I'm reading my textbook and trying to follow the math on how to minimize the action for an arbitrary Lagrangian. The author states that the action is:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

S[x(t)] = \int^{t_B}_{t_A} dt L( \dot x(t),x(t))

[/tex]

Then the author goes on to talk about finding the extrema for the action by computing [itex] \delta S[x(t)] [/itex]. The author says to compute this by substituing [itex] x(t) + \delta x(t) [/itex] into the definition for the action, expanding to 1st order and integrate by parts. The text then shows this:

[tex]

\delta S[x(t)] = \int^{t_B}_{t_A} dt [\frac {\partial L}{\partial \dot x(t)} \delta \dot x(t) + \frac {\partial L}{\partial x(t)} \delta x(t)] = [ \frac {\partial L}{\partial \dot x(t)} \delta x(t) ]^{t_B}_{t_A} + \int^{t_B}_{t_A} dt [ - \frac {d}{dt} \frac {\partial L}{\partial \dot x(t)} + \frac {\partial L}{\partial x(t)} ] \delta x(t)

[/tex]

What I don't understand is how the author got this. If all he did was substitute and expand like he said to do, what happened to the 0th order term from the expansion? I'm also unsure how he got the right most form of the equation using integration by parts.

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# Homework Help: Question on Lagrangian and Action

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