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:

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