# Routhian

The Routhian is a hybrid of the Lagrangian and the Hamiltonian developed by Edward John Routh in the branch of physics known as dynamics. The Hamiltonian can be obtained by a Legendre transform of the Lagrangian; if this transformation is performed only with regards to cyclic coordinates (those not appearing explicitly in the Lagrangian), the result is called the Routhian. This method takes advantage of the ease with which the Hamiltonian deals with cyclic variables.[1]

## Example

A good example is a particle under the influence of a central gravitational field in polar coordinates:

$L = \frac{m}{2}(\dot r^2 + r^2\dot{\theta^2}) + \frac{k}{r}.$

$\theta$ is cyclic, because it doesn't appear in the Lagrangian. Therefore, only transforming $\theta$ leads to

$p_{\theta} = \frac{\partial L}{\partial \dot {\theta}} = mr^2\dot {\theta}$
$R = p_{\theta}\dot{\theta} - L = \frac{p_{\theta}^2}{2mr^2} - \frac{1}{2}m\dot {r}^2 - \frac{k}{r}$

We can solve for the equation of motion of $r$ in the normal Lagrangian way:

$m\ddot{r}=\frac{p_{\theta}^2}{mr^3}-\frac{k}{r^2}$

And we can solve for the equation of motion of $\theta$ in the Hamiltonian way:

$\dot{p_{\theta}} = -\frac{\partial {R}}{\partial {\theta}} = 0, \quad \dot {\theta} = \frac{\partial {R}}{\partial {p_{\theta}}} = \frac{p_{\theta}}{mr^2}$
$p_{\theta} = mr^2 \dot {\theta} = l$

In this case the Routhian approach contributes nothing new; this equation for $p_{\theta}$ is the same one as derived above. However, for problems with many degrees of freedom, the Routhian can simplify calculations. This method is simply a combination of the advantages of the Hamiltonian and Lagrangian, and does not produce any new physics.

## References

1. ^ Goldstein, Herbert; Poole, Charles P., Jr.; Safko, John L. (2002). Classical Mechanics (3rd ed.). San Francisco, CA: Addison Wesley. pp. 347–349. ISBN 0-201-65702-3.