## Notes on Goldstein’s Routh’s procedure (setup).

Posted by peeterjoot on March 3, 2010

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# Motivation.

Attempting study of [1] 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.

# Guts

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

Or, explicitly,

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.

# References

[1] H. Goldstein. *Classical mechanics*. Cambridge: Addison-Wesley Press, Inc, 1st edition, 1951.

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