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 -dimensional phase space with coordinates and momenta , more general transformations are possible:
In order for such a transformation to be canonical, we require that the new variables and satisfy Hamilton’s equations, that is
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 and that determine whether or not the transformation is canonical.
The time derivative of any function can be written as a Poisson bracket:
For the transformed velocities, we have
Here, is written as a function of the original variables. If we write it as a function of the transformed variables, we can find the two derivatives of in 7 by using the chain rule:
Inserting these into 7 we get
In order for this result to satisfy 3, we must have
We can repeat the calculation for :
Requiring this to satsify 4, we have
The last equation is equivalent to
which agrees with 14. Thus in order for the transformation to be canonical, the conditions are
Note that these Poisson brackets require calculating the derivatives of the new variables and with respect to the original ones and , 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.