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 coordinateand the conjugate momentumso Hamilton's equationsare coupled together. Of course, for a free particle the Hamiltonianis independent ofThe equations of motion are immediately solvable and givewhereis a constant.

Angle action variables generalise this so that a transformationis found such that the Hamiltonian expressed in the new coordinates is independent ofHamilton'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 whereis the time andis 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 thecoordinates this will form theaxis. The phase curve in thecoordinates will be horizontal lines since

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