A double pendulum, which consists of a mass m suspended by a massless string of length l, from which is suspended another such string and mass, moves in the x-y plane.
(a) Write the Lagrangian of the system.
(b) For small oscillations, i.e. q₁,q₂<<1, derive the equations for the motion.
(c) Find the normal frequencies.

Note: For the sake of uniformity of notation, use w₀= (g/l)^1/2 as the frequency of a single pendulum.

A mass less string is stretched with constant tension between fixed supports separated by a distance 3a. Two identical masses are attached as shown in the diagram. The motion of each mass is constrained to one transverse degree of freedom (along the x-axis).

(a) Calculate the potential energy of the system for arbitrary elongations x₁ and x₂ of the two masses.
(b) Consider small transverse oscillations (x_I<<a) and find the normal mode frequencies and the normal mode eigenvectors of this system.
(c) Describe the motion represented by these solutions.

A mass is hung from a fixed support by a spring of constant k whose relaxed length is A second equal mass is hung from the first mass by an identical spring. Find the six normal coordinates and the corresponding frequencies for small vibrations of this system from its equilibrium position. Each spring exerts a force only along the line joining its two ends, but may pivot freely in any direction at its ends.