# Calculus of Variations

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 $xy^{2}z^{2}\,$ 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. $K(y)=\int _{0}^{1}y^{2}\,dx=2\,$.

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 $\int _{a}^{b}(y^{2}+y'^{2}+2ye^{x})\,dx\,$.

solution Show that the first variation $\delta J(y_{0},h)\,$ satisfies the homogeneity condition $\delta J(y_{0},\alpha h)=\alpha \delta J(y_{0},h),\alpha \in {\mathbb {R}}\,$.

solution $J:V\to R'\,$, where $V\,$ is a normed linear space, is linear if $J(y_{1}+y_{2})=J(y_{1})+J(y_{2}),y_{1},y_{2}\in V\,$ and $J(\alpha y_{1})=\alpha J(y_{1}),\alpha \in R',y_{1}\in V\,$. Which of the following are functionals on $C^{{-1}}[a,b]\,$ are linear?

(a) $J(y)=\int _{a}^{b}yy'dx\,$

(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 $y\in C[a,b]\,$ to be a local minimum of the functional

$J(y)=\iint \limits _{R}K(s,t)y(s)y(t)dsdt+\int _{a}^{b}y(t)^{2}dt-2\int _{a}^{b}y(t)f(t)dt\,$

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 $J(y)=\int _{0}^{\pi }\left[y'(x)\right]^{2}dx\,$ subject to the constraint $\int _{0}^{\pi }\left[y(x)\right]^{2}dx=1,y(0)=y(\pi )=0\,$.

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

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 $h\in V\,$, then $\delta J(y_{0},\alpha h)\,$ also exists for every real number $\alpha \,$, and $\delta J(y_{0},\alpha h)=\alpha \delta J(y_{0},h)\,$.

solution Compute the first variation $\delta J(y,h)\,$ for $y\in C[0,1]\,$: $J(y)=e^{{y(0)}}\,$

solution Compute the first variation $\delta J(y,h)\,$ for $y\in C[0,1]\,$: $J(y)=\int _{0}^{1}\int _{0}^{1}\sin(xt)y(x)y(t)dxdt\,$

solution Compute the first variation $J(y)=\int _{0}^{1}(3y^{2}+x)dx+y^{2}(0),y_{0}(x)=x,h(x)=x+1\,$