Within Hamilton-Jacobi theory, one aims to transform a set (q,p) of 2n canonical variables into a new set (q', p') of canonical variables where the equations of motion are particularly easy to solve. In the lectures, we developed Hamilton-Jacobi theory using a generating function of type 2. In this exercise, we will apply Hamilton-Jacobi theory using a generating function of type 1 instead.
Given a generating function of type 1, S = S(q,q',t), show that imposing that the new Hamiltonian is zero, i.e., H' = 0, leads to the following Hamilton-Jacobi equation:
∂S/∂t + H(q, ∂S/∂q, t) = 0
where H = H(q,p,t) is the original Hamiltonian.
Now consider a free particle of mass m = 1, so that the Hamiltonian (n = 1) is
H(p) = p^2/2
Show that the following two functions satisfy the Hamilton-Jacobi equation (1):
(a) S_a(q,q',t) = (a-q)^2/2t
(b) S_b(q,q',t) = q√(v^2q' - q't)
Next, find q(t) and p(t) using the transformation equations for both cases S_a and S_b separately. Finally, show that q' is equal to the initial position q(t = 0) in the case of S_a and to the energy H in the case of S_b.
