Editing Symplectomorphism

{{Short description|Isomorphism of symplectic manifolds}}

{{More footnotes needed|date=May 2023}}

In [[mathematics]], a '''symplectomorphism''' or '''symplectic map''' 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]].<ref>{{cite web |last=Weisstein |first=Eric W. |title=Symplectic Diffeomorphism |work=MathWorld—A Wolfram Web Resource|url=https://mathworld.wolfram.com/SymplecticDiffeomorphism.html |access-date=November 26, 2024}}</ref>

==Formal definition==

A [[diffeomorphism]] between two [[symplectic manifold]]s <math>f: (M,\omega) \rightarrow (N,\omega')</math> is called a '''symplectomorphism''' if
:<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
:<math>\mathcal{L}_X\omega=0.</math>
Also, <math>X</math> is symplectic if the flow <math>\phi_t: M\rightarrow M</math> of <math>X</math> is a symplectomorphism for every <math>t</math>.
These vector fields build a Lie subalgebra of <math>\Gamma^{\infty}(TM)</math>.
Here, <math>\Gamma^{\infty}(TM)</math> is the set of [[smooth function|smooth]] [[vector field]]s on <math>M</math>, and <math>\mathcal{L}_X</math> is the [[Lie derivative]] along the vector field <math>X.</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]].

==Flows== <!-- [[Hamiltonian isotopy]] redirects here -->
Any smooth function on a [[symplectic manifold]] gives rise, by definition, to a [[Hamiltonian vector field]] and the set of all such vector fields 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.

Since {{math|1={''H'', ''H''} = ''X''<sub>''H''</sub>(''H'') = 0,}} the flow of a Hamiltonian vector field also preserves {{math|''H''}}. In physics this is interpreted as the law of conservation of [[energy]].

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.

It can be shown that the equations for a geodesic may be formulated as a Hamiltonian flow, see [[Geodesics as Hamiltonian flows]].

== 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>\operatorname{Ham}(M,\omega)</math>.

Groups of Hamiltonian diffeomorphisms are [[simple Lie group|simple]], by a theorem of [[Augustin Banyaga|Banyaga]].<ref>McDuff & Salamon 1998, Theorem 10.25</ref> 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 curve]]s.

==Comparison with Riemannian geometry==
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. 
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.

==Quantizations==
Representations of finite-dimensional subgroups of the group of symplectomorphisms (after ħ-deformations, in general) on [[Hilbert space]]s are called ''quantizations''. 
When the Lie group is the one defined by a Hamiltonian, it is called a "quantization by energy". 
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 also| phase space formulation| geometric quantization|non-commutative geometry}}

==Arnold conjecture==
{{main|Arnold conjecture}}
A celebrated conjecture of [[Vladimir Arnold]] relates the ''minimum'' number of [[Fixed point (mathematics)|fixed points]] for a Hamiltonian symplectomorphism <math>\varphi: M \to M</math>, in case <math>M</math> is a compact [[symplectic manifold]], to [[Morse theory]] (see <ref>{{cite book |last1=Arnolʹd |first1=Vladimir |title=Mathematical methods of classical mechanics |series=Graduate Texts in Mathematics |date=1978 |volume=60 |publisher=Springer-Verlag |location=New York |doi=10.1007/978-1-4757-1693-1 |isbn=978-1-4757-1693-1 |url=https://link.springer.com/book/10.1007/978-1-4757-1693-1}}</ref>). More precisely, the conjecture states that <math>\varphi</math> has at least as many fixed points as the number of [[critical point (mathematics)|critical point]]s that a smooth function on <math>M</math> must have. Certain weaker version of this conjecture has been proved: when <math>\varphi</math> is "nondegenerate", the number of fixed points is bounded from below by the sum of [[Betti number]]s of <math>M</math> (see,<ref>{{cite journal |last1=Fukaya |first1=Kenji |last2=Ono |first2=Kaoru |title=Arnold conjecture and Gromov-Witten invariants |journal=Topology |date=September 1999 |volume=38 |issue=5 |pages=933–1048 |doi=10.1016/S0040-9383(98)00042-1 |doi-access=free }}</ref><ref>{{cite journal |last1=Liu |first1=Gang |last2=Tian |first2=Gang |title=Floer homology and Arnold conjecture |journal=Journal of Differential Geometry |date=1998 |volume=49 |issue=1 |pages=1–74 |doi=10.4310/jdg/1214460936 |doi-access=free }}</ref>). The most important development in symplectic geometry triggered by this famous conjecture is the birth of [[Floer homology]] (see <ref>{{cite journal |last1=Floer |first1=Andreas |title=Symplectic fixed points and holomorphic spheres |journal=Communications in Mathematical Physics |date=1989 |volume=120 |issue=4 |pages=575–611 |doi=10.1007/BF01260388 |s2cid=123345003 |url=https://link.springer.com/article/10.1007/BF01260388}}</ref>), named after [[Andreas Floer]].

==See also==
{{Portal|Mathematics}}

==References==
{{Reflist}}

;General:
*{{Citation |first1=Dusa |last1=McDuff |author-link=Dusa McDuff |name-list-style=amp |first2=D. |last2=Salamon |title=Introduction to Symplectic Topology |year=1998 |series=Oxford Mathematical Monographs |isbn=0-19-850451-9 }}.
*{{Citation |author-link=Ralph Abraham (mathematician) |first1=Ralph |last1=Abraham |name-list-style=amp |author-link2=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. |author-link=Mikhail Leonidovich Gromov |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 |s2cid=4983969 }}.
*{{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]]