Notes on Goldstein’s Routh’s procedure (continued)
Posted by peeterjoot on March 5, 2010
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This continues the Routhian procedure notes started previously.
Now, Goldstein defines the Routhian as
where the index is summed only over the cyclic (ignorable) coordinates. For this spherical pendulum example, this is , and , for
Now, we should also have for the non-cyclic coordinates, just like the Euler-Lagrange equations
Evaluating this we have
It would be reasonable now to compare this the Euler-Lagrange equations, but evaluating those we get
Bugger. We’ve got a sign difference on the term? But I don’t see any error made.
Simpler planar example.
Having found an inconsistency with Routhian formalism and the concrete example of the spherical pendulum which has a cyclic coordinate as desired, let’s step back slightly, and try a simpler example, artificially constructed
Our Hamiltonian and Routhian functions are
For the non-cyclic coordinate we should have
Okay, good, that’s what is expected, and exactly what we get from the Euler-Lagrange equations. This looks good, so where did things go wrong in the spherical pendulum evaluation.
 H. Goldstein. Classical mechanics. Cambridge: Addison-Wesley Press, Inc, 1st edition, 1951.