A Can0nical Integrati0n Technique

Abstract

The class of differential equations of interest to this paper is that in which the equations are derivable from a Hamiltonian by the use of Hamilton's equations. The exact solution of such a system of differential equations leads to a symplectic map from the initial conditions to the present state of the system. A characteristic feature of all explicit higher order integration methods, however, is that they are not exactly symplectic. In many applications the salient features appear only after a long time or after numerous iterations; in these applications spurious damping or excitation may lead to misleading results. The purpose of this paper is to develop an explicit third order symplectic map (i.e. a third order integration step which preserves exactly the canonical character of the equations of motion) and to indicate the method for higher order maps. For a typical numerical integration, this method can be used to eliminate the noncanonical effects while providing the accuracy corresponding to a third order integration step.