In Section 17.10 we showed that canonical transformations form a group, with group multiplication defined as successive transformation. We now show that this group is not Abelian. (The product of two canonical transformations does depend on their order.) Suppose that we make two successive active, differential canonical transformations, first with generator G2 and then with generator G1, as in
(Recall that o (δa)2 means that the dropped terms are of order smaller than (δa)2 as δa → 0. See the definitions in Section D.11.)
for the p variables. Show that
Thus two successive canonical transformations produce a different result when their order of application is reversed.
(c) Compare eqn (18.123) with eqn (18.100). Show that, not only do the canonical transformations in reversed order produce different changes, but (when the calculation is carried to second order) the difference itself is a differential canonical transformation generated by a generating function G = [G1, G2] equal to the Poisson bracket of G1 and G2.
(d) Write a short paragraph justifying the following statement (or demolishing it if you disagree): Since Lx generates rotations about the x-axis, Ly generates rotations about the y-axis, and Lz generates rotations about the z-axis, the Poisson bracket of Lx and Ly must be Lz (as is shown by direct calculation in eqn (12.75)), due to the structure of the rotation group as demonstrated in Exercise 8.11.