Nonlinear Dynamics of the 3D Pendulum

TL;DR

This paper explores the complex nonlinear dynamics of a 3D pendulum, including conserved quantities, equilibria, invariant manifolds, and chaos, highlighting its rich behavior compared to simpler models.

Contribution

It introduces full and reduced models of the 3D pendulum and analyzes their nonlinear dynamics, extending understanding beyond planar and spherical cases.

Findings

Identification of conserved quantities and equilibria.

Analysis of local dynamics near equilibria.

Evidence of chaotic motions in the 3D pendulum.

Abstract

A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force. Symmetry assumptions are shown to lead to the planar 1D pendulum and to the spherical 2D pendulum models as special cases. The case where the rigid body is asymmetric and the center of mass is distinct from the pivot location leads to the 3D pendulum. Full and reduced 3D pendulum models are introduced and used to study important features of the nonlinear dynamics: conserved quantities, equilibria, invariant manifolds, local dynamics near equilibria and invariant manifolds, and the presence of chaotic motions. These results demonstrate the rich and complex dynamics of the 3D pendulum.