Unified Approach to Hamiltonian Systems, Poisson Systems, Gradient Systems, and Systems with Lyapunov Functions or First Integrals

We show that systems with a first integral (i. e. , a constant of motion) or a Lyapunov function can be written as ``linear-gradient systems, '' \. {}x0ex{0ex}=0ex{0ex}L (x) ▽V (x), for an appropriate matrix function L, with a generalization to several integrals or Lyapunov functions. The discrete-time analog, /0ex{0ex}=0ex{0ex}L▽V, where ▽ is a ``discrete gradient, '' preserves V as an integral or Lyapunov function, respectively.