Geometric, Variational Integrators for Computer Animation

Abstract

We present a general-purpose numerical scheme for time integration of Lagrangian dynamical systems—an important
\ncomputational tool at the core of most physics-based animation techniques. Several features make this
\nparticular time integrator highly desirable for computer animation: it numerically preserves important invariants,
\nsuch as linear and angular momenta; the symplectic nature of the integrator also guarantees a correct energy
\nbehavior, even when dissipation and external forces are added; holonomic constraints can also be enforced quite
\nsimply; finally, our simple methodology allows for the design of high-order accurate schemes if needed. Two key
\nproperties set the method apart from earlier approaches. First, the nonlinear equations that must be solved during
\nan update step are replaced by a minimization of a novel functional, speeding up time stepping by more than a
\nfactor of two in practice. Second, the formulation introduces additional variables that provide key flexibility in the
\nimplementation of the method. These properties are achieved using a discrete form of a general variational principle
\ncalled the Pontryagin-Hamilton principle, expressing time integration in a geometric manner. We demonstrate
\nthe applicability of our integrators to the simulation of non-linear elasticity with implementation details.