On periodic solutions in the Whitney's inverted pendulum problem

TL;DR

This paper advances the mathematical understanding of Whitney's inverted pendulum problem by proving the existence of periodic solutions under less restrictive conditions and extending the analysis to planar carriage movements.

Contribution

It improves Polekhin's theorem by lowering regularity requirements and generalizes the existence of periodic solutions to planar carriage motions.

Findings

Existence of periodic solutions with lower regularity assumptions.

Extension of periodic solution existence to planar carriage movements.

Refinement of previous theorems on Whitney's inverted pendulum problem.

Abstract

In the book `What is Mathematics?' Richard Courant and Herbert Robbins presented a solution of a Whitney's problem of an inverted pendulum on a railway carriage moving on a straight line. Since the appearance of the book in 1941 the solution was contested by several distinguished mathematicians. The first formal proof based on the idea of Courant and Robbins was published by Ivan Polekhin in 2014. Polekhin also proved a theorem on the existence of a periodic solution of the problem provided the movement of the carriage on the line is periodic. In the present paper we slightly improve the Polekhin's theorem by lowering the regularity class of the motion and we prove a theorem on the existence of a periodic solution if the carriage moves periodically on the plane.