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.