“变分计算例子”的版本间的差异

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[[CoV1|solution]] Find the path that minimizes the [[arclength]] of the curve between <math>(x_0,y_0) = (0,0)\,</math> and <math>(x_1,y_1) = (1,1)\,</math>.<br><br>
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解答见原网页 [http://www.exampleproblems.com/wiki/index.php/Calculus_of_Variations 变分计算]
  
[[CoV2|solution]] Find the [[extrema]] of <math>x^2+y^2+z^2\,</math> subject to the constraint <math>x^2+2y^2-z^2-1=0\,</math>.<br><br>
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#Find the path that minimizes the arclength of the curve between <math>(x_0,y_0) = (0,0)\,</math> and <math>(x_1,y_1) = (1,1)\,</math>.<br><br>
  
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# Find the extrema of <math>x^2+y^2+z^2\,</math> subject to the constraint <math>x^2+2y^2-z^2-1=0\,</math>.<br><br>
[[CoV3|solution]] Find the maximum of <math>xy^2z^2\,</math> subject to the constraint <math>x+y+z=12\,</math>.<br><br>
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[[CoV4|solution]] Write the [[Euler-Lagrange equation|Euler-Lagrange equations]] for <math>L(x,y,z,y',z',y'',z'',y''',z''',...,y^{(k)},z^{(k)})\,</math>.<br><br>
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# Find the maximum of <math>xy^2z^2\,</math> subject to the constraint <math>x+y+z=12\,</math>.<br><br>
  
[[CoV5|solution]] Constraint problem: Minimize <math>T(y)=\int_0^1\left(y'^2+x^2\right)\,dx\,</math> s.t. <math>K(y)=\int_0^1y^2\,dx=2\,</math>.<br><br>
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# Write the Euler-Lagrange equation|Euler-Lagrange equations for <math>L(x,y,z,y',z',y'',z'',y''',z''',...,y^{(k)},z^{(k)})\,</math>.<br><br>
  
[[CoV6|solution]] Derive the Euler-Lagrange equation from the attempt to minimize the functional<br><br>
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# Constraint problem: Minimize <math>T(y)=\int_0^1\left(y'^2+x^2\right)\,dx\,</math> s.t. <math>K(y)=\int_0^1y^2\,dx=2\,</math>.<br><br>
  
<math>T(y)=\int_a^b L(y,y',x)\,dx\,</math><br><br>
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# Derive the Euler-Lagrange equation from the attempt to minimize the functional<br><br>
  
[[CoV7|solution]] Minimize the [[functional]] from [[classical mechanics]]: <math>\int_{t_1}^{t_2}(\mathrm{Kinetic\,Energy} - \mathrm{Potential\,Energy})\,</math><br><br>
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<math>\begin{align}T(y)=\int_a^b L(y,y',x)\,dx\,\end{align}</math><br><br>
  
[[CoV8|solution]] Find the extrema of <math>\int_a^b \frac{y'^2}{x^3}\,dx\,</math>.<br><br>
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# Minimize the functional from classical mechanics: <math>\int_{t_1}^{t_2}(\mathrm{Kinetic\,Energy} - \mathrm{Potential\,Energy})\,</math><br><br>
  
[[CoV9|solution]] Find the extrema of <math>\int_a^b (y^2 +y'^2 + 2y e^x) \,dx\,</math>.<br><br>
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# Find the extrema of <math>\int_a^b \frac{y'^2}{x^3}\,dx\,</math>.<br><br>
  
[[CoV10|solution]] Show that the [[first variation]] <math>\delta J(y_0,h)\,</math> satisfies the homogeneity condition <math>\delta J(y_0, \alpha h) = \alpha \delta J(y_0, h), \alpha \isin \mathbb{R}\,</math>.<br><br>
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# Find the extrema of <math>\int_a^b (y^2 +y'^2 + 2y e^x) \,dx\,</math>.<br><br>
  
[[CoV11|solution]] <math>J:V\to R'\,</math>, where <math>V\,</math> is a [[normed linear space]], is linear if <math>J(y_1+y_2) = J(y_1) + J(y_2), y_1,y_2\isin V\,</math> and <math>J(\alpha y_1) = \alpha J(y_1), \alpha \isin R', y_1\isin V\,</math>. Which of the following are functionals on <math>C^{-1}[a,b]\,</math> are linear?
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# Show that the first variation <math>\delta J(y_0,h)\,</math> satisfies the homogeneity condition <math>\delta J(y_0, \alpha h) = \alpha \delta J(y_0, h), \alpha \isin \mathbb{R}\,</math>.<br><br>
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# <math>J:V\to R'\,</math>, where <math>V\,</math> is a normed linear space, is linear if <math>J(y_1+y_2) = J(y_1) + J(y_2), y_1,y_2\isin V\,</math> and <math>J(\alpha y_1) = \alpha J(y_1), \alpha \isin R', y_1\isin V\,</math>. Which of the following are functionals on <math>C^{-1}[a,b]\,</math> are linear?
  
 
(a) <math>J(y)=\int_a^b y y' dx\,</math>
 
(a) <math>J(y)=\int_a^b y y' dx\,</math>
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(c) <math>J(y) = e^{y(a)}\,</math>
 
(c) <math>J(y) = e^{y(a)}\,</math>
  
(d) The set of all [[continuous functions]] on <math>[0,1]\,</math> satisfying <math>f(0)=0\,</math>
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(d) The set of all continuous functions on <math>[0,1]\,</math> satisfying <math>f(0)=0\,</math>
  
 
(e) The set of all continuous functions on <math>[0,1]\,</math> satisfying <math>f(1)=1\,</math>
 
(e) The set of all continuous functions on <math>[0,1]\,</math> satisfying <math>f(1)=1\,</math>
  
  
[[CoV12|solution]] Find the extremal for <math>J(y)=\int_1^2 \frac{\sqrt{1+y'^2}}{x} dx, y(1)=0, y(2)=1\,</math>
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# Find the extremal for <math>J(y)=\int_1^2 \frac{\sqrt{1+y'^2}}{x} dx, y(1)=0, y(2)=1\,</math>
  
[[CoV13|solution]] Compute the first variation of <math>J(y)=\int_a^b yy' dx\,</math>
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# Compute the first variation of <math>J(y)=\int_a^b yy' dx\,</math>
  
[[CoV14|solution]] Compute the first variation of <math>J(y)=\int_a^b (y'^2+2y)dx\,</math>
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# Compute the first variation of <math>J(y)=\int_a^b (y'^2+2y)dx\,</math>
  
[[CoV15|solution]] Compute the first variation of <math>J(y)=e^{y(a)}\,</math>
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# Compute the first variation of <math>J(y)=e^{y(a)}\,</math>
  
[[CoV16|solution]] Minimize <math>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\,</math>
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# Minimize <math>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\,</math>
  
[[CoV17|solution]] Find the extremals of <math>J(y) = \int_0^1(yy'+y''^2)dx, y(0)=0, y'(0)=1, y(1)=2, y'(1)=4\,</math>
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# Find the extremals of <math>J(y) = \int_0^1(yy'+y''^2)dx, y(0)=0, y'(0)=1, y(1)=2, y'(1)=4\,</math>
  
[[CoV18|solution]] Find the [[Euler-Lagrange equation|Euler equation]] for <math>J(y,z)=\int_a^b\left[ y''z' + xyz'' + z'''y^2\right] dx\,</math>
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# Find the Euler-Lagrange equation|Euler equation for <math>J(y,z)=\int_a^b\left[ y''z' + xyz'' + z'''y^2\right] dx\,</math>
  
[[CoV19|solution]] Minimize <math>J(y)=\int_0^1(1+y''^2)dx, y(0)=0,y'(0)=1,y(1)=1,y'(1)=1\,</math>
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# Minimize <math>J(y)=\int_0^1(1+y''^2)dx, y(0)=0,y'(0)=1,y(1)=1,y'(1)=1\,</math>
  
[[CoV20|solution]] Minimize <math>J(y)=\int 2\pi y \sqrt{1+y'^2} dx\,</math>
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# Minimize <math>J(y)=\int 2\pi y \sqrt{1+y'^2} dx\,</math>
  
[[CoV21|solution]] Obtain a necessary condition for a function <math>y\isin C[a,b]\,</math> to be a local minimum of the functional
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# Obtain a necessary condition for a function <math>y\isin C[a,b]\,</math> to be a local minimum of the functional
  
 
<math>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\,</math>
 
<math>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\,</math>
  
[[CoV22|solution]] Find the Euler equation for the functional
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# Find the Euler equation for the functional
  
 
<math>J(u)=\iint\limits_G\left[u_x^2+u_y^2+2f(x,y)u(x,y)\right]dxdy\,</math>
 
<math>J(u)=\iint\limits_G\left[u_x^2+u_y^2+2f(x,y)u(x,y)\right]dxdy\,</math>
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[[CoV23|solution]] Find the extremal of the functional <math>J(y)=\int_0^\pi\left[y'(x)\right]^2dx\,</math> subject to the constraint <math>\int_0^\pi \left[ y(x)\right]^2dx=1, y(0)=y(\pi)=0\,</math>.
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# Find the extremal of the functional <math>J(y)=\int_0^\pi\left[y'(x)\right]^2dx\,</math> subject to the constraint <math>\int_0^\pi \left[ y(x)\right]^2dx=1, y(0)=y(\pi)=0\,</math>.
  
[[CoV24|solution]] Determine the function <math>\hat{y}\isin C^2[0,1]\,</math> that minimizes the functional <math>J(y)=\int_0^1\left[y'(x)\right]^2dx+[y(1)]^2, y(0)=1, h(0)=0\,</math>.
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# Determine the function <math>\hat{y}\isin C^2[0,1]\,</math> that minimizes the functional <math>J(y)=\int_0^1\left[y'(x)\right]^2dx+[y(1)]^2, y(0)=1, h(0)=0\,</math>.
  
[[CoV25|solution]] Let <math>J:A\to\mathbb{R}\,</math> be a functional on a subset <math>A\,</math> of a normed linear space <math>V\,</math>.  
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# Let <math>J:A\to\mathbb{R}\,</math> be a functional on a subset <math>A\,</math> of a normed linear space <math>V\,</math>.  
  
 
(a) Define precisely the first variation <math>\delta J(y_0,h)\,</math> of <math>J\,</math> at <math>y_0\,</math> and admissible <math>h(x)\,</math>.  
 
(a) Define precisely the first variation <math>\delta J(y_0,h)\,</math> of <math>J\,</math> at <math>y_0\,</math> and admissible <math>h(x)\,</math>.  
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(b) Show that if <math>\delta J(y_0,h)\,</math> exists for a certain admissible <math>h\isin V\,</math>, then <math>\delta J(y_0,\alpha h)\,</math> also exists for every real number <math>\alpha\,</math>, and <math>\delta J(y_0,\alpha h)=\alpha \delta J(y_0,h)\,</math>.
 
(b) Show that if <math>\delta J(y_0,h)\,</math> exists for a certain admissible <math>h\isin V\,</math>, then <math>\delta J(y_0,\alpha h)\,</math> also exists for every real number <math>\alpha\,</math>, and <math>\delta J(y_0,\alpha h)=\alpha \delta J(y_0,h)\,</math>.
  
[[CoV26|solution]] Compute the first variation <math>\delta J(y,h)\,</math> for <math>y\isin C[0,1]\,</math>: <math>J(y)=e^{y(0)}\,</math>
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# Compute the first variation <math>\delta J(y,h)\,</math> for <math>y\isin C[0,1]\,</math>: <math>J(y)=e^{y(0)}\,</math>
 
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[[CoV27|solution]] Compute the first variation <math>\delta J(y,h)\,</math> for <math>y\isin C[0,1]\,</math>: <math>J(y)=\int_0^1\int_0^1\sin(xt)y(x)y(t)dxdt\,</math>
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[[CoV28|solution]] Compute the first variation <math>J(y) = \int_0^1 (3y^2 + x) dx + y^2(0), y_0(x) = x, h(x)=x+1\,</math>
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# Compute the first variation <math>\delta J(y,h)\,</math> for <math>y\isin C[0,1]\,</math>: <math>J(y)=\int_0^1\int_0^1\sin(xt)y(x)y(t)dxdt\,</math>
  
[[Main Page]]
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# Compute the first variation <math>J(y) = \int_0^1 (3y^2 + x) dx + y^2(0), y_0(x) = x, h(x)=x+1\,</math>

2016年10月18日 (二) 08:49的版本

解答见原网页 变分计算

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

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

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

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

  1. 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\, $.

  1. Derive the Euler-Lagrange equation from the attempt to minimize the functional

$ \begin{align}T(y)=\int_a^b L(y,y',x)\,dx\,\end{align} $

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

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

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

  1. 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}\, $.

  1. $ 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\, $


  1. Find the extremal for $ J(y)=\int_1^2 \frac{\sqrt{1+y'^2}}{x} dx, y(1)=0, y(2)=1\, $
  1. Compute the first variation of $ J(y)=\int_a^b yy' dx\, $
  1. Compute the first variation of $ J(y)=\int_a^b (y'^2+2y)dx\, $
  1. Compute the first variation of $ J(y)=e^{y(a)}\, $
  1. 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\, $
  1. 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\, $
  1. Find the Euler-Lagrange equation|Euler equation for $ J(y,z)=\int_a^b\left[ y''z' + xyz'' + z'''y^2\right] dx\, $
  1. Minimize $ J(y)=\int_0^1(1+y''^2)dx, y(0)=0,y'(0)=1,y(1)=1,y'(1)=1\, $
  1. Minimize $ J(y)=\int 2\pi y \sqrt{1+y'^2} dx\, $
  1. 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\, $

  1. 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.


  1. 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\, $.
  1. 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\, $.
  1. 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)\, $.

  1. Compute the first variation $ \delta J(y,h)\, $ for $ y\isin C[0,1]\, $: $ J(y)=e^{y(0)}\, $
  1. 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\, $
  1. Compute the first variation $ J(y) = \int_0^1 (3y^2 + x) dx + y^2(0), y_0(x) = x, h(x)=x+1\, $