Q.n.4 Consider two particles of masses m1 and m2. Let m1 be confined to move on a circle of radius a in the z = 0 plane, centered at x = y = 0. Let m2 be confined to move on a circle of radius b in the z =c plane, centered at x = y = 0. A light (massless) spring of spring constant k is attached between the two particles. a) Find the Lagrangian for the system.

Part b

Lagrangian Dynamics Ep = 0 A pendulum of length / and mass m is mounted on a block of mass M. The block can move freely without friction on a horizontal surface as shown in the adjacent figure H. 1. Find the velocity of mass m, w.r.t the origin O 2. Write the Lagrangian of the system 3. Derive the Euler Lagrange equations

Problem 2 The relativistic Lagrangian for a particle of rest mass m moving along the x-axis in a potential V(x) is given by 2 L = -mc² 1 V(x) c2 (a) Derive the Euler-Lagrange equation of motion. (b) Show that it reduces to Newton's equation in the limit |*| << c. (c) Compute the Hamiltonian H of the system. Eliminate ȧ from the Hamiltonian by using the equation ƏL p = ax and write H = H(p, x) as a function of x and p only.