Equations of Motion, Symmetric Matrix Operators and Determinants of a Multi-Pendula System Oscillating in a Plane

Abstract

An n-tupple chain pendulum system constrained to move in a plane was studied within the framework of a generalized coordinates by using the abridged Lagrangian formalism with a view of developing its equations of motion. This study drew numerous explorations resulting in an exhibition of natural mathematical concepts that had been hitherto either unclear or unknown. The Lagrangian is developed and further solved with ease using matrix algebra resulting in the general Energy equations, Energy symmetric matrix operators, Energy eigen functions and their respective coefficients. As the number of mass units in an n-pendula system increases, more terms are added to the kinetic and potential energy equations, making the motion of the system more dependent on initial conditions. It is generally observed that the angular acceleration for any mass is influenced by the masses and angles of the immediate neighbours.