Forest UCM CoV - Revision history

Foretony: /* Soap film (cycloid) */

2014-10-23T12:31:21Z

Soap film (cycloid)

2014-10-23T12:29:33Z

returning to Brachistochrone

2014-10-23T12:26:43Z

returning to Brachistochrone

2014-10-23T12:03:40Z

Soup film (cycloid)

2014-10-22T12:45:25Z

Soup film (cycloid)

2014-10-22T12:44:22Z

Soup film (cycloid)

2014-10-22T12:42:25Z

Soup film (cycloid)

2014-10-22T12:40:52Z

Soup film (cycloid)

2014-10-22T12:38:41Z

Soup film (cycloid)

2014-10-22T12:36:49Z

Soup film (cycloid)

← Older revision		Revision as of 12:31, 23 October 2014
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	https://www.fields.utoronto.ca/programs/scientific/12-13/Marsden/FieldsSS2-FinalSlidesJuly2012.pdf		https://www.fields.utoronto.ca/programs/scientific/12-13/Marsden/FieldsSS2-FinalSlidesJuly2012.pdf
		+
		+	http://books.google.com/books?id=4iABL2l_i98C&pg=PA323&lpg=PA323&dq=euler+lagrange+equation+alternate+form&source=bl&ots=sJfsaoONwO&sig=IiBUDYL4sOoQxMJ9tYcTHi4Agbw&hl=en&sa=X&ei=yNhDVMLKD8WGigLJroGABQ&ved=0CFoQ6AEwCQ#v=onepage&q=euler%20lagrange%20equation%20alternate%20form&f=false

	[[Forest_Ugrad_ClassicalMechanics]]		[[Forest_Ugrad_ClassicalMechanics]]

@@ Line 516: / Line 516: @@
 If<math> x=0</math> when <math>\theta =0</math> then the constant = 0
-The path of least time followed by the bead will be that traced out by the rim of a rolling circle (wheel) of radius R
+The path of least time followed by the bead will be that traced out by the rim of a rolling circle (wheel) of radius R that rolls counter clockwise with complete slipping as is moves forward.
 [[File:Forest_UCM_Brachystochrone_path.png | 400px]]

← Older revision		Revision as of 12:26, 23 October 2014
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−	If<math> x=0</math> when \theta =0~~<math>Insert formula here~~</math> then the constant = 0	+	If<math> x=0</math> when <math>\theta =0</math> then the constant = 0
		+
		+	The path of least time followed by the bead will be that traced out by the rim of a rolling circle (wheel) of radius R
		+
		+	[[File:Forest_UCM_Brachystochrone_path.png \| 400px]]

	=== Soap film (cycloid) ===		=== Soap film (cycloid) ===

@@ Line 516: / Line 516: @@
 If<math> x=0</math> when \theta =0<math>Insert formula here</math> then the constant = 0
-=== Soup film (cycloid) ===
+=== Soap film (cycloid) ===
 consider the problem of the soap bubble formed between two circular rings.  You stack two rings on top of each other in soapy water and then pull them apart after removing them from the water together where a film is created between the rings in a cylindrical shape.

← Older revision		Revision as of 12:45, 22 October 2014
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	:<math> y = C \cosh(\frac{x-K}{C})</math>		:<math> y = C \cosh(\frac{x-K}{C})</math>
		+
		+	C and K are constants used to position the area so it goes through the desired points in space

← Older revision		Revision as of 12:44, 22 October 2014
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	as long as the smallest value of y is larger than the constant C		as long as the smallest value of y is larger than the constant C

−	:<math>x = \int \frac{C}{\sqrt{y^2-C^2}} dy = C \cosh^{-1}(\frac{y}{c})</math>	+	:<math>x = \int \frac{C}{\sqrt{y^2-C^2}} dy = C \cosh^{-1}(\frac{y}{C}) + K</math>
		+
		+	or
		+
		+	:<math> y = C \cosh(\frac{x-K}{C})</math>

@@ Line 547: / Line 547: @@
 :<math> \frac{y^2}{\left ( 1 + (y^{\prime})^2 \right ) }=C^2</math>
-:<math>\Rightarrow y^{\prime} = \frac{dy}{dx} = \frac{C}{\sqrt{y^2-C^2}}</math>
+:<math>\Rightarrow y^{\prime} = \frac{dy}{dx} = \frac{\sqrt{y^2-C^2}}{C}</math>
 as long as the smallest value of y is larger than the constant C
-:<math>x = \int \frac{\sqrt{y^2-C^2}}{C} dy = C \cosh^{-1}(\frac{y}{c})</math>
+:<math>x = \int \frac{C}{\sqrt{y^2-C^2}} dy = C \cosh^{-1}(\frac{y}{c})</math>

@@ Line 548: / Line 548: @@
 :<math>\Rightarrow y^{\prime} = \frac{dy}{dx} = \frac{C}{\sqrt{y^2-C^2}}</math>
 https://www.fields.utoronto.ca/programs/scientific/12-13/Marsden/FieldsSS2-FinalSlidesJuly2012.pdf
 [[Forest_Ugrad_ClassicalMechanics]]

← Older revision		Revision as of 12:36, 22 October 2014
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	:<math>\left [f- y^{\prime}\frac{\partial f}{\partial y^{\prime} }\right ] = y\left ( 1 + (y^{\prime})^2 \right )^{\frac{1}{2}} - \frac{yy^{\prime}(y^{\prime})}{\left ( 1 + (y^{\prime})^2 \right )^{\frac{1}{2}} }</math>		:<math>\left [f- y^{\prime}\frac{\partial f}{\partial y^{\prime} }\right ] = y\left ( 1 + (y^{\prime})^2 \right )^{\frac{1}{2}} - \frac{yy^{\prime}(y^{\prime})}{\left ( 1 + (y^{\prime})^2 \right )^{\frac{1}{2}} }</math>
−	::<math>= \frac{y\left ( 1 + (y^{\prime})^2 \right ) - y(y^{\prime})^2}{\left ( 1 + (y^{\prime})^2 \right )^{\frac{1}{2}} }= \frac{y}{\left ( 1 + (y^{\prime})^2 \right )^{\frac{1}{2}} }</math>	+	::<math>= \frac{y\left ( 1 + (y^{\prime})^2 \right ) - y(y^{\prime})^2}{\left ( 1 + (y^{\prime})^2 \right )^{\frac{1}{2}} }= \frac{y}{\left ( 1 + (y^{\prime})^2 \right )^{\frac{1}{2}} }=</math> constant

		+	or

		+
		+	:<math> \frac{y^2}{\left ( 1 + (y^{\prime})^2 \right ) }=</math> contant