a. Find momenta conjugate to (§, n, o). b. Determine Hamiltonian and Hamilton's equations of motion. [6] Consider a particle executing simple harmonic motion in one dimension. The potential energy of the particle is V(x) = kr², where k is the stiffness constant. Write down the action for the system. Deduce the equation of motion using variational principle. [show the steps explicitly to arise at the Euler-Lagrange equation.] [7] The infinitesimal length on a sphere of radius R is written as ds² = dx² + dy² + dz² a. Write down ds2 in spherical polar coordinates. b. Using calculus of variation, determine the shortest length on the sphere (in terms of gener- alised coordinates) and show that it is in fact a great circle. [8] Determine following quantitics for a particle moving on a 2-D plane under potential V(r). a. Find conjugate momenta to (r, p). b. Write down Hamiltonian and Hamilton's equations of motion for the system. c. Determine the cyclic coordinate and comment on the conserved quantity.

a. Find momenta conjugate to (§, n, o). b. Determine Hamiltonian and Hamilton's equations of motion. [6] Consider a particle executing simple harmonic motion in one dimension. The potential energy of the particle is V(x) = kr², where k is the stiffness constant. Write down the action for the system. Deduce the equation of motion using variational principle. [show the steps explicitly to arise at the Euler-Lagrange equation.] [7] The infinitesimal length on a sphere of radius R is written as ds² = dx² + dy² + dz² a. Write down ds2 in spherical polar coordinates. b. Using calculus of variation, determine the shortest length on the sphere (in terms of gener- alised coordinates) and show that it is in fact a great circle. [8] Determine following quantitics for a particle moving on a 2-D plane under potential V(r). a. Find conjugate momenta to (r, p). b. Write down Hamiltonian and Hamilton's equations of motion for the system. c. Determine the cyclic coordinate and comment on the conserved quantity.