Calculus of Variations

From Exampleproblems

solution Find the path that minimizes the arclength of the curve between $(x_0,y_0) = (0,0)\,$ and $(x_1,y_1) = (1,1)\,$ .

solution Find the extrema of $x^2+y^2+z^2\,$ subject to the constraint $x^2+2y^2-z^2-1=0\,$ .

solution Find the maximum of $xy^2z^2\,$ subject to the constraint $x+y+z=12\,$ .

solution Write the Euler-Lagrange equations for $L(x,y,z,y',z',y'',z'',y''',z''',...,y^{(k)},z^{(k)})\,$ .

solution Constraint problem: Minimize $T(y)=\int_0^1\left(y'^2+x^2\right)\,dx\,$ s.t. $K(y)=\int_0^1y^2\,dx=2\,$ .

solution Derive the Euler-Lagrange equation from the attempt to minimize the functional

$T(y)=\int_a^b L(y,y',x)\,dx\,$

solution Minimize the functional from classical mechanics: $\int_{t_1}^{t_2}(\mathrm{Kinetic\,Energy} - \mathrm{Potential\,Energy})\,$

solution Find the extrema of $\int_a^b \frac{y'^2}{x^3}\,dx\,$ .

solution Find the extrema of $\int_a^b (y^2 +y'^2 + 2y e^x) \,dx\,$ .

solution Show that the first variation $\delta J(y_0,h)\,$ satisfies the homogeneity condition $\delta J(y_0, \alpha h) = \alpha \delta J(y_0, h), \alpha \isin \mathbb{R}\,$ .

solution $J:V\to R'\,$ , where $V\,$ is a normed linear space, is linear if $J(y_1+y_2) = J(y_1) + J(y_2), y_1,y_2\isin V\,$ and $J(\alpha y_1) = \alpha J(y_1), \alpha \isin R', y_1\isin V\,$ . Which of the following are functionals on $C^{-1}[a,b]\,$ are linear?

(a) $J(y)=\int_a^b y y' dx\,$

(b) $J(\alpha y) = \int_a^b (4y'^2 + 2(\alpha y))dx\,$

(c) $J(y) = e^{y(a)}\,$

(d) The set of all continuous functions on $[0,1]\,$ satisfying $f(0)=0\,$

(e) The set of all continuous functions on $[0,1]\,$ satisfying $f(1)=1\,$

solution Find the extremal for $J(y)=\int_1^2 \frac{\sqrt{1+y'^2}}{x} dx, y(1)=0, y(2)=1\,$

solution Compute the first variation of $J(y)=\int_a^b yy' dx\,$

solution Compute the first variation of $J(y)=\int_a^b (y'^2+2y)dx\,$

solution Compute the first variation of $J(y)=e^{y(a)}\,$

solution Minimize $J(y) = \int_0^\infty (y^2 + y'^2 + (y''+y')^2)dx, y(0)=1, y'(0)=2, y(\infty)=0, y'(\infty)=0\,$

solution Find the extremals of $J(y) = \int_0^1(yy'+y''^2)dx, y(0)=0, y'(0)=1, y(1)=2, y'(1)=4\,$

solution Find the Euler equation for $J(y,z)=\int_a^b\left[ y''z' + xyz'' + z'''y^2\right] dx\,$

solution Minimize $J(y)=\int_0^1(1+y''^2)dx, y(0)=0,y'(0)=1,y(1)=1,y'(1)=1\,$

solution Minimize $J(y)=\int 2\pi y \sqrt{1+y'^2} dx\,$

solution Obtain a necessary condition for a function $y\isin C[a,b]\,$ to be a local minimum of the functional

$J(y) = \iint\limits_R K(s,t) y(s) y(t) ds dt + \int_a^b y(t)^2dt-2\int_a^b y(t) f(t)dt\,$

solution Find the Euler equation for the functional

$J(u)=\iint\limits_G\left[u_x^2+u_y^2+2f(x,y)u(x,y)\right]dxdy\,$

where $G\,$ is a closed region in the $xy\,$ plane and $u\,$ has continuous second partial derivatives.

solution Find the extremal of the functional $J(y)=\int_0^\pi\left[y'(x)\right]^2dx\,$ subject to the constraint $\int_0^\pi \left[ y(x)\right]^2dx=1, y(0)=y(\pi)=0\,$ .

solution Determine the function $\hat{y}\isin C^2[0,1]\,$ that minimizes the functional $J(y)=\int_0^1\left[y'(x)\right]^2dx+[y(1)]^2, y(0)=1, h(0)=0\,$ .

solution Let $J:A\to\mathbb{R}\,$ be a functional on a subset $A\,$ of a normed linear space $V\,$ .

(a) Define precisely the first variation $\delta J(y_0,h)\,$ of $J\,$ at $y_0\,$ and admissible $h(x)\,$ .

(b) Show that if $\delta J(y_0,h)\,$ exists for a certain admissible $h\isin V\,$ , then $\delta J(y_0,\alpha h)\,$ also exists for every real number $\alpha\,$ , and $\delta J(y_0,\alpha h)=\alpha \delta J(y_0,h)\,$ .