9C (Classical Dynamics)

Solution

A one-dimensional canonical transformation is a locally invertible change of phase-space variables preserving the symplectic two-form:

d Q \land d P = d q \land d p

. Equivalently it preserves Hamilton equations for the transformed Hamiltonian. Expanding the two-form yields

d Q \land d P = (Q_{q} P_{p} - Q_{p} P_{q}) d q \land d p,

{Q, P} = Q_{q} P_{p} - Q_{p} P_{q} = 1 .

Conversely this Jacobian condition makes the transformation locally invertible and symplectic in two dimensions.

{Q, P} = α q^{β - 2}

. For it to be identically one on an open region, the exponent must vanish and the coefficient must equal one:

β = 2, α = 1 .

The transformation is defined away from

q = 0

, and its inverse there is

q = 1/ P

p = Q P^{2}

Differentiating gives

{Q, P} = α β q^{2 α - 1} (cos^{2} (βp) + sin^{2} (βp)) = α β q^{2 α - 1} .

Therefore

α = \frac{1}{2}, β = 2 .

For real coordinates take

q > 0

and a local branch of the angle. The transformation is locally canonical transformation because its determinant is one, but periodicity in

p

prevents a globally one-to-one map if all real

p

are retained.

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