Variational problems on Riemannian manifolds with constrained accelerations

TL;DR

This paper develops a framework for variational problems on Riemannian manifolds with acceleration constraints, deriving conditions for optimal curves and applying it to elastic splines with obstacle avoidance.

Contribution

It introduces a novel variational framework for constrained acceleration problems on Riemannian manifolds and derives necessary optimality conditions.

Findings

Derived necessary conditions for normal extremals in constrained acceleration problems.

Applied the framework to elastic splines with obstacle avoidance.

Provided a new approach for higher-order energy minimization on manifolds.

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.