A unified approach to Hamiltonian systems, Poisson systems, gradient systems, and systems with Lyapunov functions and/or first integrals

TL;DR

This paper presents a unified framework for representing Hamiltonian, Poisson, gradient, and Lyapunov systems using linear-gradient formulations, extending to discrete-time analogues that preserve key invariants.

Contribution

It introduces a generalized linear-gradient system formulation that unifies various classes of dynamical systems and their discrete-time counterparts.

Findings

Unified formulation for Hamiltonian, Poisson, and gradient systems.

Discrete-time methods that preserve invariants using discrete gradients.

Extension to systems with multiple integrals or Lyapunov functions.

Abstract

Systems with a first integral (i.e., constant of motion) or a Lyapunov function can be written as ``linear-gradient systems'' $\dot x= L(x)\nabla V(x)$ for an appropriate matrix function $L$, with a generalization to several integrals or Lyapunov functions. The discrete-time analogue, $\Delta x/\Delta t = L \bar\nabla V$ where $\bar\nabla$ is a ``discrete gradient,'' preserves $V$ as an integral or Lyapunov function, respectively.