## 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

which is

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.

# References

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

## Notes on Goldstein’s Routh’s procedure (continued again.) « Peeter Joot's Blog. said

[…] This continues the Routhian procedure notes from last post. […]

## Asha said

As you are taking partial derivatives of L, H and R, which are depend on multiple variables, it is helpful (and possibly clear up some confusion early on) to write the respective arguments of the functions for every circumstance.

## peeterjoot said

I ended up figuring out which variables to use in the followup to this, but you are right, that would have helped.