Calculus Small Changes, Big Ideas

Section 15.7 CAPSTONE: Calculus of variations

• In ordinary optimization, we search for a number \(x\) that minimizes a function \(f(x)\text{.}\)

  🔗
  In multivariable optimization, we search for a point \((x,y)\) that minimizes a function \(f(x,y)\text{.}\)

  🔗
  In the calculus of variations, we search for an entire function \(y(x)\) that minimizes a quantity of the form

  \begin{equation*} J[y]=\int_{a}^{b} F(x,y,y')\dd{x}\text{.} \end{equation*}

  🔗
  Such a map \(J\) is called a functional, because it assigns a number to a function.

  🔗
  🔗
• Suppose \(y\) is a candidate minimizer. Consider a small perturbation

  \begin{equation*} y_{\eps}(x)=y(x)+\eps \eta(x), \end{equation*}

  where \(\eta(a)=\eta(b)=0\) so that the endpoints remain fixed.

  🔗
  Define

  \begin{equation*} \Phi(\eps)=J[y_{\eps}]. \end{equation*}

  🔗
  If \(y\) minimizes \(J\text{,}\) then \(\Phi'(0)=0\text{.}\)

  🔗
  Thus the problem reduces to differentiating with respect to the ordinary variable \(\eps\text{.}\)

  🔗
  🔗
• Differentiating under the integral sign gives

  \begin{equation*} \Phi'(0) = \int_{a}^{b} \left( \pdv{F}{y}\eta + \pdv{F}{y'}\eta' \right)\dd{x}\text{.} \end{equation*}

  🔗
  Integrating the second term by parts,

  \begin{equation*} \int_{a}^{b} \pdv{F}{y'}\eta'\dd{x} = \left[ \pdv{F}{y'}\eta \right]_{a}^{b} - \int_{a}^{b} \dv{x} \qty(\pdv{F}{y'})\eta\dd{x}. \end{equation*}

  🔗
  Because \(\eta(a)=\eta(b)=0\text{,}\) the boundary term vanishes. Hence

  \begin{equation*} \Phi'(0) = \int_{a}^{b} \left( \pdv{F}{y}- \dv{x} \qty(\pdv{F}{y'})\right) \eta \dd{x}\text{.} \end{equation*}

  🔗
  Since this must hold for every admissible \(\eta\text{,}\) we obtain the Euler-Lagrange equation:

  \begin{equation*} \pdv{F}{y}- \dv{x} \qty(\pdv{F}{y'})=0\text{.} \end{equation*}

  🔗
  🔗
• The length of a curve from \(x=a\) to \(x=b\) is

  \begin{equation*} J[y]=\int_{a}^{b} \sqrt{1+(y')^{2}}\dd{x}\text{.} \end{equation*}

  🔗
  Here \(F(x,y,y')=\sqrt{1+(y')^{2}}\text{.}\)

  🔗
  Since \(\partial F/\partial y=0\text{,}\) the Euler-Lagrange equation reduces to

  \begin{equation*} \dv{x} \qty( \frac{y'}{\sqrt{1+(y')^{2}}} )=0\text{.} \end{equation*}

  🔗
  Therefore

  \begin{equation*} \frac{y'}{\sqrt{1+(y')^{2}}}=\text{constant}, \end{equation*}

  which implies \(y'=\text{constant}\text{.}\)

  🔗
  The shortest path between two points is a straight line.

  🔗
  🔗
• If \(z=u(x,y)\) describes a surface over a region \(R\text{,}\) its surface area is

  \begin{equation*} A[u]=\iint_{R} \sqrt{1+u_{x}^{2}+u_{y}^{2}}\dd{A}\text{.} \end{equation*}

  🔗
  Minimizing this functional leads to

  \begin{equation*} \frac{\partial}{\partial x}\qty( \frac{u_{x}}{\sqrt{1+u_{x}^{2}+u_{y}^{2}}} )+ \frac{\partial}{\partial y}\qty( \frac{u_{y}}{\sqrt{1+u_{x}^{2}+u_{y}^{2}}} )=0\text{,} \end{equation*}

  the minimal surface equation.

  🔗
  Unlike the one-dimensional case, this condition is now a partial differential equation.

  🔗
  🔗
• For a curve, curvature \(\kappa\) measures how sharply the curve bends.

  🔗
  For a surface, there are two distinguished directions of bending. The corresponding curvatures are the principal curvatures \(\kappa_{1}\) and \(\kappa_{2}\text{.}\)

  🔗
  Two natural measurements arise:

  \begin{equation*} H=\frac{\kappa_{1}+\kappa_{2}}{2}\qquad\text{and}\qquad K=\kappa_{1}\kappa_{2}\text{.} \end{equation*}

  🔗
  Here \(H\) is the mean curvature and \(K\) is the Gaussian curvature.

  🔗
  In coordinates \(z=u(x,y)\text{,}\) Gaussian curvature can be expressed as

  \begin{equation*} K=u_{xx}u_{yy}-u_{xy}^{2}\text{.} \end{equation*}

  This is in fact the quantity we used when doing the Second Derivative Test.

  🔗
  Mean curvature can be expressed as

  \begin{equation*} H = \frac{(1+u_{y}^{2})u_{xx}-2u_{x}u_{y}u_{xy}+(1+u_{x}^{2})u_{yy}}{2(1+u_{x}^{2}+u_{y}^{2})^{3/2}}\text{.} \end{equation*}

  🔗
  Expanding the minimal surface equation shows that it is exactly the condition

  \begin{equation*} H=0\text{.} \end{equation*}

  🔗
  Thus minimal surfaces are characterized by zero average bending, not by zero Gaussian curvature.

  🔗
  🔗
• • In multivariable optimization, we solve \(\Grad f=0\text{.}\)

    🔗
    🔗
  • In calculus of variations, we set the functional derivative equal to zero.

    🔗
    🔗

  🔗
  Both express stationarity, but now the object being optimized is a shape.

  🔗
  🔗