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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
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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
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shows that all symplectic manifolds of the same dimension are locally isomorphic. In contrast, isometries in
Riemannian geometry must preserve the
1001:. Moreover, Riemannian manifolds with large symmetry groups are very special, and a generic Riemannian manifold has no nontrivial symmetries.
930:
1017:. When the Lie group is the one defined by a Hamiltonian, it is called a "quantization by energy". The corresponding operator from the
813:
of a connected symplectic manifold is zero, symplectic and
Hamiltonian vector fields coincide, so the notions of Hamiltonian isotopy and
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761:
143:
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Representations of finite-dimensional subgroups of the group of symplectomorphisms (after ħ-deformations, in general) on
752:. The integration of the flow of a symplectic vector field is a symplectomorphism. Since symplectomorphisms preserve the
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883:
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follows. Symplectomorphisms that arise from
Hamiltonian vector fields are known as Hamiltonian symplectomorphisms.
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1213:(see,). The most important development in symplectic geometry triggered by this famous conjecture is the birth of
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The infinitesimal version of symplectomorphisms gives the symplectic vector fields. A vector field
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It can be shown that the equations for a geodesic may be formulated as a
Hamiltonian flow, see
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induced by any diffeomorphism of manifolds, and the coadjoint action of an element of a
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The symplectomorphisms from a manifold back onto itself form an infinite-dimensional
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is "nondegenerate", the number of fixed points is bounded from below by the sum of
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to the Lie algebra of continuous linear operators is also sometimes called the
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must have. Certain weaker version of this conjecture has been proved: when
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1290:. Graduate Texts in Mathematics. Vol. 60. New York: Springer-Verlag.
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are a (pseudo-)group, called the symplectomorphism group (see below).
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803:
802:. In physics this is interpreted as the law of conservation of
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and the set of all such vector fields form a subalgebra of the
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of the symplectomorphism group for certain simple symplectic
1510:(1985), "Pseudoholomorphic curves in symplectic manifolds",
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of a
Riemannian manifold is always a (finite-dimensional)
1025:; this is a more common way of looking at it in physics.
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The geometry of the group of symplectic diffeomorphism
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194:{\displaystyle f:(M,\omega )\rightarrow (N,\omega ')}
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109:, a symplectomorphism represents a transformation of
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1145:has at least as many fixed points as the number of
1125:(see ). More precisely, the conjecture states that
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1387:"Symplectic fixed points and holomorphic spheres"
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1321:"Arnold conjecture and Gromov-Witten invariants"
539:. These vector fields build a Lie subalgebra of
911:{\displaystyle \operatorname {Ham} (M,\omega )}
848:The group of Hamiltonian symplectomorphisms of
1467:
1445:
828:The group of (Hamiltonian) symplectomorphisms
1319:Fukaya, Kenji; Ono, Kaoru (September 1999).
430:{\displaystyle {\mathcal {L}}_{X}\omega =0.}
1546:
1287:Mathematical methods of classical mechanics
965:, symplectic manifolds are not very rigid:
704:Examples of symplectomorphisms include the
929:. They have natural geometry given by the
921:Groups of Hamiltonian diffeomorphisms are
387:{\displaystyle X\in \Gamma ^{\infty }(TM)}
1369:
1336:
1318:
69:Learn how and when to remove this message
1274:McDuff & Salamon 1998, Theorem 10.25
492:{\displaystyle \phi _{t}:M\rightarrow M}
32:This article includes a list of general
1351:
1283:
1568:
1506:
1391:Communications in Mathematical Physics
1354:"Floer homology and Arnold conjecture"
306:. The symplectic diffeomorphisms from
245:{\displaystyle f^{*}\omega '=\omega ,}
1384:
1224:
610:{\displaystyle \Gamma ^{\infty }(TM)}
571:{\displaystyle \Gamma ^{\infty }(TM)}
1551:, Basel; Boston: Birkhauser Verlag,
1065:for a Hamiltonian symplectomorphism
1042:
128:
18:
1451:Introduction to Symplectic Topology
1229:"Symplectomorphism" is a word in a
977:on a symplectic manifold defines a
957:Comparison with Riemannian geometry
16:Isomorphism of symplectic manifolds
13:
1453:, Oxford Mathematical Monographs,
668:{\displaystyle {\mathcal {L}}_{X}}
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38:it lacks sufficient corresponding
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1233:puzzle in episode 1 of the anime
519:is a symplectomorphism for every
1358:Journal of Differential Geometry
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1004:
817:of symplectomorphisms coincide.
740:gives rise, by definition, to a
121:of phase space, and is called a
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1090:{\displaystyle \varphi :M\to M}
1417:
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1352:Liu, Gang; Tian, Gang (1998).
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822:Geodesics as Hamiltonian flows
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1479:, London: Benjamin-Cummings,
1338:10.1016/S0040-9383(98)00042-1
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1547:Polterovich, Leonid (2001),
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1053:A celebrated conjecture of
989:, which exponentiates to a
873:{\displaystyle (M,\omega )}
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1449:& Salamon, D. (1998),
1284:Arnolʹd, Vladimir (1978).
1149:that a smooth function on
1046:
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460:is symplectic if the flow
1429:. Crunchyroll Collection.
1296:10.1007/978-1-4757-1693-1
941:, such as the product of
736:Any smooth function on a
706:canonical transformations
1512:Inventiones Mathematicae
1499:Symplectomorphism groups
1477:Foundations of Mechanics
1182:{\displaystyle \varphi }
1138:{\displaystyle \varphi }
1038:non-commutative geometry
979:Hamiltonian vector field
971:Riemann curvature tensor
951:pseudoholomorphic curves
945:, can be computed using
845:, modulo the constants.
750:symplectic vector fields
742:Hamiltonian vector field
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394:is called symplectic if
123:canonical transformation
1385:Floer, Andreas (1989).
1030:phase space formulation
679:along the vector field
53:more precise citations.
1371:10.4310/jdg/1214460936
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836:. The corresponding
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103:symplectic manifolds
1576:Symplectic topology
1524:1985InMat..82..307G
1473:Marsden, Jerrold E.
1119:symplectic manifold
991:one-parameter group
880:usually denoted as
762:Liouville's theorem
738:symplectic manifold
714:theoretical physics
710:classical mechanics
107:classical mechanics
1532:10.1007/BF01388806
1403:10.1007/BF01260388
1255:Mathematics portal
1225:In popular culture
1203:
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925:, by a theorem of
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815:symplectic isotopy
695:{\displaystyle X.}
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117:and preserves the
1426:Anya Gets Adopted
1305:978-1-4757-1693-1
1206:{\displaystyle M}
1162:{\displaystyle M}
1110:{\displaystyle M}
1049:Arnold conjecture
1043:Arnold conjecture
967:Darboux's theorem
754:symplectic 2-form
718:cotangent bundles
637:{\displaystyle M}
532:{\displaystyle t}
512:{\displaystyle X}
453:{\displaystyle X}
339:{\displaystyle M}
319:{\displaystyle M}
299:{\displaystyle f}
203:symplectomorphism
129:Formal definition
115:volume-preserving
87:symplectomorphism
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1518:(2): 307–347,
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1469:Abraham, Ralph
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1397:(4): 575–611.
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1215:Floer homology
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1047:Main article:
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1011:Hilbert spaces
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756:and hence the
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617:is the set of
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1219:Andreas Floer
1216:
1200:
1192:
1191:Betti numbers
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1156:
1148:
1132:
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1117:is a compact
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1015:quantizations
1012:
1005:Quantizations
1002:
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949:'s theory of
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935:homotopy type
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809:If the first
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622:vector fields
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1235:Spy × Family
1228:
1123:Morse theory
1063:fixed points
1058:
1057:relates the
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1023:quantization
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811:Betti number
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202:
201:is called a
137:between two
132:
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37:
1364:(1): 1–74.
1019:Lie algebra
1013:are called
838:Lie algebra
834:pseudogroup
746:Lie algebra
111:phase space
95:isomorphism
83:mathematics
51:introducing
1570:Categories
1508:Gromov, M.
1262:References
1097:, in case
1061:number of
1028:See also:
995:isometries
931:Hofer norm
34:references
1411:123345003
1231:crossword
1177:φ
1133:φ
1082:→
1073:φ
999:Lie group
903:ω
891:
865:ω
722:Lie group
591:∞
587:Γ
552:∞
548:Γ
484:→
469:ϕ
419:ω
368:∞
364:Γ
360:∈
268:∗
237:ω
227:ω
221:∗
182:ω
169:→
163:ω
1475:(1978),
1325:Topology
1241:See also
578:. Here,
284:pullback
230:′
185:′
113:that is
99:category
59:May 2023
1540:4983969
1520:Bibcode
1438:General
1059:minimum
961:Unlike
943:spheres
927:Banyaga
675:is the
282:is the
97:in the
47:improve
1555:
1538:
1483:
1471:&
1457:
1409:
1302:
1036:, and
947:Gromov
933:. The
923:simple
804:energy
794:) = 0,
771:Since
644:, and
619:smooth
440:Also,
255:where
93:is an
36:, but
1536:S2CID
1407:S2CID
1121:, to
732:Flows
724:on a
105:. In
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