Three balls are constrained to move on the circumference of a circle of radius R. The balls are connected to each other along the circumference of the circle by three identical springs with spring constant k so that in the equilibrium configuration, the springs all have their common rest length, and the balls are spaced evenly around the circle. The mass of the first ball is M; the other two balls have mass m, m
a) Choose as coordinates the angles a1, a2, and a3, the angular displacements of the three balls from the equilibrium configuration. Derive the Euler-Lagrange equations for the system in terms of these angles.
b) Using the result of part a, find the characteristic frequencies of the system.
c) What happens if m=M?