Lagrange - Mass under potential in spherical

[CNX]

Homework Statement

A particle of mass $m$ moves in a force field whose potential in spherical coordinates is,

$$U = \frac{-K \cos \theta}{r^3}$$

where $K$ is constant.

Identify the two constants of motion of the system.

The Attempt at a Solution

$$L = T - V = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2 \theta ~\dot{\phi}^2) + \frac{K \cos \theta}{r^3}$$

I don't see how there are two constants of motion if the Lagrangian is missing only $\phi$, i.e.,

$$\frac{ \partial L}{\partial \phi} = 0 \Rightarrow \frac{\partial L}{\partial \dot{\phi}} = constant$$

 

[gabbagabbahey]

I'm not 100% sure that this is what the questioner has in mind, but I can think of one quantity that is always a constant of motion whenever the Lagrangian has no explicit time dependence...

 

[CNX]

Energy function/Hamiltonian?

$$\frac{\partial L}{\partial t} = 0 = - \frac{dH}{dt}$$

So H = constant.

 

[gabbagabbahey]

Yup.