PX285 - C1 - the euler-lagrange equation

calculus of variations is a technique to find the function that extremizes the integral provided that the end points are fixed
taking a path, $x (t)$ , and varying it as:

\begin{matrix} x (t) \to x (t) + a (t) \\ ⟹ v (t) \to v (t) + \dot{a} (t) \end{matrix}

$x (t)$ is the extremal path from $x_{0} (t_{0})$ to $x_{1} (t_{1})$ , and $a (t)$ is a general, small excursion that the path is forced to take, but it must always pass through its fixed end points: $a (t_{0}) = a (t_{1}) = 0$
considering $L (x, v)$ , and taking a small pertubation: $L (x + a, v + \dot{a})$ like a taylor expansion, where only the first order variation is of interest:

L (x + a, v + \dot{a}) \approx L (x, v) + a \frac{\partial L}{\partial x} + \dot{a} \frac{\partial L}{\partial \dit x} + \dots

by definition, $A = \int_{t_{0}}^{t_{1}} L (x, \dot{x}) d t$ , with $x (t)$ as the extremal path, and $A$ as the extremal action:

\begin{matrix} A \to A + δ A \\ where, δ A = \int_{t_{0}}^{t_{1}} [a (t) \frac{\partial L}{\partial x} + \dot{a} (t) \frac{\partial L}{\partial \dot{x}}] d t \end{matrix}

using integration by parts on the second term:

\int d t \frac{d a}{d t} \frac{\partial L}{\partial \dot{x}} = {[a (t) \frac{\partial L}{\partial \dot{x}}]}_{t_{0}}^{t_{1}} - \int a (t) \frac{d}{d t} \frac{\partial L}{\partial \dot{x}} d t

$[a (t)]_{t_{0}}^{t_{1}}$ must be equal to zero as the variations vanish at the end points:

δ A = \int d t a (t) (\frac{\partial L}{\partial x} - \frac{d}{d t} \frac{\partial L}{\partial v})

the only way $A$ can be stationary, ie: $δ A = 0$ , is if the following euler-lagrange equation is satisfied:

\frac{\partial L}{\partial x} - \frac{d}{d t} \frac{\partial L}{\partial v} = 0