There's two basic approaches to calculus of variations. One is based around the Gateaux Variation which basically is an extension of the normal derivative. So the vector x becomes x + epsilon*v and you take the limit as epsilon goes to zero. The second is due to Lagrange and uses the symbol lower case delta. The both amount to the same thing, but the notation is different. There's epsilon in the Lagrange approach, it's just buried.
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.
> finite element analysis, and mechanics of deformable bodies
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.
Hmm, it's a Dover book. That alone makes me think it's probably a good one as they have a habit of reprinting some excellent books that would otherwise be out of print. I'll have to check it out. Thanks!
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.