Morse theory and the calculus of variations

We prove for the first time that classical Morse theory applies to functionals of the form J(u) = 1/2 integral(Omega) A(alpha beta)(ij)(x) partial derivative u(i) partial derivative u(i)/partial derivative x(a) partial derivative x(beta) dx + integral(Omega) G(x, u) dx( ) where u : Omega -> R-N, Omega subset of R-n compact with C-infinity boundary partial derivative Omega, u vertical bar(partial derivative Omega) = phi and we argue that this is the largest class to which Morse theory applies.