Higher-order discrete variational problems with constraints

TL;DR

This paper develops new higher-order variational integrators for constrained Lagrangian systems that are symplectic, conserve momenta, and handle time-dependent Lagrangians, with applications in control and interpolation.

Contribution

The paper introduces novel variational integrators for higher-order constrained systems, extending to time-dependent Lagrangians, with proven symplecticity and momentum conservation.

Findings

Methods are automatically symplectic with good energy behavior.

Discrete symmetries lead to momentum conservation.

Applications include optimal control and Riemannian interpolation.

Abstract

An interesting family of geometric integrators for Lagrangian systems can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators. In this paper, we derive new variational integrators for higher-order lagrangian mechanical system subjected to higher-order constraints. From the discretization of the variational principles, we show that our methods are automatically symplectic and, in consequence, with a very good energy behavior. Additionally, the symmetries of the discrete Lagrangian imply that momenta is conserved by the integrator. Moreover, we extend our construction to variational integrators where the lagrangian is explicitly time-dependent. Finally, some motivating applications of higher-order problems are considered; in particular, optimal control problems for explicitly time-dependent underactuated systems and an interpolation problem on Riemannian manifolds.