The GGL Variational Principle for Constrained Mechanical Systems

Abstract

We present an extension of the Livens variational principle (sometimes also referred to as Hamilton-Pontryagin principle) to mechanical systems subject to holonomic constraints. The newly proposed principle embodies an index reduction in the spirit of the often-applied GGL stabilization and thus may be termed “GGL principle”. The Euler-Lagrange equations of the GGL principle assume the form of differential-algebraic equations (DAEs) with differentiation index two. In contrast to the original GGLDAEs, the present formulation fits into the Hamiltonian framework of mechanics. Therefore, the GGL principle facilitates the design of symplectic integrators. In particular, it offers the possibility to construct variational integrators. This is illustrated with the development of a new first-order scheme which is symplectic by design. The numerical properties of the newly devised scheme are investigated in a representative example of a constrained mechanical system.