I'm trying to understand the derivation of the Euler-Lagrange equation from the classical action. This has been my main source so far. The issue I'm having is proving the following equivalence:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\int_{t_1}^{t_2} [L(x_{true} + \varepsilon, \dot{x}_{true} + \dot{\varepsilon},t) - L(x_{true}, \dot{x}_{true},t)] \mathrm{d}t = \int_{t_1}^{t_2} (\varepsilon \frac{\partial L}{\partial x} + \dot{\varepsilon} \frac{\partial L}{\partial \dot{x}}) \mathrm{d}t

[/tex]

I understand the idea behind their equivalence intuitively, The derivative of a function is the change in that function, and I see how on the left side there is a representation of a small change in the lagrangian, but I'm having a hard time proving this to myself mathematically and I'd like some help.

I understand all the other steps shown in the derivation.

Thanks to anyone that responds.

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# Euler-Lagrange equation derivation

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