# Conditions for a transformation to be canonical

References: Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Section 2.7; Exercise 2.7.3.

We’ve seen that the Euler-Lagrange equations are invariant under canonical transformations, but in the Hamiltonian formalism where the system moves in a ${2n}$-dimensional phase space with ${n}$ coordinates ${q}$ and ${n}$ momenta ${p}$, more general transformations are possible:

 $\displaystyle \overline{q}_{i}$ $\displaystyle =$ $\displaystyle \overline{q}_{i}\left(q,p\right)\ \ \ \ \ (1)$ $\displaystyle \overline{p}_{i}$ $\displaystyle =$ $\displaystyle \overline{p}_{i}\left(q,p\right) \ \ \ \ \ (2)$

In order for such a transformation to be canonical, we require that the new variables ${\overline{q}}$ and ${\overline{p}}$ satisfy Hamilton’s equations, that is

 $\displaystyle \frac{\partial H}{\partial\overline{p}_{i}}$ $\displaystyle =$ $\displaystyle \dot{\overline{q}}_{i}\ \ \ \ \ (3)$ $\displaystyle -\frac{\partial H}{\partial\overline{q}_{i}}$ $\displaystyle =$ $\displaystyle \dot{\overline{p}}_{i} \ \ \ \ \ (4)$

In principle, then, we could check the Hamiltonian in the new coordinates to see if these equations are valid, but it would seem that whether or not a set of coordinates and momenta is canonical should be determinable from the variables themselves, and not depend on the specific Hamiltonian. Here we derive a set of conditions on the ${\overline{q}}$ and ${\overline{p}}$ that determine whether or not the transformation is canonical.

The time derivative of any function ${\omega}$ can be written as a Poisson bracket:

$\displaystyle \dot{\omega}=\left\{ \omega,H\right\} \ \ \ \ \ (5)$

For the transformed velocities, we have

 $\displaystyle \dot{\overline{q}}_{j}$ $\displaystyle =$ $\displaystyle \left\{ \overline{q}_{j},H\right\} \ \ \ \ \ (6)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \sum_{i}\left(\frac{\partial\overline{q}_{j}}{\partial q_{i}}\frac{\partial H}{\partial p_{i}}-\frac{\partial\overline{q}_{j}}{\partial p_{i}}\frac{\partial H}{\partial q_{i}}\right) \ \ \ \ \ (7)$

Here, ${H}$ is written as a function ${H\left(q,p\right)}$ of the original variables. If we write it as a function of the transformed variables, we can find the two derivatives of ${H}$ in 7 by using the chain rule:

 $\displaystyle \frac{\partial H\left(\overline{q},\overline{p}\right)}{\partial p_{i}}$ $\displaystyle =$ $\displaystyle \sum_{k}\left(\frac{\partial H}{\partial\overline{q}_{k}}\frac{\partial\overline{q}_{k}}{\partial p_{i}}+\frac{\partial H}{\partial\overline{p}_{k}}\frac{\partial\overline{p}_{k}}{\partial p_{i}}\right)\ \ \ \ \ (8)$ $\displaystyle \frac{\partial H\left(\overline{q},\overline{p}\right)}{\partial q_{i}}$ $\displaystyle =$ $\displaystyle \sum_{k}\left(\frac{\partial H}{\partial\overline{q}_{k}}\frac{\partial\overline{q}_{k}}{\partial q_{i}}+\frac{\partial H}{\partial\overline{p}_{k}}\frac{\partial\overline{p}_{k}}{\partial q_{i}}\right) \ \ \ \ \ (9)$

Inserting these into 7 we get

 $\displaystyle \dot{\overline{q}}_{j}$ $\displaystyle =$ $\displaystyle \sum_{i}\sum_{k}\left[\frac{\partial\overline{q}_{j}}{\partial q_{i}}\left(\frac{\partial H}{\partial\overline{q}_{k}}\frac{\partial\overline{q}_{k}}{\partial p_{i}}+\frac{\partial H}{\partial\overline{p}_{k}}\frac{\partial\overline{p}_{k}}{\partial p_{i}}\right)-\frac{\partial\overline{q}_{j}}{\partial p_{i}}\left(\frac{\partial H}{\partial\overline{q}_{k}}\frac{\partial\overline{q}_{k}}{\partial q_{i}}+\frac{\partial H}{\partial\overline{p}_{k}}\frac{\partial\overline{p}_{k}}{\partial q_{i}}\right)\right]\ \ \ \ \ (10)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \sum_{k}\frac{\partial H}{\partial\overline{q}_{k}}\sum_{i}\left(\frac{\partial\overline{q}_{j}}{\partial q_{i}}\frac{\partial\overline{q}_{k}}{\partial p_{i}}-\frac{\partial\overline{q}_{j}}{\partial p_{i}}\frac{\partial\overline{q}_{k}}{\partial q_{i}}\right)+\sum_{k}\frac{\partial H}{\partial\overline{p}_{k}}\sum_{i}\left(\frac{\partial\overline{q}_{j}}{\partial q_{i}}\frac{\partial\overline{p}_{k}}{\partial p_{i}}-\frac{\partial\overline{q}_{j}}{\partial p_{i}}\frac{\partial\overline{p}_{k}}{\partial q_{i}}\right)\ \ \ \ \ (11)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \sum_{k}\frac{\partial H}{\partial\overline{q}_{k}}\left\{ \overline{q}_{j},\overline{q}_{k}\right\} +\sum_{k}\frac{\partial H}{\partial\overline{p}_{k}}\left\{ \overline{q}_{j},\overline{p}_{k}\right\} \ \ \ \ \ (12)$

In order for this result to satisfy 3, we must have

 $\displaystyle \left\{ \overline{q}_{j},\overline{q}_{k}\right\}$ $\displaystyle =$ $\displaystyle 0\ \ \ \ \ (13)$ $\displaystyle \left\{ \overline{q}_{j},\overline{p}_{k}\right\}$ $\displaystyle =$ $\displaystyle \delta_{jk} \ \ \ \ \ (14)$

We can repeat the calculation for ${\dot{\overline{p}}_{i}}$:

 $\displaystyle \dot{\overline{p}}_{j}$ $\displaystyle =$ $\displaystyle \left\{ \overline{p}_{j},H\right\} \ \ \ \ \ (15)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \sum_{i}\left(\frac{\partial\overline{p}_{j}}{\partial q_{i}}\frac{\partial H}{\partial p_{i}}-\frac{\partial\overline{p}_{j}}{\partial p_{i}}\frac{\partial H}{\partial q_{i}}\right)\ \ \ \ \ (16)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \sum_{i}\sum_{k}\left[\frac{\partial\overline{p}_{j}}{\partial q_{i}}\left(\frac{\partial H}{\partial\overline{q}_{k}}\frac{\partial\overline{q}_{k}}{\partial p_{i}}+\frac{\partial H}{\partial\overline{p}_{k}}\frac{\partial\overline{p}_{k}}{\partial p_{i}}\right)-\frac{\partial\overline{p}_{j}}{\partial p_{i}}\left(\frac{\partial H}{\partial\overline{q}_{k}}\frac{\partial\overline{q}_{k}}{\partial q_{i}}+\frac{\partial H}{\partial\overline{p}_{k}}\frac{\partial\overline{p}_{k}}{\partial q_{i}}\right)\right]\ \ \ \ \ (17)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \sum_{k}\frac{\partial H}{\partial\overline{q}_{k}}\sum_{i}\left(\frac{\partial\overline{p}_{j}}{\partial q_{i}}\frac{\partial\overline{q}_{k}}{\partial p_{i}}-\frac{\partial\overline{p}_{j}}{\partial p_{i}}\frac{\partial\overline{q}_{k}}{\partial q_{i}}\right)+\sum_{k}\frac{\partial H}{\partial\overline{p}_{k}}\sum_{i}\left(\frac{\partial\overline{p}_{j}}{\partial q_{i}}\frac{\partial\overline{p}_{k}}{\partial p_{i}}-\frac{\partial\overline{p}_{j}}{\partial p_{i}}\frac{\partial\overline{p}_{k}}{\partial q_{i}}\right)\ \ \ \ \ (18)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \sum_{k}\frac{\partial H}{\partial\overline{q}_{k}}\left\{ \overline{p}_{j},\overline{q}_{k}\right\} +\sum_{k}\frac{\partial H}{\partial\overline{p}_{k}}\left\{ \overline{p}_{j},\overline{p}_{k}\right\} \ \ \ \ \ (19)$

Requiring this to satsify 4, we have

 $\displaystyle \left\{ \overline{p}_{j},\overline{p}_{k}\right\}$ $\displaystyle =$ $\displaystyle 0\ \ \ \ \ (20)$ $\displaystyle \left\{ \overline{p}_{j},\overline{q}_{k}\right\}$ $\displaystyle =$ $\displaystyle -\delta_{jk} \ \ \ \ \ (21)$

The last equation is equivalent to

$\displaystyle \left\{ \overline{q}_{j},\overline{p}_{k}\right\} =\delta_{jk} \ \ \ \ \ (22)$

which agrees with 14. Thus in order for the transformation to be canonical, the conditions are

 $\displaystyle \left\{ \overline{q}_{j},\overline{q}_{k}\right\}$ $\displaystyle =$ $\displaystyle \left\{ \overline{p}_{j},\overline{p}_{k}\right\} =0\ \ \ \ \ (23)$ $\displaystyle \left\{ \overline{q}_{j},\overline{p}_{k}\right\}$ $\displaystyle =$ $\displaystyle \delta_{jk} \ \ \ \ \ (24)$

Note that these Poisson brackets require calculating the derivatives of the new variables ${\overline{q}}$ and ${\overline{p}}$ with respect to the original ones ${q}$ and ${p}$, but they don’t involve any particular Hamiltonian. Thus it’s possible to determine whether or not a transformation is canonical entirely from the transformation equations 1 and 2.