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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 '''symplectomorphism''', if | |
| :<math>f^*\omega'=\omega,</math>
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| 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).
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| The infinitesimal version of symplectomorphisms give the symplectic vector fields. A vector field <math>X \in \Gamma^{\infty}(TM)</math> is called symplectic, if
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| :<math>\mathcal{L}_X\omega=0.</math>
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| 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>.
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| These vector fields build a Lie-subalgebra of <math>\Gamma^{\infty}(TM)</math>.
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| 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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| ==Flows== <!-- [[Hamiltonian isotopy]] redirects here -->
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| 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.
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| 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]].
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| 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.
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| We [[Geodesics as Hamiltonian flows|can show]] that the equations for a geodesic may be formulated as a Hamiltonian flow.
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| == The group of (Hamiltonian) symplectomorphisms ==
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| The symplectomorphisms from a manifold back onto itself form an infinite-dimensional [[pseudogroup]]. The corresponding [[Lie algebra]] consists of symplectic vector fields.
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| The Hamiltonian symplectomorphisms form a subgroup, whose Lie algebra is given by the Hamiltonian vector fields.
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| The latter is isomorphic to the Lie algebra of smooth
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| functions on the manifold with respect to the [[Poisson bracket]], modulo the constants.
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| 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]].
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| ==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.
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| 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==
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| 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".
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| 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]].
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| ==Arnold conjecture==
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| 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]].
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| 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).
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| 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.
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| It is proven in many cases by the construction of symplectic [[Floer homology]].
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| ==See also==
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| {{Portal|Mathematics}}
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| ==References==
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| *{{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 }}.
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| *{{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''.
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| ;Symplectomorphism groups:
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| *{{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 }}.
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| *{{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 }}.
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| [[Category:Symplectic topology]]
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| [[Category:Hamiltonian mechanics]]
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Epoxy, a form of resin, is recognized as an extremely durable glue type, which renders top-notch level of bonding properties. It is usually sold in 2-different packages known as Part A (resin) and Part B (hardener). It is important to blend both the parts in exact ratios before utilizing them to avail a specific strength. This adhesive can securely bind numerous materials like plastics, wood and metals. These adhesives provide solid bonding properties and can serve as a tough layer of protection as well.
Industrial Uses
The rudiments of epoxy can be derived from petroleum products, one being Bisphenol-A, which provides this chemical its boding efficiency. Another component in this resin is Epichlorohydrin that aids the chemical and provides a strong layer, which is resistant to moisture, humidity and an array of varying temperatures.
In case you beloved this informative article along with you would want to obtain more details with regards to resinas epoxi [go to these guys] i implore you to check out our page. The qualities like strong bonding and hard coating make these adhesives a perfect choice for thermosetting epoxide polymer to be utilized with metals, which undergo great levels of stress. Some instances of such applications are, metal plating on ships and hulls of airplanes. The versatility of these adhesives facilitates it to be utilized even in the construction of wooden furniture. In this application, this hardener is able to secure sections of frames and acts as a coat of protection when it is utilized to varnish the finished product. Varnishing utilizing epoxies render the product an alluring finish and are highly potent to sustain that eye-catching look for many years.
Household Uses
Moreover, you can utilize these resins to repair defected household products. For instance, if the leg of a chair has begun to wobble, you can apply this adhesive to secure the joint.
It acts as a hardener, which creates robust joints and renders it with durability to endure any amounts of stress that it may undergo.
Some other applications of this hardener in household includes repair of broken clocks, antique picture frames, making candles and components that might have fallen off from a device. These chemicals serve as an effective substitute to any other forms of glue, which you might be utilizing to repair household stuff.
When using this hardener, keep in mind to mix the vital elements (as instructed on the pack) before applying it to the components. Different brands of epoxy come with different time limit, like, how much time is required for the chemical to dry up and provide a sturdy bond.