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In [[mathematics]], a '''symplectomorphism''' is an [[isomorphism]] in the [[category (mathematics)|category]] of [[symplectic manifold]]s. In [[classical mechanics]], a symplectomorphism represents a transformation of [[phase space]] that is [[volume-preserving]] and preserves the [[symplectic structure]] of phase space, and is called a [[canonical transformation]].
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==Formal definition==
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A [[diffeomorphism]] between two [[symplectic manifold]]s <math>f: (M,\omega) \rightarrow (N,\omega')</math> is called a '''symplectomorphism''' if
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:<math>f^*\omega'=\omega,</math>
where <math>f^*</math> is the [[pullback (differential geometry)|pullback]] of <math>f</math>. The symplectic diffeomorphisms from <math>M</math> to <math>M</math> are a (pseudo-)group, called the symplectomorphism group (see below).


The infinitesimal version of symplectomorphisms gives the symplectic vector fields. A vector field <math>X \in \Gamma^{\infty}(TM)</math> is called symplectic if
'''MathML'''
:<math>\mathcal{L}_X\omega=0.</math>
:<math forcemathmode="mathml">E=mc^2</math>
Also, <math>X</math> is symplectic iff the flow <math>\phi_t: M\rightarrow M</math> of <math>X</math> is symplectic for every <math>t</math>.
These vector fields build a Lie subalgebra of <math>\Gamma^{\infty}(TM)</math>.


Examples of symplectomorphisms include the [[canonical transformation]]s of [[classical mechanics]] and [[theoretical physics]], the flow associated to any Hamiltonian function, the map on [[cotangent bundle]]s induced by any diffeomorphism of manifolds, and the coadjoint action of an element of a [[Lie Group]] on a [[coadjoint orbit]].
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:<math forcemathmode="png">E=mc^2</math>


==Flows== <!-- [[Hamiltonian isotopy]] redirects here -->
'''source'''
Any smooth function on a [[symplectic manifold]] gives rise, by definition, to a [[Hamiltonian vector field]] and the set of all such form a subalgebra of the [[Lie Algebra]] of [[symplectic vector field]]s. The integration of the flow of a symplectic vector field is a symplectomorphism. Since symplectomorphisms preserve the [[symplectic form|symplectic 2-form]] and hence the [[symplectic form#volume form|symplectic volume form]], [[Liouville's theorem (Hamiltonian)|Liouville's theorem]] in [[Hamiltonian mechanics]] follows. Symplectomorphisms that arise from Hamiltonian vector fields are known as Hamiltonian symplectomorphisms.
:<math forcemathmode="source">E=mc^2</math> -->


Since {{nowrap|1={''H'',''H''} = ''X''<sub>''H''</sub>(''H'') = 0,}} the flow of a Hamiltonian vector field also preserves ''H''. In physics this is interpreted as the law of conservation of [[energy]].
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


If the first [[Betti number]] of a connected symplectic manifold is zero, symplectic and Hamiltonian vector fields coincide, so the notions of [[Hamiltonian isotopy]] and [[symplectic isotopy]] of symplectomorphisms coincide.
==Demos==


We [[Geodesics as Hamiltonian flows|can show]] that the equations for a geodesic may be formulated as a Hamiltonian flow.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


== The group of (Hamiltonian) symplectomorphisms ==
The symplectomorphisms from a manifold back onto itself form an infinite-dimensional [[pseudogroup]]. The corresponding [[Lie algebra]] consists of symplectic vector fields.
The Hamiltonian symplectomorphisms form a subgroup, whose Lie algebra is given by the Hamiltonian vector fields.
The latter is isomorphic to the Lie algebra of smooth
functions on the manifold with respect to the [[Poisson bracket]], modulo the constants.


The group of Hamiltonian symplectomorphisms of <math>(M,\omega)</math> usually denoted as <math>\mathop{\rm Ham}(M,\omega)</math>.
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Groups of Hamiltonian diffeomorphisms are [[simple Lie group|simple]], by a theorem of [[Augustin Banyaga|Banyaga]]. They have natural geometry given by the [[Hofer norm]]. The [[homotopy type]] of the symplectomorphism group for certain simple symplectic [[four-manifold]]s, such as the product of [[sphere]]s, can be computed using [[Mikhail Gromov (mathematician)|Gromov]]'s theory of [[pseudoholomorphic curves]].
==Test pages ==


==Comparison with Riemannian geometry==
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Unlike [[Riemannian manifold]]s, symplectic manifolds are not very rigid: [[Darboux's theorem]] shows that all symplectic manifolds of the same dimension are locally isomorphic. In contrast, isometries in Riemannian geometry must preserve the [[Riemann curvature tensor]], which is thus a local invariant of the Riemannian manifold.
*[[Displaystyle]]
Moreover, every function ''H'' on a symplectic manifold defines a [[Hamiltonian vector field]] ''X''<sub>''H''</sub>, which exponentiates to a [[one-parameter group]] of Hamiltonian diffeomorphisms. It follows that the group of symplectomorphisms is always very large, and in particular, infinite-dimensional. On the other hand, the group of [[isometry|isometries]] of a Riemannian manifold is always a (finite-dimensional) [[Lie group]]. Moreover, Riemannian manifolds with large symmetry groups are very special, and a generic Riemannian manifold has no nontrivial symmetries.
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==Quantizations==
*[[Inputtypes|Inputtypes (private Wikis only)]]
Representations of finite-dimensional subgroups of the group of symplectomorphisms (after <math>\hbar</math>-deformations, in general) on [[Hilbert space]]s are called ''quantizations''.
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When the Lie group is the one defined by a Hamiltonian, it is called a "quantization by energy".
==Bug reporting==
The corresponding operator from the [[Lie algebra]] to the Lie algebra of continuous linear operators is also sometimes called the ''quantization''; this is a more common way of looking at it in physics. See [[Weyl quantization]], [[geometric quantization]], [[non-commutative geometry]].
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
 
==Arnold conjecture==
A celebrated conjecture of [[Vladimir Arnold]] relates the ''minimum'' number of [[Fixed point (mathematics)|fixed points]] for a Hamiltonian symplectomorphism ƒ on ''M'', in case ''M'' is a [[closed manifold]], to [[Morse theory]].
More precisely, the conjecture states that ƒ has at least as many fixed points as the number of [[critical point (mathematics)|critical point]]s a smooth function on ''M'' must have (understood as for a ''generic'' case, [[Morse function]]s, for which this is a definite finite number which is at least 2).
 
It is known that this would follow from the [[Arnold–Givental conjecture]] named after Arnold and [[Alexander Givental]], which is a statement on [[Lagrangian submanifold]]s.
It is proven in many cases by the construction of symplectic [[Floer homology]].
 
==See also==
{{Portal|Mathematics}}
 
==References==
*{{Citation |first=Dusa |last=McDuff |lastauthoramp=yes |first2=D. |last2=Salamon |title=Introduction to Symplectic Topology |year=1998 |series=Oxford Mathematical Monographs |location= |publisher= |isbn=0-19-850451-9 }}.
*{{Citation |authorlink=Ralph Abraham |first=Ralph |last=Abraham |lastauthoramp=yes |authorlink2=Jerrold E. Marsden |first2=Jerrold E. |last2=Marsden |title=Foundations of Mechanics |year=1978 |publisher=Benjamin-Cummings |location=London |isbn=0-8053-0102-X }}. ''See section 3.2''.
 
;Symplectomorphism groups:
*{{Citation |last=Gromov |first=M. |title=Pseudoholomorphic curves in symplectic manifolds |journal=Inventiones Mathematicae |volume=82 |year=1985 |issue=2 |pages=307–347 |doi=10.1007/BF01388806 |bibcode = 1985InMat..82..307G }}.
*{{Citation |last=Polterovich |first=Leonid |title=The geometry of the group of symplectic diffeomorphism |location=Basel; Boston |publisher=Birkhauser Verlag |year=2001 |isbn=3-7643-6432-7 }}.
 
[[Category:Symplectic topology]]
[[Category:Hamiltonian mechanics]]

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

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