Homework Help: Lagrangian question

1. Aug 13, 2012

Maybe_Memorie

1. The problem statement, all variables and given/known data

Right I've got a relativistic particle in D dimensional space interacting with a central potential field. Writing out the entire lagrangian is a bit complicated on this but I'm sure you all know the L for a free relativistic particle. The potential term is Ae-br where r is the position vector.

(i) Find the momentum p as a function of the velocity v.

(ii) Find Lagrange's equations of motion for the particle.

(iii) Find the velocity v as a function of P and r.

2. Relevant equations

3. The attempt at a solution

(i) That would be mvi/root(1 - v2/c2)

(ii) ∂L/∂xi = d/dt(∂L/∂vi) , Pi = ∂L/∂vi

So dPi/dt = Abe-brxi/r

(iii) This is the part I'm having a problem with.

I have that v2 = P2c2/(m2c2 + P2) But I have no idea how to get r into the equation.

2. Aug 14, 2012

I like Serena

Hi Maybe_Memorie!

Did you know that f(x,y)=g(x) for any y, is not only a function of x and y, but also a function of x?

Yeah... I know, it sounds a bit lame, but it's true nonetheless.

3. Aug 14, 2012

Maybe_Memorie

Hey I Like Serena!

So are you saying that since I need to find V(r,p), my answer as a function of p is also a function of r for all r?

4. Aug 14, 2012

I like Serena

Yes. I believe that you have v(r,p)=v(p).
It's fairly typical that p and v are independent of position in potential fields.

5. Aug 14, 2012

Maybe_Memorie

So my answer is correct then?

Also I have another question.

How do the coordinates xi and momenta Pi transform under an infinitesimal rotation in the x2x4-plane?

Well, under this transformation, I know that xi, i not equal to 2 or 4 will just go to xi. In other words, invariant.

If i = 2,4, then xi -> xi + εijxj where εij is an infinitesimal parametrisation of the rotation.

Same for Pi. Is this correct?

6. Aug 14, 2012

I like Serena

I believe so.

Do you have a reasoning to support that?

7. Aug 14, 2012

Maybe_Memorie

Well for my first statement I'm just imagining a three dimensional coordinate system. It's clear to see that then when you rotate the x-y plane the z axis will remain unchanged.

But if I need to be more rigorous, if I take my other statement of xi -> xi + εijxj which I know to be true since it was proved in class, we will have in this case xi -> xi + εjkxk and the second term just wouldn't make sense, so it would be zero.

So I guess in general xi -> xi + δijεjkxk where δij is the kronecker delta.

8. Aug 15, 2012

Oxvillian

Seems like a very badly worded question

eg. presumably r is the length of the position vector.

Bearing in mind the spherical symmetry of the problem, it would make more sense if they asked for say the speed of the particle as a function of r.

I would use the matrix
$$\epsilon_{1i3j567...}$$
to generate this rotation, where ε is the completely antisymmetric symbol. Note that the i and the j go in places 2 and 4.

9. Aug 17, 2012

I like Serena

Okay, well... did you figure it out?

10. Aug 17, 2012

Maybe_Memorie

I'm not sure if my argument 3 posts up is correct or not. It makes sense to me, but I'm not sure if it's right.

Any kind of point in the right direction?

11. Aug 17, 2012

Oxvillian

I think that wouldn't make sense simply because the indices don't match up.

When you sum over j, you're back to where you started, no?

But I can heartily recommend the matrix I suggested 3 posts back

12. Aug 18, 2012

I like Serena

x and P are vectors.
Vectors themselves are independent of coordinate system.
It's only their representation into coordinates that depends on a coordinate system.

If you have a formula to represent the vector x in coordinates xi, or rather a formula to transform its coordinates to another coordinate system (as you do), then that exact same formula applies to represent the vector P in coordinates of the same coordinate system.

So yes, you are correct.