Calculus of Variations

From Example Problems
Jump to: navigation, search

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\,



Main Page