The Relativistic Hookean Spring: A Lorentz-Covariant Force

In summary: The relativistic version of an ideal Hookean spring in Newtonian mechanics is the simplest Lorentz covariant force.
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atyy
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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.
 
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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
@pervect, thanks too!

I guess what I was after is trying to think about this thread https://www.physicsforums.com/showthread.php?t=508190&page=2 .

In the cleanest version of the story, we set up the balls to have equal and opposite velocities in the train frame, and then just use velocity addition. In a sense, the balls then have always been moving that way, and the asymmetry judged from the platform has always been present.

In hprog's set-up, he says the man on the train pushes with "equal force". My intuition is that the final velocity of the balls is determined by force times time. The balls will leave the force field at the same distance (in both frames) from the center, but at the same time in the train frame, and not in the platform frame. So I wanted to know a "simple force" to enter into the calculation.
 
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FAQ: The Relativistic Hookean Spring: A Lorentz-Covariant Force

1. What is the Relativistic Hookean Spring?

The Relativistic Hookean Spring is a theoretical model that describes the behavior of a spring under the influence of the Lorentz force, which takes into account the effects of special relativity.

2. How is the Relativistic Hookean Spring different from the traditional Hooke's Law?

While traditional Hooke's Law only considers the effects of Newtonian mechanics, the Relativistic Hookean Spring takes into account the relativistic effects of time dilation and length contraction on the spring's behavior.

3. What is the Lorentz-Covariant Force and how does it relate to the Relativistic Hookean Spring?

The Lorentz-Covariant Force is a force that is consistent with the principles of special relativity. In the context of the Relativistic Hookean Spring, it describes the force that is exerted on the spring due to the effects of time dilation and length contraction.

4. Can the Relativistic Hookean Spring be applied to real-world situations?

Yes, the Relativistic Hookean Spring can be applied to real-world situations, such as the behavior of springs in high-speed or high-energy systems, where the effects of special relativity become significant.

5. What are the implications of the Relativistic Hookean Spring for our understanding of physics?

The Relativistic Hookean Spring highlights the importance of considering the effects of special relativity in our understanding of physical phenomena. It also demonstrates the need for more comprehensive and accurate models to describe the behavior of systems at high speeds or energies.

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