Are there any mathematical treatments that get at calculus of variations in a more principled way than the standard physics class that are still accessible to one with a physicsy background?
For Lagrange's approach, I recommend Lanczos' "The Variational Principles of Mechanics". It's published by Dover so if you can't find it in a library, it's only about $10 last I checked. For the other approach, we used John Troutman's "Variational Calculus and Optimal Control: Optimization with Elementary Convexity" in my Applied Math course on calculus of variations. It's good, but I don't think it's too approachable if you haven't had the lectures to go with it.
I've done a lot of work with calculus of variations including working out the math for volume integration of variational gradients which is something that appears in beam deformation problems.
One day I'll sit down and write books on calculus of variations, finite element analysis, and mechanics of deformable bodies. I was very unsatisfied with the texts I used for all of these in my undergrad and grad programs.
This sort of irks me, too. Once you get to the continuum world everyone seems to forget about the Lagrangian reformulation. These are (mostly) classical systems so the machinery of Euler-Lagrange is perfectly valid, but I've met Mech E PhDs who vaguely recall Lagrangian mechanics as a footnote. Instead of all the obtuse walls of text and hand waving you get in some of these books, it would be hella rad to get at elasticity (and plasticity? via virtual work?) variationaly, in a mainstream text.
Fermat's library looks like the perfect excuse to get back into it.
But nice work - thank you.