Multi-Multiplier Ambient-Space Formulations of Constrained Dynamical Systems, with an Application to Elastodynamics

Abstract

Various formulations of the equations of motion for both finite- and infinite-dimensional constrained Lagrangian dynamical systems are studied. The different formulations correspond to different ways of enforcing constraints through multiplier fields. All the formulations considered are posed on ambient spaces whose members are unrestricted by the need to satisfy constraint equations, but each formulation is shown to possess an invariant set on which the constraint equations and physical balance laws are satisfied. The stability properties of the invariant set within its ambient space are shown to be different in each case. We use the specific model problem of linearized incompressible elastodynamics to compare properties of three different ambient-space formulations. We establish the well-posedness of one formulation in the particular case of a homogeneous, isotropic body subject to specified tractions on its boundary.