A spring of negligible mass, and spring constant (force/extension) k, supports a mass m, and beneath it a second, identical spring, carrying a second, identical mass. Using the vertical displacements x and y of the masses from their positions with the springs unextended as generalized co-ordinates, write down the Lagrangian function. Find the position of equilibrium, and the normal modes and frequencies of vertical oscillations.

A spring of negligible mass, and spring constant (force/extension) k, supports a mass m, and beneath it a second, identical spring, carrying a second, identical mass. Using the vertical displacements x and y of the masses from their positions with the springs unextended as generalized co-ordinates, write down the Lagrangian function. Find the position of equilibrium, and the normal modes and frequencies of vertical oscillations.