Solving Lagrangian Question: Find p, Lagrange's Eq., v as fn of P and r

[Maybe_Memorie]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

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


(ii) ∂L/∂x_i = d/dt(∂L/∂v_i) , P_i = ∂L/∂v_i

So dP_i/dt = Abe^-brx_i/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.

Advice please?

 

[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.

 

[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?

 

[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.

 

[Maybe_Memorie]

So my answer is correct then?


Also I have another question.

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

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

If i = 2,4, then x_i -> x_i + ε_ijx_j where ε_ij is an infinitesimal parametrisation of the rotation.

Same for P_i. Is this correct?

 

[I like Serena]

So my answer is correct then?

I believe so.

Also I have another question.

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

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

If i = 2,4, then x_i -> x_i + ε_ijx_j where ε_ij is an infinitesimal parametrisation of the rotation.

Same for P_i. Is this correct?

Do you have a reasoning to support that?

 

[Maybe_Memorie]

Do you have a reasoning to support that?

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 x_i -> x_i + ε_ijx_j which I know to be true since it was proved in class, we will have in this case x_i -> x_i + ε_jkx_k and the second term just wouldn't make sense, so it would be zero.

So I guess in general x_i -> x_i + δ_ijε_jkx_k where δ_ij is the kronecker delta.

 

[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.

Also I have another question.

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

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

If i = 2,4, then x_i -> x_i + ε_ijx_j where ε_ij is an infinitesimal parametrisation of the rotation.

Same for P_i. Is this correct?

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.

 

[I like Serena]

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

 

[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?

 

[Oxvillian]

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 x_i -> x_i + ε_ijx_j which I know to be true since it was proved in class, we will have in this case x_i -> x_i + ε_jkx_k and the second term just wouldn't make sense, so it would be zero.

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

So I guess in general x_i -> x_i + δ_ijε_jkx_k where δ_ij is the kronecker delta.

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

 

[I like Serena]

If i = 2,4, then x_i -> x_i + ε_ijx_j where ε_ij is an infinitesimal parametrisation of the rotation.

Same for P_i. Is this correct?

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 x_i, 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.