Is there an intuition behind the Euler Lagrange equation? I am taking calculus of variations at the moment and I am curious if there is a visualization of why the EL must be satisfied for all extremals.
At first glance it's hard for me to relate:
$$\frac{d}{dx}(\frac{\partial L}{\partial y'}) = \frac{\partial L}{\partial y}$$
With a geometric/visual intuition.
 A: To my knowledge, there is no intuition behind the equation per se, but there is a qualitative heuristic that can perhaps help you see why the equation needs to take the form is takes. For starters, you should acknowledge that, here, $L$ is a function of $y$ and $y',$ both of which are functions of $x.$ Now, heuristically, $\mathrm{d}y=y'\,\mathrm{d}x.$ So there is a sense in which one would expect that differentiating with respect to $y'$ and then with respect to $x,$ somehow is "the same" as differentiating with respect to $y.$ The Euler-Lagrange equation expresses this concept. Of course, in general, it is not the case that those two derivatives are equal, so the intuition breaks down a bit there, but then this leads to the question, what are the conditions under which such an equality holds? If you try to tackle that question, then it seems that you will eventually lead to the property of extremizing the action. This is probably the best sense of an intuition I can think of.
Beyond that, you will just need to look at how the equation is derived in a textbook or in a source in the Internet.
