Reddit reviews Modern Geometry ― Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields (Graduate Texts in Mathematics) (Pt. 1)
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The only book I know that covers the calculus of variations is Modern Geometry - Methods and Applications, Part I by Dubrovin et al. The material on the calculus of variations starts at chapter 5, and depends on the preceding chapters. Take a look, maybe you'll find it useful.