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

AI Thread Summary
The discussion centers on the equation L=L(v^2) within the context of inertial reference systems and the Galilean principle of relativity. It emphasizes that the properties of space are consistent in both directions, suggesting that only the magnitude of velocity, |v|, is relevant. The relationship f(|v|)=f(√(v^2))=g(v^2) indicates that v^2 is the critical factor in this context. This understanding clarifies the role of velocity in the equation. Overall, the conversation highlights the significance of velocity magnitude in inertial frames.
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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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