Stabilization of constraints and integrals of motion in dynamical systems

Abstract

When a given system of differential equations is integrated by numerical and automatic integration it may occur that the solution at hand satisfies an analytical relation which is a corollary of the differential equations but which is unknown to the automatic computer. An example of such a relation is the energy relation in conservative systems or the analytical relation generated by an outer holonomic or non-holonomic constraint provided the Lagrange equations of the first kind are used. It is shown that, in general, the computed numerical values of the solution satisfy such analytic relations with poor accuracy. The aim of the paper is to show how the analytical relations can be satisfied in a stabilized manner in order to improve the numerical accuracy of the solution of the differential equations. The proposed method leads to a modified differential system which is often stable in the sense of Ljapunov, whereas the original system is unstable.