- #1
qinglong.1397
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- 1
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!
