Physics 31600 R. Wald Classical Mechanics Autumn, 2002

Problem Set VII

Due in class on Tuesday, December 3

1) (a) Show that the transformation on 2n-dimensional phase space associated with a coordinate transformation on configuration space, namely:

q_{i} → Q_{i}(q)

p_{i} → P_{i}(q, p) = ∑_{j} p_{j} \frac{∂q_{j}}{∂Q_{i}}

is a canonical transformation.

(b) On a 2-dimensional phase space, show that the transformation

q → Q = ln[\frac{1}{q} sin p]

p → P = q cot p

is canonical.

2) Give an “elementary” proof of Liouville’s theorem as follows: Introduce local canonical coordinates (q_{1}, p_{1}, q_{2}, p_{2}, ..., q_{n}, p_{n}) and pretend that these are Cartesian coordinates in 2n-dimensional Euclidean space, R^{2n}. Consider a bounded region, R, of phase space covered by these coordinates. Let R_{t} denote the image of R under dynamical evolution by time t. Argue that the volume, V (R_{t}), of R_{t} must satisfy

\frac{dV}{dt} = ∫_{∂R_{t}} h⃗ · n̂dS

where ∂R_{t} denotes the boundary of R_{t}, h⃗ denotes the Hamiltonian vector field, and n̂ denotes the unit outward pointing normal to ∂R_{t}. Then show that dV/dt = 0.

3) (a) Find the generating function, F_{2}(q, P), for the canonical transformation of problem 1(a) above.

(b) Find the generating functions, F_{1}(q, Q) and F_{2}(q, P), for the canonical transformation of problem 1(b) above.

(c) Explicitly choose a function of two variables, f(x, y). Then obtain the canonical transformations on a 2-dimensional phase space that it generates via the generating functions (i) F_{1}(q, Q) = f(q, Q) and F_{2}(q, P) = f(q, P).