Chapter 7: Invariant Measures in Phase Space and Various Forms of Development in Time

The two equivalent ways of defining canonical transformations using either PB's or LB's. Eqs.(5.28) and (5.29), are both differential in character. These conditions involving the fundamental brackets have been shown to be completely equivalent to the invariance properties, under canonical transformations, of the PB's and LB's among arbitrary functions on phase space. There is another type of entity, namely, the collection of volume and surface elements of various dimensions in phase space, which are also invariant under canonical transformations. We turn now to a discussion of these expressions…