A Hamiltonian-Gradient System for Multiple Conservation Laws

A hamiltonian (H) system with a gradient term is studied from the viewpoint of the temporal behavior of the conserved quantities. If we take a suitable potential G, the conserved quantities approach constants which are given as free parameters. This property holds even if there is another conserved quantity in addition to the hamiltonian. Two examples are given in order to demonstrate this hamiltonian-gradient (HG) system. One is a simple harmonic oscillation and the other is a system of many point vortices. The latter is a typical hamiltonian system with two conserved quantities. Analytical and numerical results of the H and HG system are given and compared to each other, by using various explicit finite difference schemes, including the Euler's, Heun's, and the fourth-order Runge-Kutta methods.