Maxwell strata in Euler's elastic problem

TL;DR

This paper analyzes Euler's elastic problem as an optimal control problem on the Euclidean group, describing attainable sets, extremals, symmetries, and providing bounds on cut points using elliptic functions and symmetry analysis.

Contribution

It introduces a novel geometric and control-theoretic framework for Euler's elastic problem, including symmetry group analysis and explicit parametrization of extremals.

Findings

Attainable set characterized

Existence and boundedness of optimal controls proved

Upper bound on cut points established

Abstract

The classical Euler's problem on stationary configurations of elastic rod with fixed endpoints and tangents at the endpoints is considered as a left-invariant optimal control problem on the group of motions of a two-dimensional plane $\E(2)$. The attainable set is described, existence and boundedness of optimal controls are proved. Extremals are parametrized by Jacobi's elliptic functions of natural coordinates induced by the flow of the mathematical pendulum on fibers of the cotangent bundle of $\E(2)$. The group of discrete symmetries of Euler's problem generated by reflections in the phase space of the pendulum is studied. The corresponding Maxwell points are completely described via the study of fixed points of this group. As a consequence, an upper bound on cut points in Euler's problem is obtained.