Requirements
A powerful technique for systems that have as many constants of the motion as degrees of freedom is called the action-angle variables. Integrable systems in
dimensions have
constants of the motion that we will call
. We would like to transform the Hamiltonian from
and
to a new set of momenta and coordinates,
and
such that
- The equations of motion are preserved
- The new momenta
are functions of the constants of the motion
.
- The new coordinates
are ignorable. They don't appear in the Hamiltonian.
With the Lagrangian technique we were free to use different coordinates to simplify the equations of motion. In the Hamiltonian picture we can define the coordinates and the momenta. Let's do an example. Let's say that we have a particular set of
and
that satisfy the equations of motion, then we must have
from the principle of least action. If we want new set of coordinates and momenta that satisfy the equations of motion we have
The New Variables
For both of these to be true in general it is sufficient (but not necessary) that we have
so the transformation preserves the area. For motion that is periodic in the coordinates, we can take the integrals over a period of the motion in one of the coordinates. To make life easy for ourselves, we would like for new coordinate
to increase linearly in time. For example it could go from 0 to
in one oscillation, so
where
is the time for the particle to oscillation in the coordinate once.
Momenta
To achieve this we could pick
which is clearly a constant of the motion so we have
and the principle of least action is preseved coordinate by coordinate. We can use Hamilton's equations to determine
and system is solved at least formally.
Coordinates
We may like to know the old coordinates in terms of the new ones. We have
Let's take the derivative of both sides with respect to
to yield
where the final equation holds because the initial value of
is arbitrary.
Some Examples
Harmonic Oscillator
For a harmonic oscillator we have
The energy is clearly conserved. Let's take the energy equal to
where
are the bounds of the motion, so we have
What is the area of the path through phase space as
goes from
to
and back again? The equation above clearly defines an ellipse in phase space
of area
, so
and
and
We have found the period of the oscillation without really solving any differential equations and only some basic geometry and algebra. We didn't get any information about the shape of the oscillation
but possibly we weren't interested in that in the first place.
We can find
geometrically as well
that can be solved with
where
.
We can also do this using the integral relationship; we have
so
as before. We can obtain the value of
in terms of
using the conservation of energy.
Gravity
Let's write out the Hamilonian for a particle in a gravitational field of a point mass. Let's use polar coordinates to get
is clearly a constant of the motion so we have
We get this simple result because
is already an angular coordinate (it is periodic in
) and it is cyclic.
The Hamiltonian is clearly constant; let's take its value to be
to give
Let's solve this for
to yield
so
where
and
are the minimum and maximum radii respectively. Continuing we have
We don't know
and
but we don't have to. Let's multiply out the two terms to give
so
Let's solve for
in terms of
and
to yield
so we find that
so the orbits must be closed and we recovered Kepler's third law. We don't know the shape but we do know that the two non-linear frequencies are equal to each other.
Canonical Transformations
The action-angle variables are specific case of a general class of canonical transformations. Canonical transformations are changes in the coordinates and momenta that preserve the equations of motion. Specifcally, we have
from the principle of least action. If we want new set of coordinates and momenta that satisfy the equations of motion we have
The most general set of transformations that satisfy these requirements are called canonical transformations. We can satisfy these if