Variational problems on Riemannian manifolds with constrained accelerations

Abstract

We introduce variational problems on Riemannian manifolds with constrained acceleration and derive necessary conditions for normal extremals in the constrained variational problem. The problem consists on minimizing a higher-order energy functional, among a set of admissible curves defined by a constraint on the covariant acceleration. In addition, we use this framework to address the elastic splines problem with obstacle avoidance in the presence of this type of contraints.

A Coordinate-Free Variational Approach to Fourth-Order Dynamical Systems on Manifolds: A System and Control Theoretic Viewpoint

“…A variety of alternative formulations may be found in the specialized literature. We mention that the authors of [44] studied forcing terms arising from non-holonomic constraints on the trajectory of a fourth-order dynamical system on manifold. It is also worth mentioning the alternative method to account for some sorts of non-conservative forces that may be found in the work of Herglotz (see, for example, [45]) based on action-dependent Lagrangian functions.…”