  We present an extension of the classical theory of calculus of variations to
generalized functions. The framework is the category of generalized smooth
functions, which includes Schwartz distributions while sharing many nonlinear
properties with ordinary smooth functions. We prove full connections between
extremals and Euler-Lagrange equations, classical necessary and sufficient
conditions to have a minimizer, the necessary Legendre condition, Jacobi's
theorem on conjugate points and Noether's theorem. We close with an application
to low regularity Riemannian geometry.
