Lagrangian Mechanics, Conservation of Momentum problem

[fisica1988]

Hello, I am just starting Lagrangian mechanics and on a conservation problem but I am stuck. I have part of the solution (i think) but I am not sure how to complete it.

Homework Statement

In an infinite homogeneous plane. Find all the components of momentum and angular momentum that are conserved.

Homework Equations

Obviously there are:
$$\frac{\partial L}{\partial q_i} = f_i$$
$$p_i = \frac{\partial L}{\partial \dot{q}_i}$$
$$f_i = {\dot{p}_i}$$

The Attempt at a Solution

I said that in the infinite homogeneous plane (in the xy plane) since the potential energy doesn't depend on x or y displacement alone these coordinates did not change the Lagrangian. Therefore:

$$L = L(x, y, z, \dot{x}, \dot{y}, \dot{z}) = L(x+e, y+e, z, \dot{x}, \dot{y}, \dot{z})$$
Where e is a change in the coordinates (I'm assuming you can displace them equally and still get back the Lagrangian. If I'm wrong please correct me) .

So since the Lagrangian is translationally invariant I said:

$$f_x = \frac{\partial L}{\partial x} = 0 =\dot{p}_x$$

If $$\dot{p}_x = 0$$, the rate of change of the momentum in the x-direction is constant ( $$p_x = constant$$ ).

Likewise for the y-direction of momentum:

$$f_y = \frac{\partial L}{\partial y} = 0 =\dot{p}_y$$
$$p_y = constant$$
Does this show, sufficently, mathematically that the x and y components of momentum are conserved?

My central difficulty is showing that, since the infinite homogeneous plane is centrally symmetric about the z-axis, show the z-component of angular momentum, M_z is conserved. I'm unclear on how to do that.

-Thanks

Edit: P.S. the reason I have trouble with the M_z comonent is because I think of it as M_z = x*p_y - y*p_x and since p_x and p_y are constant and the coordinates are invariant under translation that means M_z is constant. But I don't know if that right because it doesn't seem to make sense for the field of 2 points (whos like connecting them is on the z-axis). p_y and p_x are not conserved there. I don't know hwo to show this, or show it with a general Lagrangian like L = T - U or something.

 

[BigJDubs]

The answer to your question is yes, your argument does show that the x and y components of momentum are conserved. To show that the z-component of angular momentum is conserved, you can use the fact that the Lagrangian is invariant under rotations about the z-axis. This means that the derivative of the Lagrangian with respect to the angle of rotation (which is equal to the z-component of angular momentum) must be zero, i.e. $\frac{\partial L}{\partial \phi} = 0$, where $\phi$ is the angle of rotation. This implies that the z-component of angular momentum is conserved.