be Hamiltonian without this condition

Emphasized the importance of symplectic form conservation

Abstract

We derived a condition under which a coupled system consisting of two finite-dimensional Hamiltonian systems becomes a Hamiltonian system. In many cases, an industrial system can be modeled as a coupled system of some subsystems. Although it is known that symplectic integrators are suitable for discretizing Hamiltonian systems,the composition of Hamiltonian systems may not be Hamiltonian. In this paper, focusing on a property of Hamiltonian systems, that is, the conservation of the symplectic form, we provide a condition under which two Hamiltonian systems coupled with interactions compose a Hamiltonian system.