5.9  Lie Transforms

The evolution of a system under any Hamiltonian generates a continuous family of canonical transformations. To study the behavior of some system governed by a Hamiltonian H, it is sometimes appropriate to use a canonical transformation generated by evolution governed by another Hamiltonian-like function W on the same phase space. Such a canonical transformation is called a Lie transform.

The functions H and W are both Hamiltonian-shaped functions defined on the same phase space. Time evolution for an interval governed by H is a canonical transformation _{, H}. Evolution by W for an interval is a canonical transformation '_{, W}:

The independent variable in the H evolution is time, and the independent variable in the W evolution is an arbitrary parameter of the canonical transformation. We chose ' for the W evolution so that the canonical transformation induced by W does not change the time in the system governed by H.

Figure 5.9 shows how a Lie transform is used to transform a trajectory. We can see from the diagram that the canonical transformations obey the relation

For generators W that do not depend on the independent variable, the resulting canonical transformation '_{, W} is time independent and symplectic. A time-independent symplectic transformation is canonical if the Hamiltonian transforms by composition:³²

We will use only Lie transforms that have generators that do not depend on the independent variable.

Lie transforms of functions

The value of a phase-space function F changes if its arguments change. We define the function E'_{, W} of a function F of phase-space coordinates (t, q, p) by

We say that E'_{, W} F is the Lie transform of the function F.

In particular, the Lie transform advances the coordinate and momentum selector functions Q = I₁ and P = I₂:

So we may restate equation (5.422) as

More generally, Lie transforms descend into compositions:

In terms of E'_{, W} we have the canonical transformation

We can also say

where I is the phase-space identity function: I(t, q, p) = (t, q, p).

Note that E'_{, W} has the property:³³

The identity I is

We can define the inverse function

with the property

Simple Lie transforms

For example, suppose we are studying a system for which a rotation would be a helpful transformation. To concoct such a transformation we note that we intend a configuration coordinate to increase uniformly with a given rate. In this case we want an angle to be incremented. The Hamiltonian that consists solely of the momentum conjugate to that configuration coordinate always does the job. So the angular momentum is an appropriate generator for rotations.

The analysis is simple if we use polar coordinates r, with conjugate momenta p_r, p. The generator W is just:

The family of transformations satisfies Hamilton's equations:

The only variable that appears in W is p, so is the only variable that varies as is varied. In fact, the family of canonical transformations is

So angular momentum is the generator of a canonical rotation.

The example is simple, but it illustrates one important feature of Lie transformations -- they give one set of variables entirely in terms of the other set of variables. This should be contrasted with the mixed-variable generating function transformations, which always give a mixture of old and new variables in terms of a mixture of new and old variables, and thus require an inversion to get one set of variables in terms of the other set of variables. This inverse can be written in closed form only for special cases. In general, there is considerable advantage in using a transformation rule that generates explicit transformations from the start. The Lie transformations are always explicit in the sense that they give one set of variables in terms of the other, but for there to be explicit expressions the evolution governed by the generator must be solvable.

Let's consider another example. This time consider a three-degree-of-freedom problem in rectangular coordinates, and take the generator of the transformation to be the z component of the angular momentum:

The evolution equations are

We notice that z and p_z are unchanged, and that the equations governing the evolution of x and y decouple from those of p_x and p_y. Each of these pairs of equations represents simple harmonic motion, as can be seen by writing them as second-order systems. The solutions are

So we see that again a component of the angular momentum generates a canonical rotation. There was nothing special about our choice of axes, so we can deduce that the component of angular momentum about any axis generates rotations about that axis.

Example

Suppose we have a system governed by the Hamiltonian

Hamilton's equations couple the motion of x and y:

We can decouple the system by performing a coordinate rotation by /4. This is generated by

which is similar to the generator for the coordinate rotation above but without the z degree of freedom. Evolving (; x, y; p_x, p_y) by W for an interval of /4 gives a canonical rotation:

Composing the Hamiltonian H with this time-independent transformation gives the new Hamiltonian

which is a Hamiltonian for two uncoupled harmonic oscillators. So the original coupled problem has been transformed by a Lie transform to a new form for which the solution is easy.