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jbergman

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- TL;DR Summary
- I have a question about choosing proper time for the parametrization of the Lagrangian in special relativity.

According to @vanhees71 and his notes at https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf under certain conditions one can choose ##\tau## as the parameter to parametrize the Lagrangian in special relativity.

For instance if we have,

$$A[x^{\mu}]=\int d\lambda \left[-mc\sqrt{\eta_{\mu\nu}\dot{x}^{\mu} \dot{x}^{\nu}} - \frac{q}{c}\eta_{\mu\nu}\dot{x}A^{\nu}(x) \right]$$

then we can choose ##\lambda=\tau##.

I am trying to follow the proof in the above mentioned notes and I get hung up on the following line of reasoning.

I am not seeing how the above equation implies that.

For instance if we have,

$$A[x^{\mu}]=\int d\lambda \left[-mc\sqrt{\eta_{\mu\nu}\dot{x}^{\mu} \dot{x}^{\nu}} - \frac{q}{c}\eta_{\mu\nu}\dot{x}A^{\nu}(x) \right]$$

then we can choose ##\lambda=\tau##.

I am trying to follow the proof in the above mentioned notes and I get hung up on the following line of reasoning.

vanhees71 said:Since ##\dot{x}^{\mu}\frac{d}{d\lambda}\frac{\partial L}{\partial \dot{x}^{\mu}} = \dot{x}^{\mu}\frac{\partial L}{\partial x^{\mu}}## holds forany word line, only three of the four space-time variables, ##x^{\mu}## are independent.

I am not seeing how the above equation implies that.

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