Section A
Q1. Define generalized coordinates. How many generalized coordinates are required to describe the dynamics of Write down the expression for kinetic energy of a relativistic particle with rest mass m0, moving with a speed v. The Lagrangian of a system is given by L (\vecr, \vecv) = 1/2m (\vecv + \veca)2 + \vecb \cdot \vecv - \vecc \cdot \vecr where, \veca, \vecb and \vecc are constant vectors. Construct the Hamiltonian of the system and find the canonical equations of motion. Explain the phenomenon of double-refraction in calcite crystal with special reference to Ordinary Ray (OR), Extraordinary Ray (ER) and uniaxial negative crystal. Draw the energy level diagram of He-Ne laser. The metastable state of ruby laser is at 1.786 eV. Calculate the wavelength of the light emitted.
(a) a free rigid body in 3 dimension ?
(b) a solid cylinder rolling in an inclined plane without slipping ?
(a) Show that the kinetic energy reduces to 1/2 m0 v2 in the non-relativistic limit.
(b) Find the first relativistic correction to the non-relativistic kinetic energy.