Symplectic Manifolds / Geometry  

In General > s.a. symplectic vector space.
* History: Symplectic geometry arose from physics problems, as the n-body problem was studied by Poincaré; Now it underlies our understanding of the geometry of classical and quantum mechanics in the Hamiltonian formulation, and it is fruitful in other areas of mathematics (knot theory, number theory, etc); V Arnold has suggested the symplectization of most mathematical concepts.
@ History: Marle LMP(09) [beginnings, Lagrange and Poisson in 1808−1810].
> Online resources: see Wikipedia page.

Presymplectic Structure on a Manifold
* Idea: A manifold Γ with a closed, but possibly degenerate, 2-form Ω; This can occur, e.g., if one comes from a TQ by a degenerate Legendre transformation, i.e., if ∂2L / ∂vivj is not invertible (we then have to add constraint terms to H).
@ General references: Gotay et al JMP(78) [and constraints]; Cariñena et al JMP(85) [canonical transformations]; Dubrovin et al IJMPA(93) [Poisson brackets from action]; Saavedra et al JMP(01)ht/00 [evolution and phase space structure].
@ Special types of systems: Künzle JMP(72) [spinning particle]; Sharapov a1602 [non-Lagrangian dynamics of free, massive higher-spin fields].

Almost Symplectic Structure
$ Def: An n-manifold M has an almost symplectic (or Hamiltonian) structure if there exists an everywhere non-degenerate 2-form Ω on it; Equivalently, if its frame bundle is reducible to an Sp(n, \(\mathbb R\))-bundle.
* Necessary conditions: Even dimension n, orientable M; If M is compact, H2(M, \(\mathbb R\)) ≠ 0.
* Special case: If the 2-form Ω is closed, we have a manifold with a symplectic (or Hamiltonian) structure.

Symplectic Structure on a Manifold > s.a. Gromov-Witten Invariants; poisson structure.
$ Def: A pair (Γ, Ω) of a differentiable manifold Γ and a closed non-degenerate 2-form Ω on Γ.
* Remark: The fact that the form is closed guarantees the local existence of bases such that Ω assumes the form "dp ∧ dq" and of a symplectic potential θ, Ω = dθ, and the Jacobi identity for the Poisson brackets; Non-degeneracy guarantees the existence of the inverse, to form Poisson brackets and Hamiltonian vector fields.
* Darboux's theorem: If ω is a symplectic form on a Banach manifold P, about any xP there is a local chart such that ω = constant; Corollary: If P is finite-dimensional, then it is even (2n)-dimensional and there are local canonical coordinates (xi, yi), i = 1, ..., n, such that ω = ∑i dxi ∧ dyi.
* Non-squeezing theorem: (Gromov, 1985) The symplectic camel can pass through the symplectic needle only in 2D; > s.a. Symplectic Capacity.
* Example: Given any manifold M, T*M is an even-dimensional manifold; A canonical choice is

θ = pi dqi ,   Ω = dθ = dpi ∧ dqi .

* Necessary conditions: Even dimension n, orientable M; If M is compact, H2(M, \(\mathbb R\)) ≠ 0.
* Poisson brackets: Defined by {f, g}:= Ωaba fb g = Xfaa g = \(\cal L\)Xf g.
@ Darboux's theorem: Moser TAMS(65); Weinstein BAMS(69), AiM(71); Lang 72; Marsden PAMS(72).

Related Concepts and Structure > s.a. clifford algebra; group types [metaplectic]; Momentum Map; Polarization; Reduction [symmetry reduction].
* Volume element: On a symplectic manifold, it is given by εΩ = (−1)n/2n/n!).
* Hamiltonian vector fields: Every function f on Γ defines a vector field Xf by Xfa:= Ωabb f.
@ Moduli space of symplectic structures: Fricke et al DG&A(05) [pseudo-Riemannian metric].
@ Connection: Ghaboussi JMP(93) [symplectic form as connection]; > s.a. deformation quantization [Hitchin's connection]; Star Product.

References > s.a. generalized symplectic structure; hamiltonian systems; symmetry; symplectic structures in physics.
@ I: Stewart ThSc(90)may.
@ III, short: in Hermann 70 [geometrical meaning]; Weinstein BAMS(81); in Ashtekar 88.
@ General: Godbillon 69; Souriau 70; Weinstein 77; Aldaya & Azcárraga RNC(80); Guillemin & Sternberg 84; Fomenko 88; Arnold 89; Lucey & Newman JMP(88); Sławianowski 91; Banyaga & Houenou 16; Moshayedi a2012-ln.
@ Infinite-dimensional: in Lang 72; Chernoff & Marsden 74; in Marsden 74; in Choquet-Bruhat et al 82, VII.A.2; Schmid 87.
@ Special cases: in Souriau 70 [non cotangent bundle]; Sharipov m.DG/01, m.DG/01, m.DG/01 [on Riemannian manifold].
@ Quantization: Emmrich & Weinstein ht/93 [Fedosov deformation]; Donin AiM(97)qa/95.
@ Symplectic topology: McDuff & Salamon 99; Arnold JMP(00); Cardin 15.
@ Generalizations: Bates in(90) [exotic]; in Valach PLB(20)-a2001 [graded geometry, NQ symplectic manifolds].
@ Related topics: Vaisman 87 [characteristic classes]; Schleich et al FP(88) [phase space volume]; Ozorio de Almeida PRS(90) [generating functions]; Xia CMP(96) [symplectic diffeomorphisms]; Joyce AGAG(04)m.DG/02, AGAG(04)m.DG/02 [Lagrangian submanifolds]; Golovnev & Ushakov JPA(08)-a0710 [non-exact symplectic forms]; Forger & Yepes JDG(13)-a1202 [Lagrangian distributions and connections].

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