If you can think of an infinitismal transformation of fields that vanishes at the endpoints, then doesn't the action automatically vanish by the Euler-Lagrange equations?(adsbygoogle = window.adsbygoogle || []).push({});

For example take the Lagrangian:

L=.5 m v^{2}

and the transformation:

x'(t)=x(t)+ε*(1/t^{2})

At t±∞, x'(±∞)=x(±∞), so the field x(t) doesn't change at the endpoints t=±∞ under this transformation. Since the transformation is infinitismal and the endpoints don't change, then the change in the action should be zero: hence there should be a conserved quantity, namely:

Q=mv*(1/t^{2})

But this quantity Q is not conserved for a free particle.

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# Trouble with noether's theorem

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