Symplectic-energy-momentum preserving variational integrators

Abstract

The purpose of this paper is to develop variational integrators for conservative mechanical systems that are symplectic and energy and momentum conserving. To do this, a space-time view of variational integrators is employed and time step adaptation is used to impose the constraint of conservation of energy. Criteria for the solvability of the time steps and some numerical examples are given.

Variational integrators and time-dependent lagrangian systems

“…By similar arguments to those used in [6], it is easy to prove that for solutions of the algorithm (1) without imposing endpoints conditions:…”

“…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.…”

Asynchronous Variational Integrators

“…This resultsin time adaption, in as much as the time set is not prescribed at the outset but is determined as part of the solution instead. The resulting method generalizes that proposed by Kane, Marsden & Ortiz [1999], which allows for one adaptable time variable only and thus results in global energy conservation only. An alternative interpretation of (58) and (59) is as joint discrete Euler-Lagrange equations corresponding to a spacetime discretization of the spacetime domain B.…”

“…However, Kane, Marsden & Ortiz [1999] pointed out that it is not always possible to determine a positive time step from the discrete energy-conservation equation, especially near turning points where velocities are small. Kane, Marsden & Ortiz [1999] overcame this difficulty by formulating a minimization problem that returns the exact spacetime solution whenever one exists.…”

“…The construction of an analogous symplectic-energy-momentum time integrator for finite degree-of-freedom mechanical systems (such as the N -body problem or rigid body mechanics) was carried out in Kane, Marsden & Ortiz [1999], where the time step of the complete system was computed in order to preserve the total energy. Conditions for the solvability of the time step were investigated there, and some of these features also appear in the PDE context developed here.…”