A unified approach to Hamiltonian systems, Poisson systems, gradient systems, and systems with Lyapunov functions and/or first integrals
Abstract
Systems with a first integral (i.e., constant of motion) or a Lyapunov function can be written as ``linear-gradient systems'' x= L(x)∇ V(x) for an appropriate matrix function L, with a generalization to several integrals or Lyapunov functions. The discrete-time analogue, Δx/Δt = L ∇ V where ∇ is a ``discrete gradient,'' preserves V as an integral or Lyapunov function, respectively.
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