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Apr
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comment Deriving the Lagrangian for a free particle
@Qmechanic, thanks a lot for such a detailed update! You derived the form of $L$ from the condition that the equations should not change, which is a weaker condition than $\Delta L$ being a time derivative, as far as I know. So my last question is why did you mention that form-invariance implies $\Delta L$ being a time derivative? I looked up Noether's theorem, but didn't find there an answer. Could you please point out to me, from where exactly does that statement about $\Delta L$ follow? And why did you even mention it, since you used the condition of form-invariance?
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comment Deriving the Lagrangian for a free particle
@altertoby, I didn't understand your argument, as $$4v^3=\text{const}$$ seems to imply $v=\text{const}$ (one dimentional case). In 3 dimensions it would be $$4v^2{\bf v}=\text{const},$$ what lead me to the same result ($v_i^3=\text{const}$). Also, how can $v$ (hence ${\bf v}$ as well) have no physical meaning? Would ${\bf v}$ not had the meaning of velocity, how could we apply the Galileo's relativity principle and come to the conclusion that $L=L(v^2)$?
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comment Deriving the Lagrangian for a free particle
Finally, did I get it right that Landau only proved, that $\frac{m v^2}{2}$ will fit the requirements? Is there a proof that this is the only possible expression?
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comment Deriving the Lagrangian for a free particle
Thank you, the first question is now clear! It's a shame, the authors didn't bother to mention the smoothness there. But I'm still confused with the second part. I didn't find a clear definition of form-invariance here. Does form-invariance mean that the families of solutions will be the same? If so, do there exist second-order DEs that generate the same sets of solutions, but are different (linear-independent)?
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revised Deriving the Lagrangian for a free particle
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