Editing Calculus of variations (section)

=== Sturm–Liouville problems ===
{{See also|Sturm–Liouville theory}}
The Sturm–Liouville [[eigenvalue problem]] involves a general quadratic form
<math display="block">Q[y] = \int_{x_1}^{x_2} \left[ p(x) y'(x)^2 + q(x) y(x)^2 \right] \, dx, </math>
where <math>y</math> is restricted to functions that satisfy the boundary conditions
<math display="block">y(x_1)=0, \quad y(x_2)=0. </math>
Let <math>R</math> be a normalization integral
<math display="block">R[y] =\int_{x_1}^{x_2} r(x)y(x)^2 \, dx.</math>
The functions <math>p(x)</math> and <math>r(x)</math> are required to be everywhere positive and bounded away from zero. The primary variational problem is to minimize the ratio <math>Q/R</math> among all <math>y</math> satisfying the endpoint conditions, which is equivalent to minimizing <math>Q[y]</math> under the constraint that <math>R[y]</math> is constant. It is shown below that the Euler–Lagrange equation for the minimizing <math>u</math> is
<math display="block">-(p u')' +q u -\lambda r u = 0, </math>
where <math>\lambda</math> is the quotient
<math display="block">\lambda = \frac{Q[u]}{R[u]}. </math>
It can be shown (see Gelfand and Fomin 1963) that the minimizing <math>u</math> has two derivatives and satisfies the Euler–Lagrange equation. The associated <math>\lambda</math> will be denoted by <math>\lambda_1</math>; it is the lowest eigenvalue for this equation and boundary conditions. The associated minimizing function will be denoted by <math>u_1(x).</math> This variational characterization of eigenvalues leads to the [[Rayleigh–Ritz method]]: choose an approximating <math>u</math> as a linear combination of basis functions (for example trigonometric functions) and carry out a finite-dimensional minimization among such linear combinations. This method is often surprisingly accurate.

The next smallest eigenvalue and eigenfunction can be obtained by minimizing <math>Q</math> under the additional constraint
<math display="block">\int_{x_1}^{x_2} r(x) u_1(x) y(x) \, dx = 0. </math>
This procedure can be extended to obtain the complete sequence of eigenvalues and eigenfunctions for the problem.

The variational problem also applies to more general boundary conditions. Instead of requiring that <math>y</math> vanish at the endpoints, we may not impose any condition at the endpoints, and set
<math display="block">Q[y] = \int_{x_1}^{x_2} \left[ p(x) y'(x)^2 + q(x)y(x)^2 \right] \, dx + a_1 y(x_1)^2 + a_2 y(x_2)^2, </math>
where <math>a_1</math> and <math>a_2</math> are arbitrary. If we set <math>y = u + \varepsilon v</math>, the first variation for the ratio <math>Q/R</math> is
<math display="block">V_1 = \frac{2}{R[u]} \left( \int_{x_1}^{x_2} \left[ p(x) u'(x)v'(x) + q(x)u(x)v(x) -\lambda r(x) u(x) v(x) \right] \, dx + a_1 u(x_1)v(x_1) + a_2 u(x_2)v(x_2) \right), </math>
where λ is given by the ratio <math>Q[u]/R[u]</math> as previously.
After integration by parts,
<math display="block">\frac{R[u]}{2} V_1 = \int_{x_1}^{x_2} v(x) \left[ -(p u')' + q u -\lambda r u \right] \, dx + v(x_1)[ -p(x_1)u'(x_1) + a_1 u(x_1)] + v(x_2) [p(x_2) u'(x_2) + a_2 u(x_2)]. </math>
If we first require that <math>v</math> vanish at the endpoints, the first variation will vanish for all such <math>v</math> only if
<math display="block">-(p u')' + q u -\lambda r u =0 \quad \hbox{for} \quad x_1 < x < x_2.</math>
If <math>u</math> satisfies this condition, then the first variation will vanish for arbitrary <math>v</math> only if
<math display="block">-p(x_1)u'(x_1) + a_1 u(x_1)=0, \quad \hbox{and} \quad p(x_2) u'(x_2) + a_2 u(x_2)=0.</math>
These latter conditions are the '''natural boundary conditions''' for this problem, since they are not imposed on trial functions for the minimization, but are instead a consequence of the minimization.