Principles of Quantum Mechanics · Problem 2.7.8

All Is Not Well with Classical Mechanics — R. Shankar

Physics — Quantum Mechanics · canonical-transformations-point-transformation

Problem

We would like to derive here Eq. (2.7.9), which gives the transformation of the momenta under a coordinate transformation in configuration space:

q_i \to \bar{q}_i(q_1, \ldots, q_n)

(1) Argue that if we invert the above equation to get

q = q(\bar{q})

, we can derive the following counterpart of Eq. (2.7.7):

\dot{q}_i = \sum_j \frac{\partial q_i}{\partial \bar{q}_j} \dot{\bar{q}}_j

(2) Show from the above that

\left(\frac{\partial \dot{q}_i}{\partial \dot{\bar{q}}_j}\right)_q = \frac{\partial q_i}{\partial \bar{q}_j}

(3) Now calculate

\bar{p}_i = \left[\frac{\partial \mathcal{L}(\bar{q}, \dot{\bar{q}})}{\partial \dot{\bar{q}}_i}\right]_q = \left[\frac{\partial \mathcal{L}(q, \dot{q})}{\partial \dot{\bar{q}}_i}\right]_q

Use the chain rule and the fact that

q = q(\bar{q})

and not

q(\bar{q}, \dot{\bar{q}})

to derive Eq. (2.7.9). (4) Verify, by calculating the PB in Eq. (2.7.18), that the point transformation is canonical.