In Landau's Mechanics, if an inertial frame [itex]\textit{K}[/itex] is moving with an infinitesimal velocity [itex]\textbf{ε}[/itex] relative to another inertial frame [itex]\textit{K'}[/itex], then [itex]\textbf{v}'=\textbf{v}+\textbf{ε}[/itex]. Since the equations of motion must have the same form in every frame, the Lagrangian [itex]L(v^2)[/itex] must be converted by this transformation into a function [itex] L'[/itex] which differs from [itex]L(v^2)[/itex], if at all, only by the total time derivative of a function of co-ordinates and time. Then he gave the formula [itex]L'=L(v'^2)=L(v^2+2\textbf{v}\bullet\textbf{ε} + \textbf{ε}^2)[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

So my question is what does the sentence 'the equations of motion must have the same form in every frame' mean? Whether [itex]L'(v'^2)=L(v'^2)[/itex] or [itex]L'(v'^2)=L(v^2)[/itex]? Why?

And what is the variable in the two Lagrangians,[itex]\textbf{v}[/itex] or [itex]\textbf{v}'[/itex]?

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# Lagrangian for a free particle

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