Variational Integrators and Time-dependent Lagrangian Systems

There are several numerical integration methods [16] that preserve some of the invariants of an autonomous mechanical system. In [8], T.D. Lee studies the possibility that time can be regarded as a bona fide dynamical variable giving a discrete time formulation of mechanics (see also [9, 10]). From other point of view (integrability aspects) Veselov [19] uses a discretization of the equations of classical mechanics. Both approaches can be characterized as the creation of integrators based on a discretization of the variational principle determined by a lagrangian function. These integration methods have usually better long term simulation properties and computational efficiency than the conventional ones. The main geometrical invariants that these integrators preserve are symplecticity, energy or/and momentum. Ge and Marsden [4] have proved that a constant time stepping integrator cannot preserve the symplectic form, energy and momentum, simultaneously, unless it coincides with the exact solution of the initial system up to a time reparametrization. However, Kane, Marsden, Ortiz and West [6] show that using an appropriate definition of symplecticity and an adaptative time stepping it is possible to construct a variational integrator which is simultaneously symplectic, momentum and energy preserving.