Variational Integrators for Hamiltonizable Nonholonomic Systems

We report on new applications of the Poincaré and Sundman timetransformations to the simulation of nonholonomic systems. These transformations are here applied to nonholonomic mechanical systems known to be Hamiltonizable (briefly, nonholonomic systems whose constrained mechanics are Hamiltonian after a suitable time reparameterization). We show how such an application permits the usage of variational integrators for these nonvariational mechanical systems. Examples are given and numerical results are compared to the standard nonholonomic integrator results. Introduction. It is well known that the dynamical equations of motion of unconstrained mechanical systems follow from a variational principle, namely Hamilton’s principle of stationary action [1, 26]. In the 1970s and 1980s several researchers discretized this continuous variational principle and developed the discrete EulerLagrange equations (see [27] and references therein for a historical account). Like its continuous counterpart, this discrete variational mechanics preserves many of the constants of motion between timestep increments, like the energy, momentum map, or symplectic form, under appropriate assumptions [21, 27]. The resulting numerical integrators, termed mechanical integrators, have found application in molecular dynamics simulations [23, 24, 34] and planetary motion [23], as well as in satellite dynamics [23]. For fixed timesteps, it was shown in [18] that a mechanical integrator for a non-integrable mechanical system with symmetry can at best preserve two of the three aforementioned constants of motion. For this reason, fixed timestep mechanical integrators are named according to what invariants they do preserve. In 2000 Mathematics Subject Classification. Primary: 37J60; Secondary: 34K28.