Asking for hints to Goldstein chapter 7, problem 9

[qinglong.1397]

Homework Statement

A generalized potential suitable for use in a covariant Lagrangian for a single particle


$U=-A_{\lambda\nu}(x_\mu)u^\lambda u^\nu$


where $A_{\lambda\nu}$ stands for a symmetric world tensor of the second rank and $u^\nu$ are the components of the world velocity. If the Lagrangian is made up of $\frac{1}{2}mu_\nu u^\nu$ minus U, obtain the Lagrange equations of motion. What is the Minkowski force? Give the components of the force as observed in some Lorentz frame.

Homework Equations

I've got the Lagrange equations:

$m\frac{d u_\nu}{d \tau}+2A_{\lambda\nu}\frac{d u^\lambda}{d \tau}+2\frac{\partial A_{\lambda\nu}}{\partial x^\rho}u^\lambda u^\rho-\frac{\partial A_{\lambda\rho}}{\partial x^\nu}u^\lambda u^\rho =0$

The Attempt at a Solution

Now, the problem is following. The 4-acceleration appears twice in the Lagrange equations. So which one represents the Minkowski force?

I think the first term definitely represents one component of the Minkowski force. If you rewrite the second term in the following form,

$2A_{\lambda\nu}\frac{d u^\lambda}{d \tau}=2\frac{A_{\lambda\nu}}{m}(m\frac{d u^\lambda}{d \tau})$

then you find the Minkowski force. Then what we have to do is to solve a set of linear equations, which is hopeless at least at the first sight.

So which should be the force? Just the first term? Or, try to solve the set of linear equations? And how to solve it? I need your help. Thank you!

 

[QFT25]

I'm also in the same situation and would like some points on where to proceed once I have Lagrange's equation's of motion. Can Minkowski force be just the mass times the four velocity? Also to find the force in a Lorenz frame, that refers to just the spatial component of the Minkowski force right?

 

[QFT25]

Can this be moved to the homework help section?