# Simplest force

1. Jun 20, 2011

### atyy

What is the simplest Lorentz covariant force - the relativistic version of an ideal Hookean spring in Newtonian mechanics?

"Simplest" as an easy to do an ideal calculation with.

2. Jun 20, 2011

### bcrowell

Staff Emeritus
3. Jun 20, 2011

### atyy

Last edited: Jun 20, 2011
4. Jun 20, 2011

### pervect

Staff Emeritus
I would say that a Lagrangian density of -rho is the simplest relativistic equivalent to a spring - but I also suspect that's not quite what you're after, it's not formulated as a particle-particle force, and to turn it into one, you'd have to define the motion of the spring which starts to get non-simple.

http:http://www.gregegan.net/SCIENCE/Rindler/SimpleElasticity.html talks about it a bit, though he doesn't use Lagrangians.

I think https://www.physicsforums.com/showpost.php?p=1365762&postcount=157 was what I finally came up with for the Lagrangian density of a hoop, not that you're doing hoops. It should also work for a wire. You'll see some added terms there, related to the terms due to the motion of the wire, and the energy involved in stretching it.

Getting the volume element right was a painful process. And the stretch-factor computation isn't trivial, either - as you'll see if you have the patience to read the very very long thread. And the last caveat is that the speed of sound in the wire goes up when you stretch it, and when that speed gets to be faster than 'c' bad things happen, bad things including singular Lagrangians.

On the other hand,I suspect it's not quite what your'e after, so you might not want to bother.

5. Jun 20, 2011

### atyy

@pervect, thanks too!