Notes on Goldstein’s Routh’s procedure (setup).
Posted by peeterjoot on March 3, 2010
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Attempting study of  section 7-2 on Routh’s procedure has been giving me some trouble. It wasn’t “sinking in”, indicating a fundamental misunderstanding, or at least a requirement to work some examples. Here I pick a system, the spherical pendulum, which has the required ignorable coordinate, to illustrate the ideas for myself with something less abstract.
The Lagrangian for the pendulum is
and our conjugate momenta are therefore
That’s enough to now formulate the Hamiltonian , which is
We’ve got the ignorable coordinate here, since the Hamiltonian has no explicit dependence on it. In the Hamiltonian formalism the constant of motion associated with this comes as a consequence of evaluating the Hamiltonian equations. For this system, those are
The second of these provides the integration constant, allowing us to write, . Once this is done, our Hamiltonian example is reduced to one complete set of conjugate coordinates,
Goldstein notes that the behaviour of the cyclic coordinate follows by integrating
In this example , so this is really just one of our Hamiltonian equations
Okay, good. First part of the mission is accomplished. The setup for Routh’s procedure no longer has anything mysterious to it. Next step for another day.
 H. Goldstein. Classical mechanics. Cambridge: Addison-Wesley Press, Inc, 1st edition, 1951.