The Dynamics of Pendulums on Surfaces of Constant Curvature

Abstract

In [1], the first and third authors investigated the motion of barbells on surfaces of constant curvature. It is natural to extend this study to pendulums.We define the notion of a pendulum on a surface of constant curvature and study the motion of a mass at a fixed distance from a pivot. We consider some special cases for the pendulum.Case 1: a pivot that moves with constant speed along a fixed geodesic. Case 2: a pivot that undergoes acceleration along a fixed geodesic.

An analytical approach to the external force-free motion of pendulums on surfaces of constant curvature

“…The introduction of Riemannian manifolds and the geometry of their geodesics were motivated by the mechanics of constrained particle systems. In the last few years, the study of force-free motion systems on Riemannian manifolds of constant sectional curvature has attracted the interest of several authors [1], [2], [3], [6], [7] and [8].…”

“…In particular, in [2] the authors define the notion of a pendulum on a surface of constant Gaussian curvature K and they study the motion of a mass at a fixed distance from a pivot. So, a pendulum problem on a surface of constant curvature is defined as a pivot point and a mass connected to that point by a rigid massless rod of fixed length ρ.…”

“…1). In [2] it is studied the pendulum problem when the pivot moves along a geodesic path, and the space is a surface of constant (not zero) curvature. It is considered the surface immersed in R 3 or L 3 and it is obtained the differential motion equation by Newtonian procedure doing a laborious calculation.…”