Abstract: These notes discuss those partial differential equations that can be formulated as a variational problem ("minimization problem"). They were written for a graduate course in nonlinear analysis. They begin with a discussion of critical point of an action functional and the associated Euler-Lagrange equations. Variations in the action under transformation groups acting on the function space is then considered (these give rise to virial relations). Next we discuss symmetries of differential equations and of action functionals. The relation between symmetries and conservation laws (Noether's Theorem) is explained. The last section specializes to virial relations for nonlinear wave equations. References are provided at the end.