Polyconic projection - Revision history

31.55.40.133: corrected formula for x when phi = 0

2014-11-16T19:51:34Z

corrected formula for x when phi = 0

← Older revision		Revision as of 21:51, 16 November 2014
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	~~The '''[[Constantin Carathéodory\|Carathéodory]]–[[Carl Gustav Jakob Jacobi\|Jacobi]]–[[Sophus Lie\|Lie]] theorem''~~' is a ~~[[theorem]] in [[symplectic geometry]] which generalizes [[Darboux~~'s ~~theorem]].~~		Myrtle Benny is how I'm known as and I feel comfortable when people use the full title. My day job is a meter reader. It's not a typical factor but what she likes performing is foundation jumping and now she is attempting to make cash with it. South Dakota is where I've always been living.<br><br>Feel free to visit my site; [http://nxnn.info/blog/284637 over the counter std test]

	~~==Statement==~~
	~~Let ''M'' be~~ a ~~2''n''-dimensional [[symplectic manifold]]~~ with ~~symplectic form ω~~. For ''p'' ∈ ''M'' and ''r'' ≤ ''n'', let ''f''<sub>1</sub>, ''f''<sub>2</sub>, ..., ''f''<sub>r</sub> be [[smooth function]]s defined on an [[open neighborhood]] ''V'' of ''p'' whose [[differential form\|differential]]s are [[linearly independent]] at each point, or equivalently

	~~:<math>df_1(p) \wedge \ldots \wedge df_r(p) \neq 0,</math>~~

	where ~~{f<sub>i</sub>, f<sub>j</sub>} = 0. (In other words they are pairwise in involution.) Here {–,–} is the [[Poisson bracket]]. Then there are functions ''f~~'~~'<sub>r+1</sub>,~~ .~~.., ''f''~~<~~sub~~>n<~~/sub~~>~~, ''g''<sub>1</sub>, ''g''<sub>2</sub>, ..., ''g''<sub>n</sub> defined on an open neighborhood ''U'' ⊂&nbsp~~;~~''V'' of ''p'' such that (f<sub>i</sub>, g<sub>i</sub>) is a~~ [~~[symplectic chart]] of ''M'', i.e., ω is expressed on ''U'' as~~

	:~~<math>\omega = \sum_{i=1}^n df_i \wedge dg_i.<~~/~~math>~~

	~~==Applications==~~
	~~As a direct application we have the following~~. ~~Given a [[Hamiltonian system]] as <math>(M,\omega,H)<~~/~~math> where ''M'' is a symplectic manifold with symplectic form <math>\omega<~~/~~math> and ''H'' is~~ the ~~[[Hamiltonian mechanics\|Hamiltonian function]], around every point where <math>dH \neq 0</math> there is a symplectic chart such that one of its coordinates is ''H''.~~

	~~==References==~~
	* Lee, John M., ''Introduction to Smooth Manifolds'', Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbook on smooth manifolds.

	~~{{DEFAULTSORT:Caratheodory-Jacobi-Lie theorem}}~~
	~~[[Category:Symplectic geometry]]~~
	~~[[Category:Theorems in differential geometry]~~]


	~~{{differential-geometry-stub}}~~

en>Chienlit: ==See also== {{Portal|Atlas}} * List of map projections

2013-12-22T17:36:06Z

==See also== {{Portal|Atlas}} * List of map projections

← Older revision		Revision as of 19:36, 22 December 2013
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	The ~~title of the author~~ is ~~Numbers but it~~'s ~~not the most masucline title out there~~. ~~One of the things she enjoys most is to read comics and she~~'ll be ~~starting some thing else along~~ with it. ~~Minnesota is where he~~'s ~~been living for many years~~. ~~Hiring~~ is ~~her day occupation now and she will not alter it anytime quickly~~.<br><br>~~my web site - [http:~~//~~www~~.~~hard-ass-porn~~.~~com~~/~~blog~~/~~111007 http~~://~~www~~.~~hard~~-~~ass~~-~~porn~~.~~com/~~]		The '''[[Constantin Carathéodory\|Carathéodory]]–[[Carl Gustav Jakob Jacobi\|Jacobi]]–[[Sophus Lie\|Lie]] theorem''' is a [[theorem]] in [[symplectic geometry]] which generalizes [[Darboux's theorem]].

			==Statement==
			Let ''M'' be a 2''n''-dimensional [[symplectic manifold]] with symplectic form ω. For ''p'' ∈ ''M'' and ''r'' ≤ ''n'', let ''f''<sub>1</sub>, ''f''<sub>2</sub>, ..., ''f''<sub>r</sub> be [[smooth function]]s defined on an [[open neighborhood]] ''V'' of ''p'' whose [[differential form\|differential]]s are [[linearly independent]] at each point, or equivalently

			:<math>df_1(p) \wedge \ldots \wedge df_r(p) \neq 0,</math>

			where {f<sub>i</sub>, f<sub>j</sub>} = 0. (In other words they are pairwise in involution.) Here {–,–} is the [[Poisson bracket]]. Then there are functions ''f''<sub>r+1</sub>, ..., ''f''<sub>n</sub>, ''g''<sub>1</sub>, ''g''<sub>2</sub>, ..., ''g''<sub>n</sub> defined on an open neighborhood ''U'' ⊂ ''V'' of ''p'' such that (f<sub>i</sub>, g<sub>i</sub>) is a [[symplectic chart]] of ''M'', i.e., ω is expressed on ''U'' as

			:<math>\omega = \sum_{i=1}^n df_i \wedge dg_i.</math>

			==Applications==
			As a direct application we have the following. Given a [[Hamiltonian system]] as <math>(M,\omega,H)</math> where ''M'' is a symplectic manifold with symplectic form <math>\omega</math> and ''H'' is the [[Hamiltonian mechanics\|Hamiltonian function]], around every point where <math>dH \neq 0</math> there is a symplectic chart such that one of its coordinates is ''H''.

			==References==
			* Lee, John M., ''Introduction to Smooth Manifolds'', Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbook on smooth manifolds.

			{{DEFAULTSORT:Caratheodory-Jacobi-Lie theorem}}
			[[Category:Symplectic geometry]]
			[[Category:Theorems in differential geometry]]


			{{differential-geometry-stub}}

en>EdwardLane: /* External links */ added map projection template

2012-07-18T16:31:53Z

External links: added map projection template

New page

The title of the author is Numbers but it's not the most masucline title out there. One of the things she enjoys most is to read comics and she'll be starting some thing else along with it. Minnesota is where he's been living for many years. Hiring is her day occupation now and she will not alter it anytime quickly.<br><br>my web site - [http://www.hard-ass-porn.com/blog/111007 http://www.hard-ass-porn.com/]