Linking Systems

Many problems arising in science and engineering call for the solving of the Euler equations of functionals, i.e., equations of the form Euler equations G ′ ( u ) = 0 , $$\displaystyle G'(u)=0, $$ where Functional G(u) is a C 1 functional Energy (usually representing the energy) arising from the given data. As an illustration, the equation − Δ u ( x ) = f ( x , u ( x ) ) $$\displaystyle -\Delta u(x)=f(x,u(x)) $$ is the Euler equation of the functional G ( u ) = 1 2 ∥ ∇ u ∥ 2 − ∫ F ( x , u ( x ) ) d x $$\displaystyle G(u)=\frac {1}{2}\|\nabla u\|{ }^2-\int F(x,u(x)) \,dx $$ on an appropriate space, where F ( x , t ) = ∫ 0 t f ( x , s ) d s , $$\displaystyle F(x,t) = \int ^{t}_{0}f(x,s)\,ds, $$ and the norm is that of L 2. The solving of the Euler equations is tantamount to finding critical points of the corresponding functional. The history of this approach goes back to the calculus of variations. Calculus of variations Then the desire was to find extrema of certain expressions G (functionals). Following the approach of calculus, one tried to find all critical points of G, substitute them back in G, and see which one gives the required extremum. Critical points This worked fairly well in one dimension where G′(u) = 0 is an ordinary differential equation. However, in higher dimensions, it turned out that it was easier to find the extrema of G than solve G′(u) = 0. This led to the approach of solving equations of the form G′(u) = 0 by finding extrema of G.