An inverted pendulum with a moving pivot point: examples of topological approach

TL;DR

This paper explores the dynamics of an inverted pendulum with a moving pivot using topological methods, demonstrating the existence of solutions and periodic motions without falling, based on advanced fixed point theories.

Contribution

It applies topological principles, including Wa{\.z}ewski and Lefschetz-Hopf theories, to establish existence results for solutions and periodic motions of a moving-pivot inverted pendulum.

Findings

Existence of solutions without falling for arbitrary pivot motions.

Existence of periodic solutions when the pivot motion is periodic.

Periodic solutions where the pendulum never becomes horizontal.

Abstract

Two examples concerning an application of topology in the study of the dynamics of an inverted plain mathematical pendulum with a pivot point moving along a horizontal straight line are considered. The first example is an application of the Wa{\.z}ewski principle to the problem of the existence of a solution without falling in the case of a arbitrary prescribed law of motion of the pivot point. The second example is a proof of the existence of periodic solution in the same system when the law of motion is periodic as well. Moreover, in the second case it is also shown that along the obtained periodic solution the pendulum never becomes horizontal (falls). The proof is an example of application of the recent developments in the fixed point theory based on the Lefschetz-Hopf theory.