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,'' ̇ \.x0ex0ex=0ex0exL(x)▽V(x), for an appropriate matrix function L, with a generalization to several integrals or Lyapunov functions. The discrete-time analog, Δx/Δt0ex0ex=0ex0exL▽̄V, where ▽̄ is a ``discrete gradient,'' preserves V as an integral or Lyapunov function, respectively.