Inherently Energy Conserving Time Finite Elements for Classical Mechanics

In this paper, we develop a finite element method for the temporal discretization of the equations of motion. The continuous Galerkin method is based upon a weighted-residual statement of Hamiltonʹs canonical equations. We show that the proposed finite element formulation is energy conserving in a natural sense. A family of implicit one-step algorithms is generated by specifying the polynomial approximation in conjunction with the quadrature formula used for the evaluation of time integrals. The numerical implementation of linear, quadratic, and cubic time finite elements is treated in detail for the model problem of a circular pendulum. In addition to that, concerning dynamical systems with several degrees of freedom, we address the design of nonstandard quadrature rules which retain the energy conservation property. Our numerical investigations assess the effect of numerical quadrature in time on the accuracy and energy conservation property of the time-stepping schemes.