Forest UCM Energy CentralForce - Revision history

Foretony: /* Spherical symmetry */

2014-09-27T18:38:01Z

Spherical symmetry

2014-09-27T18:36:33Z

Spherical symmetry

2014-09-27T18:35:50Z

Spherical symmetry

2014-09-27T18:35:32Z

Spherical symmetry

2014-09-27T18:32:50Z

Spherical symmetry

2014-09-27T18:28:27Z

Spherical symmetry

2014-09-27T18:27:01Z

Spherical symmetry

2014-09-27T18:26:50Z

Spherical symmetry

2014-09-27T18:26:01Z

The Central Force

2014-09-27T13:58:47Z

Gradient in spherical coordinates

← Older revision		Revision as of 18:38, 27 September 2014
Line 74:		Line 74:

	If <math>\vec F</math> is a conservative central force then <math>\vec F</math> and <math>U</math> are spherically symmetric.		If <math>\vec F</math> is a conservative central force then <math>\vec F</math> and <math>U</math> are spherically symmetric.
		+
		+	The converse is also true
		+	: if the central force is spherically symmetric then it is a conservative force.

	[[Forest_UCM_Energy#Central_Forces]]		[[Forest_UCM_Energy#Central_Forces]]

@@ Line 65: / Line 65: @@
 :<math>\vec F = -\vec \nabla U</math>
-::<math>   = \left ( \hat r \frac{d}{dr}  + \hat \theta \frac{1}{r} \frac{d}{d \theta}  +  \hat \phi \frac{1}{ r \sin \theta } \frac{d}{d\phi} \right ) U </math>
+::<math>   = - \left ( \hat r \frac{d}{dr}  + \hat \theta \frac{1}{r} \frac{d}{d \theta}  +  \hat \phi \frac{1}{ r \sin \theta } \frac{d}{d\phi} \right ) U </math>
 in order to be only a function of <math>r</math> then

@@ Line 72: / Line 72: @@
 :<math>  \hat \phi \frac{1}{ r \sin \theta } \frac{d}{d\phi} U  = 0 </math>
-Both If  <math>\vec F</math> is a conservative central force then <math>\vec F</math> and <math>U</math> are spherically symmetric.
 [[Forest_UCM_Energy#Central_Forces]]

@@ Line 72: / Line 72: @@
 :<math>  \hat \phi \frac{1}{ r \sin \theta } \frac{d}{d\phi} U  = 0 </math>
-Both <math>\vec F</math> and <math>U</math> are spherically symmetric is the central force is to be conservative.
+Both If  <math>\vec F</math> is a conservative central force then <math>\vec F</math> and <math>U</math> are spherically symmetric.
 [[Forest_UCM_Energy#Central_Forces]]

@@ Line 67: / Line 67: @@
 ::<math>   = \left ( \hat r \frac{d}{dr}  + \hat \theta \frac{1}{r} \frac{d}{d \theta}  +  \hat \phi \frac{1}{ r \sin \theta } \frac{d}{d\phi} \right ) U </math>
-in order to be only a function of r then
+in order to be only a function of <math>r</math> then
 :<math>   \hat \theta \frac{1}{r} \frac{d}{d \theta}  U  = 0 </math>
 :<math>  \hat \phi \frac{1}{ r \sin \theta } \frac{d}{d\phi} U  = 0 </math>
+Both <math>\vec F</math> and <math>U</math> are spherically symmetric is the central force is to be conservative.
 [[Forest_UCM_Energy#Central_Forces]]

← Older revision		Revision as of 18:28, 27 September 2014
Line 67:		Line 67:
	::<math> = \left ( \hat r \frac{d}{dr} + \hat \theta \frac{1}{r} \frac{d}{d \theta} + \hat \phi \frac{1}{ r \sin \theta } \frac{d}{d\phi} \right ) U </math>		::<math> = \left ( \hat r \frac{d}{dr} + \hat \theta \frac{1}{r} \frac{d}{d \theta} + \hat \phi \frac{1}{ r \sin \theta } \frac{d}{d\phi} \right ) U </math>

		+	in order to be only a function of r then
		+
		+	:<math> \hat \theta \frac{1}{r} \frac{d}{d \theta} U = 0 </math>
		+	:<math> \hat \phi \frac{1}{ r \sin \theta } \frac{d}{d\phi} U = 0 </math>



	[[Forest_UCM_Energy#Central_Forces]]		[[Forest_UCM_Energy#Central_Forces]]

← Older revision		Revision as of 18:27, 27 September 2014
Line 65:		Line 65:

	:<math>\vec F = -\vec \nabla U</math>		:<math>\vec F = -\vec \nabla U</math>
−	::<math> = \left ( \hat r \frac{d}{dr} + \hat \theta \frac{1}{r} \frac{d}{d \theta} + \hat \phi \frac{1}{ r \sin \theta } \frac{d}{d\phi} \~~rigt~~ ) U </math>	+	::<math> = \left ( \hat r \frac{d}{dr} + \hat \theta \frac{1}{r} \frac{d}{d \theta} + \hat \phi \frac{1}{ r \sin \theta } \frac{d}{d\phi} \right ) U </math>

← Older revision		Revision as of 18:26, 27 September 2014
Line 65:		Line 65:

	:<math>\vec F = -\vec \nabla U</math>		:<math>\vec F = -\vec \nabla U</math>
		+	::<math> = \left ( \hat r \frac{d}{dr} + \hat \theta \frac{1}{r} \frac{d}{d \theta} + \hat \phi \frac{1}{ r \sin \theta } \frac{d}{d\phi} \rigt ) U </math>

← Older revision		Revision as of 18:26, 27 September 2014
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	=The Central Force=		=The Central Force=
		+
		+	==Spherical symmetry==
		+
		+	A central force needs to be spherically symmetric in order to be conservative.
		+
		+	:<math>\vec F = f(\vec r) \hat r</math>
		+
		+	If the force is conservative then a potential exists such that
		+
		+	:<math>\vec F = -\vec \nabla U</math>
		+



	[[Forest_UCM_Energy#Central_Forces]]		[[Forest_UCM_Energy#Central_Forces]]

← Older revision		Revision as of 13:58, 27 September 2014
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	:<math> \vec \nabla = \hat r \frac{d}{dr} + \hat \theta \frac{1}{r} \frac{d}{d \theta} + \hat \phi \frac{1}{ r \sin \theta } \frac{d}{d\phi} </math>		:<math> \vec \nabla = \hat r \frac{d}{dr} + \hat \theta \frac{1}{r} \frac{d}{d \theta} + \hat \phi \frac{1}{ r \sin \theta } \frac{d}{d\phi} </math>

		+
		+	=The Central Force=



	[[Forest_UCM_Energy#Central_Forces]]		[[Forest_UCM_Energy#Central_Forces]]