Numerical Integration of Nearly-Hamiltonian System

Abstract

Consideration is given to the solution by numerical integration of systems of differential equations that are derived from a Hamiltonian function in the extended phase space plus additional forces not included in the Hamiltonian (that is, nearly-Hamiltonian systems). An extended phase space Hamiltonian which vanishes initially will vanish on any solution of the system differential equations. Furthermore, it vanishes in spite of the additional forces, and defines a surface in the extended phase space upon which the solution is constrained.

Direct numerical comparisons are made between (1) nearly-Hamiltonian systems having vanishing Hamiltonians and (2) those having nonvanishing Hamiltonians. It is seen that for some problems, numerical solutions are more stable when computed from systems of the type (1). The problems considered are the harmonic oscillator with the van der Pol perturbation and perturbed Keplerian motion.