Calculus of Variations

Notation

• $$C_n[a,b]​$$: the set of continuous functions defined on $$[a,b]​$$, which are continuous and have derivatives up to order $$n​$$. Specifically, $$C[a,b]​$$ is the set of continuous functions.
• $$D_n[a,b]​$$: the set of functions defined on $$[a,b]​$$ which are continuous and have continuous derivatives up to order $$n​$$.

Fundamental lemma in the calculus of variations

If $$\alpha(x)$$ is continuous in $$[a,b]$$, and if $\int_a^b \alpha(x) h(x) = 0$ for every function $$h(x) \in \mathcal{l}(a,b)$$ such that $$h(a) = h(b) = 0$$, then $$\alpha(x)=0$$ for all $$x$$ in $$[a,b]$$.

Euler's Equation

All the functionals in following cases should have boundary conditions specified.

Single variate, single functional case

A necessary condition for $$J[y]$$:

$J[y] = \int_a^b F(x,y,y') dx$

to have an extremum for $$y(x)\in D_1[a,b]​$$ is:

$F_y - \frac{\partial}{\partial x} \frac{\partial F}{\partial (\partial_x y)} = 0$

where $$\partial_x y = \partial y / \partial x = y'$$.

Singla variate, multi functional case

A necessary condition for $$J[y_1,\dots,y_n]​$$: $J[y_1,\dots,y_n] = \int_a^b F(x,y_1,\dots,y_n,y_1',\dots,y_n') \,dx$ to have an extremum is : $F_{y_i} - \frac{\partial}{\partial x} \frac{\partial F}{\partial(\partial_x y_i)} = 0 \qquad (i=1,\dots,n)$

Multi variate, single functional case

A necessary condition for $$J[u]​$$:

$J[u] = \int\cdots\int_R F(x_1, \dots, x_n, u, \partial_{x_1} u, \dots, \partial_{x_n} u) \,dx_1 \cdots dx_n$

to have an extremum is:

$F_u - \sum_{i=1}^n \frac{\partial}{\partial x_i} \frac{\partial F}{\partial(\partial_{x_i} u)} = 0$