I'm trying to understand the derivation of the Euler-Lagrange equation from the classical action. http://en.wikipedia.org/wiki/Action...93Lagrange_equations_for_the_action_integral" 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Euler-Lagrange equation derivation

**Physics Forums | Science Articles, Homework Help, Discussion**