Is THIS what Lagrangians are about?
So I’ve read about lagrangians a lot, did some practice problems, and it all still seemed sort of mysterious and arcane to me . In my head I was doing that thing where I just solved the problems with no intuition as to how it applied to reality, but now I think I get what Lagrangians are doing, and I’m embarrassed at how obvious it sounds. But just in case I want to check to make sure I’m not misleading myself.
Basically, how I understand if, there are 2 ways of looking at a system: equations of motion and Lagrangians. I’m sure there are more but there are at least 2.
Equations of motion look at the current state and ask what happens next. So, an object with momentum in the x direction will, after an infinitesimal amount of time passing, move an infinitesimal amount to the right.
Now, this obviously only allows certain series of outcomes. Situations where an object flies around the sun at twice the speed of light and then instantly halting for no reason won’t happen under Newtonian physics, for example. And Lagrangians are a way of assessing whether or not they comply with the laws of physics by looking at the path, so to speak.
Basically, I have always been concerned by how equations of motion deal with completely different quantities to Lagrangians and their “action”, and I never got how they connect, but now I think I do. Equations of motions are bottom up recipes for constructing paths that a system can take, and Lagrangians are ways of assessing if the paths are valid, in a kind of top-down way. And the Euler-Lagrange equation lets you swap between them. For every recipe, there is a corresponding assessment to see if it was made with the right recipe, and vice versa.
Apologies if this is heavily poetic language, talking about recipes and stuff. I’m trying to build an intuition - someone once told me that I’d eventually learn how to think in terms of Lagrangians, and this is where trying to do that led me, and I’m not confident it led me to the right place.