Introduction to Angle Action Variables
It is desirable to construct a coordinate system in which the Hamiltonian has the simplest possible form. In the case of rotational or librational motion one of those coordinates behaves like an angle and the conjugate variable is called the action – in fact it is only for rotational or librational motion that angle action variables are defined. The coordinates taken together are called angle action variables. Angle action variables are of fundamental importance in quantum mechanics, where they are used for example to model the radiation emitted from quantum systems.
Most Hamiltonians
for functions of both the generalised coordinate
and the conjugate momentum
so Hamilton's equations
are coupled together. Of course, for a free particle the Hamiltonian
is independent of
The equations of motion are immediately solvable and give
where
is a constant.
Angle action variables generalise this so that a transformation
is found such that the Hamiltonian expressed in the new coordinates is independent of
Hamilton's equations can be solved immediately in the new coordinate system and transformed back to express the solution in the original coordinates.
If the Hamiltonian system is conservative, then we can find a transformation
where
is the time and
is the (constant) energy.andcan be defined by choosing a phase curve, and a point on that curve, to be regarded as the initial point. The collection of initial points forms a curve AB at right angles to each phase curve and in the
coordinates this will form the
axis. The phase curve in thecoordinates will be horizontal lines since
