12-267/Derivation of Euler-Lagrange - Revision history

Twine: Added special cases (without derivations)

2012-10-24T22:18:13Z

Added special cases (without derivations)

← Older revision		Revision as of 18:18, 24 October 2012
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	As this must be equal to 0 for all h satisfying the endpoint constraints, we must have that <math>F_2 = \frac{d}{dx} F_3</math>, or in other terms, <math>F_y - \frac{d}{dx} F_y' = 0</math>.		As this must be equal to 0 for all h satisfying the endpoint constraints, we must have that <math>F_2 = \frac{d}{dx} F_3</math>, or in other terms, <math>F_y - \frac{d}{dx} F_y' = 0</math>.

			Special cases (without derivations):

			In the case that F does not depend on y', we have <math>F_y = 0</math>

			In the case that F does not depend on y, we have <math>F_{y'} = c</math>

			In the case that F does not depend on x, we have <math>F - y'F_{y'} = c</math>

Twine: Added navigation

2012-10-24T21:54:59Z

Added navigation

← Older revision		Revision as of 17:54, 24 October 2012
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			{{12-267/Navigation}}

	Disclamer: This is a student prepared note based on [http://drorbn.net/dbnvp/12-267-121002-2.php the lecure of Tuesday October 2nd].		Disclamer: This is a student prepared note based on [http://drorbn.net/dbnvp/12-267-121002-2.php the lecure of Tuesday October 2nd].

Twine: Corrected typo

2012-10-24T21:24:31Z

Corrected typo

← Older revision		Revision as of 17:24, 24 October 2012
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	<math> = \int_a^b (F_2(x, y, y') \cdot h + F_3(x, y, y') \cdot h') dx </math>		<math> = \int_a^b (F_2(x, y, y') \cdot h + F_3(x, y, y') \cdot h') dx </math>

	<math> = \int_a^b (F_2 \cdot h + [\frac{d}{dx} F_3] \cdot h) dx + F_3 \cdot h \|_a^b</math> (integrating by parts)		<math> = \int_a^b (F_2 \cdot h - [\frac{d}{dx} F_3] \cdot h) dx + F_3 \cdot h \|_a^b</math> (integrating by parts)

	Due to the constraints of <math>h(a) = 0</math> and <math>h(b) = 0</math>, <math>F_3 \cdot h \|_a^b = 0</math>.		Due to the constraints of <math>h(a) = 0</math> and <math>h(b) = 0</math>, <math>F_3 \cdot h \|_a^b = 0</math>.

Twine: corrected date of lecture

2012-10-12T23:07:00Z

corrected date of lecture

← Older revision		Revision as of 19:07, 12 October 2012
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	Disclamer: This is a student prepared note based on [http://drorbn.net/dbnvp/12-267-121002-2.php the lecure of Tuesday October ~~1st~~].		Disclamer: This is a student prepared note based on [http://drorbn.net/dbnvp/12-267-121002-2.php the lecure of Tuesday October 2nd].

	For a function <math>y(x)</math> defined on <math>[a, b]</math> to be an extremum of <math>J(y) = \int_a^b F(x, y, y') dx</math>, it must be that for any function <math>h(x)</math> defined on <math>[a, b]</math> that preserves the endpoints of <math>y</math> (that is, <math>h(a) = 0</math> and <math>h(b) = 0</math>), we have <math> \frac{d}{d \epsilon } J(y + \epsilon h) </math><math>\|_{\epsilon = 0} = 0 </math>.		For a function <math>y(x)</math> defined on <math>[a, b]</math> to be an extremum of <math>J(y) = \int_a^b F(x, y, y') dx</math>, it must be that for any function <math>h(x)</math> defined on <math>[a, b]</math> that preserves the endpoints of <math>y</math> (that is, <math>h(a) = 0</math> and <math>h(b) = 0</math>), we have <math> \frac{d}{d \epsilon } J(y + \epsilon h) </math><math>\|_{\epsilon = 0} = 0 </math>.

Twine: Created page

2012-10-12T23:01:07Z

Created page

New page

Disclamer: This is a student prepared note based on [http://drorbn.net/dbnvp/12-267-121002-2.php the lecure of Tuesday October 1st].

For a function <math>y(x)</math> defined on <math>[a, b]</math> to be an extremum of <math>J(y) = \int_a^b F(x, y, y') dx</math>, it must be that for any function <math>h(x)</math> defined on <math>[a, b]</math> that preserves the endpoints of <math>y</math> (that is, <math>h(a) = 0</math> and <math>h(b) = 0</math>), we have <math> \frac{d}{d \epsilon } J(y + \epsilon h) </math><math>|_{\epsilon = 0} = 0 </math>.

<math> \frac{d}{d \epsilon } J(y + \epsilon h) </math><math>|_{\epsilon = 0} </math>

<math> = \frac{d}{d \epsilon } \int_a^b F(x, y + \epsilon h, y' + \epsilon h') dx |_{\epsilon = 0} </math>

Let <math>F_n</math> signify F differentiated with respect to its nth variable.

<math> = \int_a^b (F_1 \cdot 0 + F_2 \cdot h + F_3 \cdot h') dx |_{\epsilon = 0} </math>

<math> = \int_a^b (F_2(x, y, y') \cdot h + F_3(x, y, y') \cdot h') dx </math>

<math> = \int_a^b (F_2 \cdot h + [\frac{d}{dx} F_3] \cdot h) dx + F_3 \cdot h |_a^b</math> (integrating by parts)

Due to the constraints of <math>h(a) = 0</math> and <math>h(b) = 0</math>, <math>F_3 \cdot h |_a^b = 0</math>.

<math> = \int_a^b (F_2 - \frac{d}{dx} F_3) \cdot h dx</math>

As this must be equal to 0 for all h satisfying the endpoint constraints, we must have that <math>F_2 = \frac{d}{dx} F_3</math>, or in other terms, <math>F_y - \frac{d}{dx} F_y' = 0</math>.