Conserved quantities in the cart and pendulum problem

[ct1993]

A problem on an assignment I'm doing deals with a cart of mass m1 which can slide frictionlessly along the x-axis. Suspended from the cart by a string of length l is a mass m2, which is constrained to move in the x-y plane. The angle between the pendulum and vertical is notated as phi. The question asks me to first derive the Lagrangian for the system which I got as:

It then asks me to find any continuous symmetrys in the problem and the corresponding conserved quantities. I found the Euler-Lagrange equations for x:

and for phi:

From the E-L equation for x, I know that:

Which I believe indicates the momentum in the x-direction is conserved, but I'm not sure. I think I'm supposed to do this with some application of Noether theorem, which was supposed to be covered today, but class was canceled due to snow and the assignment is still due Wednesday. Can anyone help me?

 

[BvU]

Hello ct, welcome to PF. :)

Something went wrong with the template: it shows up when you start a thread but now it's gone for some inexplicable (?) reason. Not good, because its use is mandatory in PF -- for good reasons !

Anyway, your post is quite extensive and merits a response. Symmetry is incredibly important in physics. Did you notice that in your Lagrangian there is no x (that's why the template is so important: I can guess what your x stands for, but misunderstandings are lurking just around the corner. Thousands of examples in PF, all a waste of time, goodwill, energy and what else :) ).

There is $\ddot x$, but $L$ is invariant under a change of coordinate x (I mean a coordinate transformation: picking a different origin). That means momentum conservation (Emmy N's theorem is a fancy way of deriving that -- "the canonical momentum associated with x, $\partial L\over \partial \dot x\$"). As you show.

I think you did just fine.