3. (10 points) The Hamiltonian for a particle moving in a vertical uniform gravitational field g is 2 Hmgq where q is the altitude above the ground. We want to find any canonical transformation from...
3. (10 points) The Hamiltonian for a particle moving in a vertical uniform gravitational field g is 2 Hmgq where q is the altitude above the ground. We want to find any canonical transformation from old variables (q,p) to new variables (Q,P) which provides a cyclic coordinate. To do this, define new variables as where a, b are constants. a) Determine any combination of constants a and b, which provides a canonical transformation b) Find the type 1 generating function, Fi (a, Q) c) Use the relation F2lq, P)- F1 +PQ to find the type 2 generating function and check your result by showing that F2 indeed generates the same transformation d) Find the new Hamiltonian Kfor the new canonical variables Q, P Are there any cyclic variables? e) Solve Hamilton equations for the new canonical variables f) Find the original variables q,p as a function of time
3. (10 points) The Hamiltonian for a particle moving in a vertical uniform gravitational field g is 2 Hmgq where q is the altitude above the ground. We want to find any canonical transformation from old variables (q,p) to new variables (Q,P) which provides a cyclic coordinate. To do this, define new variables as where a, b are constants. a) Determine any combination of constants a and b, which provides a canonical transformation b) Find the type 1 generating function, Fi (a, Q) c) Use the relation F2lq, P)- F1 +PQ to find the type 2 generating function and check your result by showing that F2 indeed generates the same transformation d) Find the new Hamiltonian Kfor the new canonical variables Q, P Are there any cyclic variables? e) Solve Hamilton equations for the new canonical variables f) Find the original variables q,p as a function of time