Multiple pendulum and nonuniform distribution of average kinetic energy

TL;DR

This paper analyzes the distribution of kinetic energy in multiple pendulums, showing nonuniformity due to system constraints and deriving explicit formulas for average energies.

Contribution

It provides analytical expressions for average kinetic energies in multiple pendulums, including exact solutions for the double pendulum case, linking energy distribution to system constraints.

Findings

Average kinetic energy varies along the pendulum length.

Energy distribution aligns with the generalized equipartition principle.

Exact formulas are derived for double pendulum energy distribution.

Abstract

Multiple pendulums are investigated numerically and analytically to clarify the nonuniformity of average kinetic energies of particles. The nonuniformity is attributed to the system having constraints and it is consistent with the generalized principle of the equipartition of energy. With the use of explicit expression for Hamiltonian of a multiple pendulum, approximate expressions for temporal and statistical average of kinetic energies are obtained, where the average energies are expressed in terms of masses of particles. In a typical case, the average kinetic energy is large for particles near the end of the pendulum and small for those near the root. Moreover, the exact analytic expressions for the average kinetic energy of the particles are obtained for a double pendulum.