- Symplectomorphism
In

mathematics , a**symplectomorphism**is anisomorphism in the category ofsymplectic manifold s.**Formal definition**Specifically, let ("M"

_{1}, ω_{1}) and ("M"_{2}, ω_{2}) be symplectic manifolds. A map:"f" : "M"

_{1}→ "M"_{2}is a symplectomorphism if it is a

diffeomorphism and the pullback of ω_{2}under "f" is equal to ω_{1}::$f^\{*\}omega\_2\; =\; omega\_1.,$Examples of symplectomorphisms include the

canonical transformation s ofclassical mechanics andtheoretical physics , the flow associated to any Hamiltonian function, the map oncotangent bundle s induced by any diffeomorphism of manifolds, and the coajoint action of an element of aLie Group on acoadjoint orbit .**Examples***Translations in $mathbf\{R\}^n$ are symplectomorphisms.

**Flows**Any smooth function on a

symplectic manifold gives rise, by definition, to aHamiltonian vector field and the set of all such form a subalgebra of theLie Algebra ofsymplectic vector field s. The integration of the flow of a symplectic vector field is a symplectomorphism. Since symplectomorphisms preserve the symplectic 2-form and hence the symplectic-volumeform, Liouville's theorem inHamiltonian mechanics follows. Symplectomorphisms that arise from Hamiltonian vector fields are known as Hamiltonian symplectomorphisms.Since

:{"H","H"} = "X"

_{"H"}("H") = 0the flow of a Hamiltonian vector field also preserves "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 ofHamiltonian isotopy andsymplectic isotopy of symplectomorphisms coincide.The equations for a geodesic may be formulated as a Hamiltonian flow.

**The group of (Hamiltonian) symplectomorphisms**The symplectomorphisms from a manifold back onto itself form an infinite-dimensional

pseudogroup . The correspondingLie 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 smoothfunctions on the manifold with respect to thePoisson bracket , modulo the constants.Groups of Hamiltonian diffeomorphisms are simple, by a theorem of Banyaga. They have natural geometry given by the

Hofer norm . Thehomotopy type of the symplectomorphism group for certain simple symplecticfour-manifold s, such as the product ofsphere s, can be computed using Gromov's theory ofpseudoholomorphic curves .**Comparison with Riemannian geometry**Unlike

Riemannian manifold s, symplectic manifolds are not very rigid:Darboux's theorem shows that all symplectic manifolds are locally isomorphic. In contrast, isometries in Riemannian geometry must preserve theRiemann curvature tensor , which is thus a local invariant of the Riemannian manifold. Moreover, every function "H" on a symplectic manifold defines aHamiltonian vector field "X"_{"H"}, which exponentiates to aone-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 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 $hbar$-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 theLie 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. SeeWeyl quantization ,geometric quantization ,non-commutative geometry .**Arnold Conjecture**A celebrated conjecture of

V. I. Arnold relates the "minimum" number offixed point s for a Hamiltonian symplectomorphism "f" on "M", in case "M" is aclosed manifold , toMorse theory . More precisely, the conjecture states that "f" has at least as many fixed points as the number ofcritical 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 V.I Arnold andAlexander Givental , which is a statement onLagrangian submanifold s. It is proven in many cases by the construction of symplecticFloer homology .**References*** Dusa McDuff and D. Salamon: "Introduction to Symplectic Topology" (1998) Oxford Mathematical Monographs, ISBN 0-19-850451-9.

*Ralph Abraham andJerrold E. Marsden , "Foundations of Mechanics", (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X "See section 3.2".Symplectomorphism groups:

* Gromov, M. Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), no. 2, 307--347.

* Polterovich, Leonid. The geometry of the group of symplectic diffeomorphism. Basel ; Boston : Birkhauser Verlag, 2001.

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