Why L=L（v^2) in inertial reference system?

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SUMMARY

The discussion centers on the equation L = L(v^2) within the context of inertial reference systems, emphasizing the Galilean principle of relativity. Participants agree that the properties of space remain consistent in both directions, indicating that only the magnitude of velocity |v| is significant. The conclusion drawn is that the function f(|v|) can be expressed as f(√(v²)) = g(v²), reinforcing that v² is the critical factor in this equation. This understanding is pivotal for grasping the implications of motion in physics.

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  • Understanding of the Galilean principle of relativity
  • Familiarity with basic physics concepts of velocity and motion
  • Knowledge of mathematical functions and their properties
  • Concept of inertial reference frames in physics
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  • Research the implications of the Galilean principle of relativity in classical mechanics
  • Study the mathematical derivation of L = L(v^2) in various inertial frames
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This discussion is beneficial for physics students, educators, and anyone interested in the foundational concepts of motion and relativity in classical mechanics.

Dr turtle
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TL;DR
Why Landau pointed out that Lagrange function shall only be affected by v square in inertial reference system?
Why he said that beacause space's propertiy is the same in both direction, so L=L(v^2), or do I misunderstand him incorrectly?
btw this conclusion appears in somewhere like page 5 and its about Galilean principle of relativity.
 
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The direction should not matter, so only magnitude of velocity |v| should matter.
f(|v|)=f(\sqrt{v^2})=g(v^2)
So we can say only v^2 matters.
 
That's really helpful, lots of thanks
 

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