On this problem, we'll get a little bit of practice using Hamiltonian mechanics with a simple but somewhat richer example than the free particle and simple harmonic oscillator that we studied in class. Consider a particle with mass m moving in the plane with polar coordinates (r, θ), in the presence of a central potential V(r).
a. The canonical momenta p conjugate to a canonical coordinate q can be defined for a time-independent system as p = ∂L/∂q, where L = T - V is the Lagrangian for this system. Find expressions for the canonical momenta pr and pθ. Do they have units of momentum? Why or why not? Interpret each physically. (Hint: recall that the kinetic energy in polar coordinates can be expressed as the sum of the radial and angular kinetic energies.)
b. Write down the Hamiltonian using r and θ as canonical coordinates: H = pr²/2m + pθ²/(2mr²) + V(r), where T is the kinetic energy of the particle.
c. Use Hamilton's equations to derive the equations of motion for pr and pθ.
d. Recall, as we discussed in class, a canonical coordinate which does not appear in the Hamiltonian is called cyclical and produces a conserved quantity. Are either of the two coordinates cyclical? Why? If so, what is the conserved quantity associated with it?
e. Note that pr = -∂Veff/∂r is the radial force and is given by Hamilton's equations in terms of an effective potential Veff. What is the effective potential for this problem? Does it consist solely of the potential V(r)? Give a physical interpretation for each of the terms in the effective potential.