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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]?