Reduction for constrained variational problems on 3D null curves

TL;DR

This paper develops a reduction method for solving constrained variational problems on null curves in de Sitter 3-space, leading to explicit solutions via elliptic functions.

Contribution

It introduces a novel reduction approach combining moving frames and exterior differential systems for null curves in de Sitter space.

Findings

Explicit solutions expressed through elliptic functions and integrals.

Reduction procedure simplifies integration of extremals.

Method applicable to similar geometric variational problems.

Abstract

We consider the optimal control problem for null curves in de Sitter 3-space defined by a functional which is linear in the curvature of the trajectory. We show how techniques based on the method of moving frames and exterior differential systems, coupled with the reduction procedure for systems with a Lie group of symmetries lead to the integration by quadratures of the extremals. Explicit solutions are found in terms of elliptic functions and integrals.