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

Robert I McLachlan·GRW Quispel·Nicolas Robidoux
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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.

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Robert I McLachlan, GRW Quispel, Nicolas Robidoux. 1998-05-25. A unified approach to Hamiltonian systems, Poisson systems, gradient systems, and systems with Lyapunov functions and/or first integrals. https://doi.org/10.1103/physrevlett.81.2399

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