Integrals of the motion

From QED

Within a mechanical system with s degree of freedom (which is completely described by s generalised coordinates and s velocities), there exist functions of these quantities which remain constant in time. Such functions, if they can be identified, are often extraordinarily useful in practice. Examples include the energy and the momentum.

There are s equations of motion of the system. These are second order differential equations. The solutions to thse equations have 2 constants each, which would be completely specified by the 2s generalised variables. However, since each of the 2s variables has a time dependence, the time may always be eliminated from the 2s constants, as the starting point is completely arbitrary. Thus there are 2s − 1 independent integrals of the motion.

It may be helpful to note that the 2s − 1 integrals of the motion completely specify which path in phase space the system is on, and the only remaining variable to describe the system is the time, or some other coordinate, to describe the point on the path in phase space which the system currently occupies.

This page was recovered in October 2009 from the Plasmagicians page on Integrals_of_the_motion dated 18:36, 11 October 2006.