Variational calculus deals with local extremization of functionals employing some aspects of the differential calculus? on spaces of functionals.
Specifically, it studies the critical points, i.e. the points where the first variational derivative of a functional vanishes, for functionals on spaces of sections of jet bundle?s. The kinds of equations specifying these critical points are Euler-Lagrange equations.
This applies to, and is largely motivated from, the study of action functionals in physics. In classical physics the critical points of a specified action functional on the space of field configurations encode the physically observable configurations.
There are strong cohomological tools for studying variational calculus, such as the variational bicomplex and BV-BRST formalism.
Ian Anderson, The variational bicomplex (pdf)
Peter Olver, Applications of Lie groups to differential equations, Springer; Equivalence, invariants, and symmetry, Cambridge Univ. Press 1995.
Olga Krupková, The geometry of ordinary variational equations, Springer 1997, 251 p.