Unfortunately it is very hard to find a lucid explanation of the idea .. anywhere.. that isn't a bunch of 'just so' explanations.
The 'short version' is that a Lagrangian is a device for book-keeping the interactions in a physical theory. It consists of terms like aBa, where 'a' is some physical quantity, and (Ba) is the 'cost' to change it. B in general is an operator that can depend on all of the other physical quantities around, possibly including derivatives. So you end up with something like:
L = aBa + cDc + eFe + gHg + ...
Derivatives of this function with respect to a given variable -- for instance, a particular value `a` or `c` -- tell you the 'physical law' for that field. The simplest example in classical mechanics is L = 1/2mv^2 - V(x), and the derivative rule is dL/dx - d/dt dL/dv = - V_x - m dv/dt = (F - ma) = 0.
In general including a term like ... + cD(a)c +... causes the c field to couple to the a field, which is how interactions between different fields (represented by the lower-case variables a, c, etc here) are encoded in field theory.
I haven't found out a cogent explanation of why the terms have the form aBa, though, or why this is the 'simplest' way to express the concepts that it encodes, but it seems to be. But it works well in part because of it expresses physical laws in terms of scalar functions (instead of a vector-equality like F=ma, which becomes a complicated differential equation in practice), which somehow means it is more naturally encoding the underlying symmetries of the system (because, I guess, one equation is simpler than three).
I'm grateful for your attempt at explaining! I've read it twice, and I'll read it again.
Your explanation relies on calculus; at school, I grokked differentiation, but I got a huge mental block on integration. My maths teacher was very good, he wrote famous maths textbooks, but he had no insight into my difficulties. He couldn't teach me, because he was too clever. That's why I never tackled graduate-level maths. I resent that old maths teacher; he meant well, but being super-clever wasn't his job. His job was to teach me.
It's not going to be useful to understand Lagrangians in detail unless you're actually doing the physics yourself -- especially because the whole framework is super problematic, pedagogically (I hope it gets 'refactored' someday into something more intuitive).
The important part is: a Lagrangian is a function that encodes the relationship between all of objects in a physical model, in such a way that you can 'add terms' to the function to add new interactions in your model. That much, at least, is quite elegant, and it's kinda miraculous that such a thing exists at all.
For what it's worth, integration is really not that tricky. If I had to guess what really deterred you was the obsession with annoying proofs about dividing functions into little rectangles and summing them up and taking their limit. That stuff is totally unnecessary, imo.
It's easier to think of integration as the inverse of differentiation. If differentiation tells you how fast a function is changing, then differentiating your position gives your velocity. Integration is the opposite: integrating your velocity over a length of time tells you how the position changes. That's a pretty intuitive idea: if I told you the velocity vector for the whole duration of a road trip, you can probably figure out where I ended up, if I also tell you where I started. That's all integration is.
What annoyed me about integral calculus was that it all seemed to be a bunch of heuristic rules. They showed me how to differentiate "from scratch", as it were, like from first principles; but integration seemed to be working backwards from the results of differentiation (which is roughly what you just said).
That is, it didn't look like maths to me. It didn't seem to have the rigour of, say, trig.
I expected maths to be deterministic, and I hated having heuristics in there. It was a gut reaction, it happened when I was about 17, and I'm sure that integration isn't heuristic in that way; I wonder if my mental block might not have ocurred if it had been presented to me differently.
The 'short version' is that a Lagrangian is a device for book-keeping the interactions in a physical theory. It consists of terms like aBa, where 'a' is some physical quantity, and (Ba) is the 'cost' to change it. B in general is an operator that can depend on all of the other physical quantities around, possibly including derivatives. So you end up with something like:
L = aBa + cDc + eFe + gHg + ...
Derivatives of this function with respect to a given variable -- for instance, a particular value `a` or `c` -- tell you the 'physical law' for that field. The simplest example in classical mechanics is L = 1/2mv^2 - V(x), and the derivative rule is dL/dx - d/dt dL/dv = - V_x - m dv/dt = (F - ma) = 0.
In general including a term like ... + cD(a)c +... causes the c field to couple to the a field, which is how interactions between different fields (represented by the lower-case variables a, c, etc here) are encoded in field theory.
I haven't found out a cogent explanation of why the terms have the form aBa, though, or why this is the 'simplest' way to express the concepts that it encodes, but it seems to be. But it works well in part because of it expresses physical laws in terms of scalar functions (instead of a vector-equality like F=ma, which becomes a complicated differential equation in practice), which somehow means it is more naturally encoding the underlying symmetries of the system (because, I guess, one equation is simpler than three).