Mechanical Integrators Derived from a Discrete Variational Principle
Wendlandt, J. M. and J. E. Marsden
Physica D, 106, 223-246.
Abstract:
Many numerical integrators for mechanical system simulation are created
by using discrete algorithms to approximate the continuous equations of
motion. In this paper, we present a procedure to construct time-stepping
algorithms that approximate the flow of continuous ODE's for mechanical
systems by discretizing Hamilton's principle rather than the equations of
motion. The discrete equations share similarities to the continuous
equations by preserving invariants, including the symplectic form and the
momentum map. We first present a formulation of discrete mechanics along
with a discrete variational principle. We then show that the resulting
equations of motion preserve the symplectic form and that this formulation
of mechanics leads to conservation laws from a discrete version of
Noether's theorem. We then use the discrete mechanics formulation to
develop a procedure for constructing mechanical integrators for continuous
Lagrangian systems. We apply the construction procedure to the
rigid body and the double spherical pendulum to demonstrate numerical
properties of the integrators.