变分计算例子 - 版本历史

2016年10月18日 (二) 01:02 宋玉坤

2016-10-18T01:02:29Z

2016年10月18日 (二) 00:54 宋玉坤

2016-10-18T00:54:51Z

2016年10月18日 (二) 00:53 宋玉坤

2016-10-18T00:53:51Z

2016年10月18日 (二) 00:52 宋玉坤

2016-10-18T00:52:09Z

2016年10月18日 (二) 00:51 宋玉坤

2016-10-18T00:51:46Z

2016年10月18日 (二) 00:49 宋玉坤

2016-10-18T00:49:26Z

宋玉坤：创建页面，内容为“solution Find the path that minimizes the arclength of the curve between $(x_0,y_0) = (0,0)\,$ and $(x_1,y_1) = (1,1)\,$ .

...”

2016-10-18T00:41:50Z

创建页面，内容为“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>. ...”

新页面

[[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>. 

[[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>. 

 
[[CoV3|solution]] Find the maximum of <math>xy^2z^2\,</math> subject to the constraint <math>x+y+z=12\,</math>. 

[[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>. 

[[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>. 

[[CoV6|solution]] Derive the Euler-Lagrange equation from the attempt to minimize the functional 

<math>T(y)=\int_a^b L(y,y',x)\,dx\,</math> 

[[CoV7|solution]] Minimize the [[functional]] from [[classical mechanics]]: <math>\int_{t_1}^{t_2}(\mathrm{Kinetic\,Energy} - \mathrm{Potential\,Energy})\,</math> 

[[CoV8|solution]] Find the extrema of <math>\int_a^b \frac{y'^2}{x^3}\,dx\,</math>. 

[[CoV9|solution]] Find the extrema of <math>\int_a^b (y^2 +y'^2 + 2y e^x) \,dx\,</math>. 

[[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>. 

[[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?

(a) <math>J(y)=\int_a^b y y' dx\,</math>

(b) <math>J(\alpha y) = \int_a^b (4y'^2 + 2(\alpha y))dx\,</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>

(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>

[[CoV13|solution]] 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>

[[CoV15|solution]] 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>

[[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>

[[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>

[[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>

[[CoV20|solution]] 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

<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

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

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

 
[[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>.

[[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>.

[[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>.

(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>.

(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>

[[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>

[[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>

[[Main Page]]

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 解答见原网页 [http://www.exampleproblems.com/wiki/index.php/Calculus_of_Variations 变分计算]
-# 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>
+# 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>.
 # 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>.
-# 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>
+# Find the maximum of <math>xy^2z^2\,</math> subject to the constraint <math>x+y+z=12\,</math>.
 # Write the Euler-Lagrange equation for <math>L(x,y,z,y',z',y'',z'',y''',z''',...,y^{(k)},z^{(k)})\,</math>.
-# Find the maximum of <math>xy^2z^2\,</math> subject to the constraint <math>x+y+z=12\,</math>.<br><br>
+# 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>.
 # Derive the Euler-Lagrange equation from the attempt to minimize the functional
-# Write the Euler-Lagrange equation for <math>L(x,y,z,y',z',y'',z'',y''',z''',...,y^{(k)},z^{(k)})\,</math>.<br><br>
+#: <math>T(y)=\int_a^b L(y,y',x)\,dx\,</math>
 # Minimize the functional from classical mechanics: <math>\int_{t_1}^{t_2}(\mathrm{Kinetic\,Energy} - \mathrm{Potential\,Energy})\,</math>
-# 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>
+# Find the extrema of <math>\int_a^b \frac{y'^2}{x^3}\,dx\,</math>.<br>
 # Find the extrema of <math>\int_a^b (y^2 +y'^2 + 2y e^x) \,dx\,</math>.<br>
-# Derive the Euler-Lagrange equation from the attempt to minimize the functional<br><br>
+# 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>. <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>
-<math>T(y)=\int_a^b L(y,y',x)\,dx\,</math><br><br>
+#:(b) <math>J(\alpha y) = \int_a^b (4y'^2 + 2(\alpha y))dx\,</math>
+#:(c) <math>J(y) = e^{y(a)}\,</math>
-# Minimize the functional from classical mechanics: <math>\int_{t_1}^{t_2}(\mathrm{Kinetic\,Energy} - \mathrm{Potential\,Energy})\,</math><br><br>
+#:(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>
-# Find the extrema of <math>\int_a^b \frac{y'^2}{x^3}\,dx\,</math>.<br><br>
-−
-# Find the extrema of <math>\int_a^b (y^2 +y'^2 + 2y e^x) \,dx\,</math>.<br><br>
-−
-# 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>
-−
-# <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>
-−
 (b) <math>J(\alpha y) = \int_a^b (4y'^2 + 2(\alpha y))dx\,</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>
-−
 (e) The set of all continuous functions on <math>[0,1]\,</math> satisfying <math>f(1)=1\,</math>
-−
-−
 # Find the extremal for <math>J(y)=\int_1^2 \frac{\sqrt{1+y'^2}}{x} dx, y(1)=0, y(2)=1\,</math>
 # Compute the first variation of <math>J(y)=\int_a^b yy' dx\,</math>
 # Compute the first variation of <math>J(y)=\int_a^b (y'^2+2y)dx\,</math>
 # Compute the first variation of <math>J(y)=e^{y(a)}\,</math>
 # 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>
 # 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>
 # Find the Euler-Lagrange equation for <math>J(y,z)=\int_a^b\left[ y''z' + xyz'' + z'''y^2\right] dx\,</math>
 # Minimize <math>J(y)=\int_0^1(1+y''^2)dx, y(0)=0,y'(0)=1,y(1)=1,y'(1)=1\,</math>
 # Minimize <math>J(y)=\int 2\pi y \sqrt{1+y'^2} dx\,</math>
 # 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>
-−
 # 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>
+#: where <math>G\,</math> is a closed region in the <math>xy\,</math> plane and <math>u\,</math> has continuous second partial derivatives.<br>
-−
-where <math>G\,</math> is a closed region in the <math>xy\,</math> plane and <math>u\,</math> has continuous second partial derivatives.<br>
-−
 <br>
 # 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>.
 # 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>.
 # 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>.
+#:(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>.
-−
 # 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>
 # 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>
 # 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>

@@ 第7行： / 第7行： @@
 # Find the maximum of <math>xy^2z^2\,</math> subject to the constraint <math>x+y+z=12\,</math>.<br><br>
-# 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>
+# Write the Euler-Lagrange equation for <math>L(x,y,z,y',z',y'',z'',y''',z''',...,y^{(k)},z^{(k)})\,</math>.<br><br>
 # 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>
@@ 第48行： / 第48行： @@
 # 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>
-# 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>
+# Find the Euler-Lagrange equation for <math>J(y,z)=\int_a^b\left[ y''z' + xyz'' + z'''y^2\right] dx\,</math>
 # Minimize <math>J(y)=\int_0^1(1+y''^2)dx, y(0)=0,y'(0)=1,y(1)=1,y'(1)=1\,</math>

@@ 第13行： / 第13行： @@
 # Derive the Euler-Lagrange equation from the attempt to minimize the functional<br><br>
-#;<math>T(y)=\int_a^b L(y,y',x)\,dx\,</math><br><br>
+<math>T(y)=\int_a^b L(y,y',x)\,dx\,</math><br><br>
 # Minimize the functional from classical mechanics: <math>\int_{t_1}^{t_2}(\mathrm{Kinetic\,Energy} - \mathrm{Potential\,Energy})\,</math><br><br>

@@ 第1行： / 第1行： @@
 解答见原网页 [http://www.exampleproblems.com/wiki/index.php/Calculus_of_Variations 变分计算]
-#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>
+# 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>
 # 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>

@@ 第13行： / 第13行： @@
 # Derive the Euler-Lagrange equation from the attempt to minimize the functional<br><br>
-<math>\begin{align}T(y)=\int_a^b L(y,y',x)\,dx\,\end{align}</math><br><br>
+#;<math>T(y)=\int_a^b L(y,y',x)\,dx\,</math><br><br>
 # Minimize the functional from classical mechanics: <math>\int_{t_1}^{t_2}(\mathrm{Kinetic\,Energy} - \mathrm{Potential\,Energy})\,</math><br><br>