Consider a single particle of mass m in spherical coordinates, with the kinetic energy P² p²+ + 7.2 p² r2 sin² 0 IT= 1 2m (a) Write the hamiltonian H and find Hamilton's equations in the case of a central potential V(r). Discuss how the z axis of spherical coordinates should be chosen to simplify the problem to a 2D situation. Write the corresponding 2D hamiltonian. Recover the standard expression for dt in terms of dr, r, E, V(r), l, m. (b) Consider now V(r) = V₁(r)âà f, where â is a fixed unit vector and is the unit vector along the direction of F. V₁ (r) <0 is a function of the distance r = r. (i) Choose an appropriate direction for the z axis, write the hamiltonian H, and find Hamilton's equations. Show that one component of angular momentum is conserved. (ii) Propose a circular motion solution for the equations of motion, with its axis of rotation along â. Find a condition of the form fr(ro) = fe(o)

Consider a single particle of mass m in spherical coordinates, with the kinetic energy P² p²+ + 7.2 p² r2 sin² 0 IT= 1 2m (a) Write the hamiltonian H and find Hamilton's equations in the case of a central potential V(r). Discuss how the z axis of spherical coordinates should be chosen to simplify the problem to a 2D situation. Write the corresponding 2D hamiltonian. Recover the standard expression for dt in terms of dr, r, E, V(r), l, m. (b) Consider now V(r) = V₁(r)âà f, where â is a fixed unit vector and is the unit vector along the direction of F. V₁ (r) <0 is a function of the distance r = r. (i) Choose an appropriate direction for the z axis, write the hamiltonian H, and find Hamilton's equations. Show that one component of angular momentum is conserved. (ii) Propose a circular motion solution for the equations of motion, with its axis of rotation along â. Find a condition of the form fr(ro) = fe(o)