An Introduction to Hamiltonian Mechanics by Gerardo F. Torres del Castillo

Author:Gerardo F. Torres del Castillo
Language: eng
Format: epub, pdf
ISBN: 9783319952253
Publisher: Springer International Publishing

where m and ω are constants. From (5.18) and (5.27) we have

(which is, indeed, a function of P and t only). Choosing

we have K = 0, and, according to (5.28), , which has to be a constant of motion. This last expression gives q in terms of the constants of motion P and Q.

It may be noticed that if we already have a constant of motion, the solution of the equations of motion can be obtained in an elementary manner by solving a first-order ODE (of course, this is true only when there is one degree of freedom). In the present case, from the Hamilton equations we have and, therefore, , which is a linear equation for q that can be readily solved. However, here we want to emphasize the applications of the canonical transformations.

Exercise 5.24

The function p + mgt is a constant of motion if the Hamiltonian is given by

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