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,' ' (x) over dot = L(x)del V(x), for an appropriate matrix function L, w ith a generalization to several integrals or Lyapunov functions. The d iscrete-time analog, Delta x/Delta t = L<(del)over bar>V, where V is a ''discrete gradient,'' preserves V as an integral or Lyapunov functio n, respectively. [S0031-9007(98)07076-8].