Calculus of Variations

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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)\,.

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

solution Compute the first variation \delta J(y,h)\, for y\isin 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\,



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