29 Hamilton-Jacobi Equation : Action Angle Variables

Ashok Goyal

  1. Introduction

 

In many physical situations where the system is complex but exhibits some kind of regularity that is some kind of periodic motion, we may not be as much interested in the details of the motion as in knowing the frequencies of the individual independent motions of the complex system. A very powerful and elegant method which is particularly useful in the field of astronomy is provided by a variation of the Hamilton–Jacobi method. In this technique instead of choosing integrating constants as the new momenta which appear directly in the solution of Hamilton–Jacobi equation, one defines a set of constants which form a set of n independent functions of the integration constants. These independent variables are called Action Variables. The method of action variables also provided application in the early days of Bohr and Sommerfeld to quantum mechanics.

  1. Periodic Motion

Consider a system with one degree of freedom.  For this system, the phase space is a two dimensional ( ) plane. There are two types of periodic motion one can distinguish.

 

a) The first type of periodic motion occurs when the system point returns to its initial position after every period of its motion. The system point traces its steps after every period and the orbit in phase space is closed.This is a truly periodic motion and one–dimensional simple harmonic oscillator is an example. In the field of astronomy such a periodic motion is called ‘libration’.

 

b) In the second type of periodic motion, itself is not periodic but the motion in  is periodic what it means is that as  increases by same value of  say,  returns back to its original position. The simplest example of such periodic motion is rotation of a top about some axis where the generalized coordinate  is taken to be the angle of rotation.

Both types of periodicity may occur in the same physical system.

  1. Action Angle Variables

 

We consider a system with n degree of freedom and that the Hamilton–Jacobi equation is separable in a suitable set of variables  In order to study the periodicity of such a system, we introduce the action variables  of the system as

 

where the integral is over one complete period of oscillation or rotation of  as the case may be. We have one action variable for each degree of freedom. We consider the system to be conservative so that the Hamilton–Jacobi function  can be written as  where  is the Hamilton’s characteristic function. The canonical momenta  are given by

Remember that for holonomic time independent constants, the kinetic energy T is a homogeneous quadratic function of generalized velocities, so that

Thus only one action variable is independent and the Hamiltonian or the energy in terms of action variables is given by

  1. Adiabatic Theorem

 

An adiabatic invariant is defined as a quantity which remains unchanged when the parameters occurring in the Hamiltonian are changed slowly (adiabatically). The Adiabatic theorem states that the action variables are the adiabatic invariants of the system. We will not attempt to give the proof of the theorem. As the action variables are adiabatic invariants, they were found to be suitable candidates for quantization in the early days of Quantum mechanics leading to the Sommerfeld–Wilson quantization rules. In statistical mechanics, the entropy remains unchanged during an adiabatic process.

  1. Summary
  • The action variables constitute important independent variables which are constants of motion. They play important role in the field of astronomy and historically played important role in the early days of quantum mechanics.
  • The fundamental frequencies of oscillations in a system with periodicity can be directly