In a review of the book Mathematics for Physics: A Guided Tour for Graduate Students by Michael Stone and Paul Goldbart (2009), David Khmelnitskii perceptively writes:

There is perhaps no other branch of applied mathematics that is more in need of such a “repackaging” than the calculus of variations.

A glance through the reference list at the end of the book will reveal that there are a number of now-classic texts on the calculus of variations from up through the mid-1960s, such as Weinstock (1952) and Gelfand and Fomin (1963), with a sharp drop subsequent to that period. The classic texts that emphasize applications, such as Morse and Feshback (1953) and Courant and Hilbert (1953), typically focus the majority of their discussion of variational methods on classical mechanics, for example, statics, dynamics, elasticity, and vibrations. Since that time, it has been more common to simply include the necessary elements of variational calculus in books dedicated to specific topics, such as analytical dynamics, dynamical systems, mechanical vibrations, elasticity, finite-element methods, and optimal control theory. More recently, the trend has been to avoid treating these subjects from a variational point of view altogether.