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

Posted by peeterjoot on March 3, 2010

[Click here for a PDF of this post with nicer formatting]Note that this PDF file is formatted in a wide-for-screen layout that is probably not good for printing.

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

\begin{aligned}\mathcal{L} = \frac{1}{{2}} m r^2 \left( \dot{\theta}^2 + \dot{\phi}^2 \sin^2 \theta \right) - m g r ( 1 + \cos\theta ),\end{aligned} \hspace{\stretch{1}}(2.1)

and our conjugate momenta are therefore

\begin{aligned}p_\theta &= \frac{\partial {\mathcal{L}}}{\partial {\dot{\theta}}} = m r^2 \dot{\theta} \\ p_\phi &= \frac{\partial {\mathcal{L}}}{\partial {\dot{\phi}}} = m r^2 \sin^2\theta \dot{\phi}.\end{aligned} \hspace{\stretch{1}}(2.2)

That’s enough to now formulate the Hamiltonian H = \dot{\theta} p_\theta + \dot{\phi} p_\phi - \mathcal{L}, which is

\begin{aligned}H = H(\theta, p_\theta, p_phi) = \frac{1}{{2 m r^2 }} (p_\theta)^2 + \frac{1}{{2 m r^2 \sin^2\theta}} (p_\phi)^2 + m g r ( 1 + \cos\theta ).\end{aligned} \hspace{\stretch{1}}(2.4)

We’ve got the ignorable coordinate \phi 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

\begin{aligned}\frac{\partial {H}}{\partial {\theta}} &= - \dot{p}_\theta \\ \frac{\partial {H}}{\partial {\phi}} &= - \dot{p}_\phi \\ \frac{\partial {H}}{\partial {p_\theta}} &= \dot{\theta} \\ \frac{\partial {H}}{\partial {p_\phi}} &= \dot{\phi},\end{aligned} \hspace{\stretch{1}}(2.5)

Or, explicitly,

\begin{aligned}- \dot{p}_\theta &= -m g r \sin\theta - \frac{\cos\theta}{2 m r^2 \sin^3 \theta} (p_\phi)^2  \\ - \dot{p}_\phi &= 0 \\ \dot{\theta} &= \frac{1}{{m r^2 }} p_\theta \\ \dot{\phi} &= \frac{1}{{m r^2 \sin^2 \theta}} p_\phi.\end{aligned} \hspace{\stretch{1}}(2.9)

The second of these provides the integration constant, allowing us to write, p_\phi = \alpha. Once this is done, our Hamiltonian example is reduced to one complete set of conjugate coordinates,

\begin{aligned}H(\theta, p_\theta, \alpha) = \frac{1}{{2 m r^2 }} (p_\theta)^2 + \frac{1}{{2 m r^2 \sin^2\theta}} \alpha^2 + m g r ( 1 + \cos\theta ).\end{aligned} \hspace{\stretch{1}}(2.13)

Goldstein notes that the behaviour of the cyclic coordinate follows by integrating

\begin{aligned}\dot{q}_n = \frac{\partial {H}}{\partial {\alpha}}.\end{aligned} \hspace{\stretch{1}}(2.14)

In this example \alpha = p_\theta, so this is really just one of our Hamiltonian equations

\begin{aligned}\dot{\phi} = \frac{\partial {H}}{\partial {p_\phi}}.\end{aligned} \hspace{\stretch{1}}(2.15)

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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One Response to “Notes on Goldstein’s Routh’s procedure (setup).”

  1. […] Notes on Goldstein’s Route’s procedure (setup). […]

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