4. Consider a harmonic oscillator in two dimensions described by the Lagrangian m mw? L = "( +r*ở*) - 2 where (r, o) are the polar coordinates, m > 0 is the mass, and w > 0 the oscillation frequency of the oscillator. (a) Find the canonical momenta p, and po, and show that the Hamiltonian is given by mw? H(r, ø, Pr: Po) 2m' 2mr2 2 (b) Show that p, is a constant of motion. (c) In the following we will perform a variable transformation (r, 6, pr; Ps) → (Qr, Qó, Pr, Po defined by Q, = lp, P, Po Po = 2 Qo = 20, 2r where l is an arbitrary constant length that just ensures that Q, has also the dimension of length. Show that this transformation is canonical.

4. Consider a harmonic oscillator in two dimensions described by the Lagrangian m mw? L = "( +rở) - 2 where (r, o) are the polar coordinates, m > 0 is the mass, and w > 0 the oscillation frequency of the oscillator. (a) Find the canonical momenta p, and po, and show that the Hamiltonian is given by mw? H(r, ø, Pr: Po) 2m' 2mr2 2 (b) Show that p, is a constant of motion. (c) In the following we will perform a variable transformation (r, 6, pr; Ps) → (Qr, Qó, Pr, Po defined by Q, = lp, P, Po Po = 2 Qo = 20, 2r where l is an arbitrary constant length that just ensures that Q, has also the dimension of length. Show that this transformation is canonical.