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en>Billinghurst: Remove link to dab page Index (mathematics) using popups
2015-01-11T08:10:46Z
<p>Remove link to dab page <a href="/index.php?title=Index_(mathematics)&action=edit&redlink=1" class="new" title="Index (mathematics) (page does not exist)">Index (mathematics)</a> using <a href="/index.php?title=En:Wikipedia:Tools/Navigation_popups&action=edit&redlink=1" class="new" title="En:Wikipedia:Tools/Navigation popups (page does not exist)">popups</a></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 09:10, 11 January 2015</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">In mathematics</del>, <del style="font-weight: bold; text-decoration: none;">a '''superintegrable Hamiltonian system''' is a [[Hamiltonian system]] on a </del>2<del style="font-weight: bold; text-decoration: none;">''n''</del>-<del style="font-weight: bold; text-decoration: none;">dimensional [[symplectic manifold]] </del>for <del style="font-weight: bold; text-decoration: none;">which </del>the <del style="font-weight: bold; text-decoration: none;">following conditions hold:</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">It involves expertise and knowledge of various tools and technologies used for creating websites. This means you can setup your mailing list and auto-responder on your wordpress site and then you can add your subscription form to any other blog</ins>, <ins style="font-weight: bold; text-decoration: none;">splash page, capture page or any other site you like. * A community forum for debate of the product together with some other customers in the comments spot. </ins>2- <ins style="font-weight: bold; text-decoration: none;">Ask </ins>for the <ins style="font-weight: bold; text-decoration: none;">designs and graphics that will be provided along with the Word - Press theme. 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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">(i) There exist ''n''&nbsp;≤&nbsp;''k'' independent integrals ''F''<sub>&nbsp;''i''</sub> </del>of <del style="font-weight: bold; text-decoration: none;">motion. Their level surfaces (invariant submanifolds) form </del>a <del style="font-weight: bold; text-decoration: none;">fibered manifold <math>F:Z\to N=F(Z)</math> over </del>a <del style="font-weight: bold; text-decoration: none;">connected open subset <math>N\subset\mathbb R^k</math></del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">(ii) There exist smooth real functions </del><<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">s_{ij}</del><<del style="font-weight: bold; text-decoration: none;">/math</del>> <del style="font-weight: bold; text-decoration: none;">on <math>N</math> such that </del>the <del style="font-weight: bold; text-decoration: none;">[[Poisson manifold|Poisson bracket]] of integrals of motion reads</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"><math>\{F_i</del>,<del style="font-weight: bold; text-decoration: none;">F_j\}= s_{ij}\circ F</math>. </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">(iii) The matrix function <math>s_{ij}</math> </del>is of <del style="font-weight: bold; text-decoration: none;">constant corank <math>m=2n</del>-<del style="font-weight: bold; text-decoration: none;">k</math> on <math>N</math>.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">If <math>k=n</math></del>, <del style="font-weight: bold; text-decoration: none;">this </del>is <del style="font-weight: bold; text-decoration: none;">the case </del>of <del style="font-weight: bold; text-decoration: none;">a [[integrable system|completely integrable Hamiltonian system]]</del>. <del style="font-weight: bold; text-decoration: none;">The Mishchenko</del>-<del style="font-weight: bold; text-decoration: none;">Fomenko theorem for superintegrable Hamiltonian systems generalizes </del>the <del style="font-weight: bold; text-decoration: none;">Liouville-Arnold theorem on [[action-angle coordinates]] </del>of <del style="font-weight: bold; text-decoration: none;">completely integrable Hamiltonian system </del>as <del style="font-weight: bold; text-decoration: none;">follows.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Let invariant submanifolds </del>of <del style="font-weight: bold; text-decoration: none;">a superintegrable Hamiltonian system be connected compact </del>and <del style="font-weight: bold; text-decoration: none;">mutually diffeomorphic</del>. <del style="font-weight: bold; text-decoration: none;">Then the fibered manifold <math>F</math> is </del>a [<del style="font-weight: bold; text-decoration: none;">[fiber bundle]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">in tori <math>T^m<</del>/<del style="font-weight: bold; text-decoration: none;">math></del>. <del style="font-weight: bold; text-decoration: none;">Given its fiber <math>M<</del>/<del style="font-weight: bold; text-decoration: none;">math>, there exists an open neighbourhood <math>U<</del>/<del style="font-weight: bold; text-decoration: none;">math> of <math>M<</del>/<del style="font-weight: bold; text-decoration: none;">math> which is a trivial fiber bundle provided with the bundle (generalized action</del>-<del style="font-weight: bold; text-decoration: none;">angle) coordinates <math>(I_A,p_i,q^i, \phi^A)<</del>/<del style="font-weight: bold; text-decoration: none;">math></del>,</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">A=1,\ldots, m</del><<del style="font-weight: bold; text-decoration: none;">/math</del>>, <del style="font-weight: bold; text-decoration: none;"><math>i=1</del>,<del style="font-weight: bold; text-decoration: none;">\ldots,n-m</math> such that <math>(\phi^A)</math> are coordinates </del>on <del style="font-weight: bold; text-decoration: none;"><math>T^m</math></del>. <del style="font-weight: bold; text-decoration: none;">These coordinates are the [[Darboux</del>'<del style="font-weight: bold; text-decoration: none;">s theorem|Darboux coordinates]] on a symplectic manifold <math>U</math></del>. <del style="font-weight: bold; text-decoration: none;">A Hamiltonian of a superintegrable system depends only on </del>the <del style="font-weight: bold; text-decoration: none;">action variables <math>I_A</math> </del>which are the <del style="font-weight: bold; text-decoration: none;">Casimir functions </del>of the <del style="font-weight: bold; text-decoration: none;">coinduced [[Poisson manifold|Poisson structure]] on <math>F(U)</math></del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The Liouville-Arnold theorem for [[Integrable system|completely integrable systems]] and the Mishchenko-Fomenko theorem for the superintegrable ones are generalized </del>to the <del style="font-weight: bold; text-decoration: none;">case </del>of <del style="font-weight: bold; text-decoration: none;">non-compact invariant submanifolds</del>. <del style="font-weight: bold; text-decoration: none;">They are diffeomorphic to a toroidal cylinder </del><<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">T^{m-r}\times\mathbb R^r</del><<del style="font-weight: bold; text-decoration: none;">/math</del>>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== See also ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[[Integrable system]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*[[Action-angle coordinates]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== References ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* Mishchenko, A</del>., <del style="font-weight: bold; text-decoration: none;">Fomenko,A., Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl. </del>'<del style="font-weight: bold; text-decoration: none;">''12''' (1978) 113</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* Bolsinov</del>, <del style="font-weight: bold; text-decoration: none;">A., Jovanovic, B., Noncommutative integrability, moment map </del>and <del style="font-weight: bold; text-decoration: none;">geodesic flows</del>, <del style="font-weight: bold; text-decoration: none;">Ann. Global Anal. Geom</del>. <del style="font-weight: bold; text-decoration: none;">'''23''' (2003) 305; {{arxiv|math</del>-<del style="font-weight: bold; text-decoration: none;">ph/0109031}}.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* Fasso, F., Superintegrable Hamiltonian systems: geometry </del>and <del style="font-weight: bold; text-decoration: none;">applications</del>, <del style="font-weight: bold; text-decoration: none;">Acta Appl. Math. '''87'''(2005) 93.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* Fiorani</del>, <del style="font-weight: bold; text-decoration: none;">E</del>.<del style="font-weight: bold; text-decoration: none;">, [[Gennadi Sardanashvily|Sardanashvily, G.]], Global action</del>-<del style="font-weight: bold; text-decoration: none;">angle coordinates </del>for <del style="font-weight: bold; text-decoration: none;">completely integrable systems with non</del>-<del style="font-weight: bold; text-decoration: none;">compact invariant manifolds, J. Math. Phys</del>. '<del style="font-weight: bold; text-decoration: none;">''48''' (2007) 032901; {{arxiv|math/0610790}}</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* Giachetta</del>, <del style="font-weight: bold; text-decoration: none;">G</del>.<del style="font-weight: bold; text-decoration: none;">, Mangiarotti, L., [[Gennadi Sardanashvily|Sardanashvily, G.]], </del>'<del style="font-weight: bold; text-decoration: none;">'Geometric Methods in Classical and Quantum Mechanics'' (World Scientific, Singapore, 2010) ISBN 978-981-4313-72-8; [http://xxx.lanl</del>.<del style="font-weight: bold; text-decoration: none;">gov/abs/1303.5363 arXiv: 1303.5363].</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Hamiltonian mechanics]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Dynamical systems]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
</table>
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In case you loved this information and you would want to receive more details relating to </del>[http://<del style="font-weight: bold; text-decoration: none;">snipitfor</del>.<del style="font-weight: bold; text-decoration: none;">me</del>/<del style="font-weight: bold; text-decoration: none;">backup_plugin_7514713 wordpress backup plugin</del>] <del style="font-weight: bold; text-decoration: none;">generously visit the web site</del>. <del style="font-weight: bold; text-decoration: none;">Article Source</del>: <del style="font-weight: bold; text-decoration: none;"> Hostgator discount coupons for your Wordpress site here.</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In mathematics</ins>, a <ins style="font-weight: bold; text-decoration: none;">'''superintegrable Hamiltonian system''' is a [[Hamiltonian system]] </ins>on a <ins style="font-weight: bold; text-decoration: none;"> 2''n''-dimensional [[symplectic manifold]] for which the following conditions hold:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">(i) There exist ''n''&nbsp;≤&nbsp;''k'' independent integrals ''F''<sub>&nbsp;''i''</sub> </ins>of <ins style="font-weight: bold; text-decoration: none;">motion</ins>. <ins style="font-weight: bold; text-decoration: none;">Their level surfaces (invariant submanifolds) form a fibered manifold <math>F:Z\</ins>to <ins style="font-weight: bold; text-decoration: none;">N=F(Z)</math> over a connected open subset <math>N\subset\mathbb R^k</math></ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">(ii) There exist smooth real functions <math>s_{ij}</math> on <math>N</math> such that </ins>the <ins style="font-weight: bold; text-decoration: none;">[[Poisson manifold|Poisson bracket]] of integrals of motion reads</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><math>\{F_i</ins>,<ins style="font-weight: bold; text-decoration: none;">F_j\}= s_{ij}\circ F</math>. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">(iii) The matrix function <math>s_{ij}</math> is </ins>of <ins style="font-weight: bold; text-decoration: none;">constant corank <math>m=2n</ins>-<ins style="font-weight: bold; text-decoration: none;">k</math> on <math>N</math></ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If <math>k=n</math>, this is </ins>the <ins style="font-weight: bold; text-decoration: none;">case </ins>of <ins style="font-weight: bold; text-decoration: none;">a [[integrable system|completely integrable Hamiltonian system]]. The Mishchenko-Fomenko theorem for superintegrable Hamiltonian systems generalizes </ins>the <ins style="font-weight: bold; text-decoration: none;">Liouville-Arnold theorem on [[action</ins>-<ins style="font-weight: bold; text-decoration: none;">angle coordinates]] of completely integrable Hamiltonian system as follows.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Let invariant submanifolds of </ins>a <ins style="font-weight: bold; text-decoration: none;">superintegrable Hamiltonian system be connected compact </ins>and <ins style="font-weight: bold; text-decoration: none;">mutually diffeomorphic. Then </ins>the <ins style="font-weight: bold; text-decoration: none;">fibered manifold <math>F</math> is a [[fiber bundle]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">in tori <math>T^m</math></ins>. <ins style="font-weight: bold; text-decoration: none;">Given its fiber </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">M</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>><ins style="font-weight: bold; text-decoration: none;">, there exists an open neighbourhood <math>U</math> </ins>of <ins style="font-weight: bold; text-decoration: none;"><math>M</math> which </ins>is <ins style="font-weight: bold; text-decoration: none;">a trivial fiber bundle provided with </ins>the <ins style="font-weight: bold; text-decoration: none;">bundle (generalized action-angle) coordinates <math>(I_A</ins>,<ins style="font-weight: bold; text-decoration: none;">p_i,q^i, \phi^A)</math>,</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><math></ins>A<ins style="font-weight: bold; text-decoration: none;">=1,\ldots, m</math>, <math>i=1,\ldots,n-m</math> such </ins>that <ins style="font-weight: bold; text-decoration: none;"><math>(\phi^A)</math> are coordinates on <math>T^m</math></ins>. <ins style="font-weight: bold; text-decoration: none;"> These coordinates are the [[Darboux</ins>'s <ins style="font-weight: bold; text-decoration: none;">theorem|Darboux coordinates]] on </ins>a <ins style="font-weight: bold; text-decoration: none;">symplectic manifold <math>U</math>. A Hamiltonian </ins>of a <ins style="font-weight: bold; text-decoration: none;">superintegrable system depends only on the action variables <math>I_A</math> </ins>which <ins style="font-weight: bold; text-decoration: none;">are </ins>the <ins style="font-weight: bold; text-decoration: none;">Casimir functions of </ins>the <ins style="font-weight: bold; text-decoration: none;">coinduced [[Poisson manifold|Poisson structure]] on </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">F(U)</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>><ins style="font-weight: bold; text-decoration: none;">.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The Liouville-Arnold theorem for [[Integrable system|completely integrable systems]] and </ins>the <ins style="font-weight: bold; text-decoration: none;">Mishchenko-Fomenko theorem for </ins>the <ins style="font-weight: bold; text-decoration: none;">superintegrable ones are generalized </ins>to the <ins style="font-weight: bold; text-decoration: none;">case </ins>of <ins style="font-weight: bold; text-decoration: none;">non-compact invariant submanifolds</ins>. <ins style="font-weight: bold; text-decoration: none;">They </ins>are <ins style="font-weight: bold; text-decoration: none;">diffeomorphic to </ins>a <ins style="font-weight: bold; text-decoration: none;">toroidal cylinder </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">T^{m-r}\times\mathbb R^r</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>><ins style="font-weight: bold; text-decoration: none;">.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== See also ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[[Integrable system]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*[[Action</ins>-<ins style="font-weight: bold; text-decoration: none;">angle coordinates]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== References ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Mishchenko, A., Fomenko,A., Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl</ins>. <ins style="font-weight: bold; text-decoration: none;">'''12''' </ins>(<ins style="font-weight: bold; text-decoration: none;">1978</ins>) <ins style="font-weight: bold; text-decoration: none;">113.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Bolsinov, A</ins>., <ins style="font-weight: bold; text-decoration: none;">Jovanovic, B</ins>., <ins style="font-weight: bold; text-decoration: none;">Noncommutative integrability, moment map </ins>and <ins style="font-weight: bold; text-decoration: none;">geodesic flows</ins>, <ins style="font-weight: bold; text-decoration: none;">Ann. Global Anal. Geom</ins>. <ins style="font-weight: bold; text-decoration: none;">'''23''' (2003) 305; {{arxiv|math</ins>-<ins style="font-weight: bold; text-decoration: none;">ph/0109031}}.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Fasso, F</ins>.<ins style="font-weight: bold; text-decoration: none;">, Superintegrable Hamiltonian systems: geometry </ins>and <ins style="font-weight: bold; text-decoration: none;">applications</ins>, <ins style="font-weight: bold; text-decoration: none;">Acta Appl. Math</ins>. ''<ins style="font-weight: bold; text-decoration: none;">'87'''(2005) 93.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Fiorani, E., [[Gennadi Sardanashvily|Sardanashvily, G</ins>.<ins style="font-weight: bold; text-decoration: none;">]]</ins>, <ins style="font-weight: bold; text-decoration: none;">Global action-angle coordinates </ins>for <ins style="font-weight: bold; text-decoration: none;">completely integrable systems </ins>with <ins style="font-weight: bold; text-decoration: none;">non</ins>-<ins style="font-weight: bold; text-decoration: none;">compact invariant manifolds, J. Math. Phys. '''48''' (2007) 032901; {{arxiv|math/0610790}}.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Giachetta, G., Mangiarotti, L., [[Gennadi Sardanashvily|Sardanashvily, G.]], ''Geometric Methods in Classical </ins>and <ins style="font-weight: bold; text-decoration: none;">Quantum Mechanics'' (World Scientific, Singapore, 2010) ISBN 978-981-4313</ins>-<ins style="font-weight: bold; text-decoration: none;">72</ins>-<ins style="font-weight: bold; text-decoration: none;">8; </ins>[http://<ins style="font-weight: bold; text-decoration: none;">xxx.lanl</ins>.<ins style="font-weight: bold; text-decoration: none;">gov/abs</ins>/<ins style="font-weight: bold; text-decoration: none;">1303.5363 arXiv: 1303.5363</ins>].</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category</ins>:<ins style="font-weight: bold; text-decoration: none;">Hamiltonian mechanics]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Dynamical systems]]</ins></div></td></tr>
</table>
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en>Jarble at 21:32, 3 June 2012
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