Canonical transformation theory

The Hamiltonian phase space approach to mechanics has many fundamental advantages over Lagrangians in addition to providing a path to canonical quantization. One of these is the insight it gives into coordinate transformations in phase space.

In many situation, the initial phase-space coordinates (p, q), referred to as the old coordinates, are not the best coordinates to use. Often the observer will consider transforming to new phase-space coordinates (P, Q) related in some specific way to the old coordinates by a rule of the form

where we include the possibility of time dependence in the coordinate transformation.

Not all such transformations are useful. Canonical transformations are those for which Hamilton’s equations of motion (14.7) are satisfied in the new coordinates as well as in the old, and these will be of primary interest to us for the following reasons:

1. Restricting transformations to be canonical ensures that we have a unified approach to mechanics.

2. Quantum mechanics is often obtained from a classical mechanical description based on Hamilton’s equations of motion.

Restricted canonical transformations are such that there is no explicit time dependence in the transformation, that is, the phase-space coordinate transformations takes the form

Point transformations are even more restrictive, being transformations of the form

Although Lagrangians and Hamiltonians are related, they are fundamentally different objects. In special relativistic mechanics, for example, a point particle Lagrangian is usually a Lorentz scalar,38 whereas the corresponding Hamiltonian is the zeroth (or time) component of the energy-momentum four-vector. With this in mind, we shall assume that Lagrangians are invariant to canonical transformations. If L' is the Lagrangian at a point on a trajectory as seen in the new coordinates, we assume

We note that in this discussion, time is being treated as absolute, that is, it is the same parameter in old and new coordinate descriptions.

Again, taking a lead from special relativity, we cannot insist that the Hamiltonian H' = H'(P, Q, t) in the new phase-space coordinates is the same as the Hamiltonian in the old coordinates. However, since we are using a canonical transformation, we can say that

where

Imposing the condition that the transformation is canonical means that

38 A Lorentz scalar is a quantity in SR, the value of which is independent of inertial frame.

Standard analysis [Goldstein et al., 2002; Leech, 1965] then leads to the relation

for some function F of the variables q, Q, p, P, and t. Because of the transformation equations (14.10), there are only 2n +1 variables. Standard transformation theory classifies four main transformation function types. We shall be interested in the type known as F2 = F2 (q, P, t), for which the rules are

 
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