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Applications of the calculus of variations include:

Solutions to the brachistochrone problem, tautochrone problem, catenary problem, and Newton's minimal resistance problem;
Finding minimal surfaces of a given boundary, or solving Plateau's problem;

Analytical mechanics, or reformulations of Newton's laws of motion, most notably Lagrangian and Hamiltonian mechanics;
Geometric optics, especially Lagrangian and Hamiltonian optics;

Variational method (quantum mechanics), one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states;
Variational Bayesian methods, a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning;
Variational methods in general relativity, a family of techniques using calculus of variations to solve problems in Einstein's general theory of relativity;
Finite element method is a variational method for finding numerical solutions to boundary-value problems in differential equations;

Undergraduate Texts in Mathematics

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