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 T*Q* by a degenerate Legendre
transformation, i.e., if ∂^{2}*L*/∂*v*_{i}∂*v*_{j}
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, H^{2}(*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 "d*p* ∧ d*q*" and
of a symplectic potential *θ*, Ω = d*θ*, 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 *x* ∈ *P*
there is a local chart such that *ω* = constant; __Corollary__:
If *P* is finite-dimensional, then it is even (2*n*)-dimensional
and there are local canonical coordinates (*x*^{i}, *y*_{i}), *i* =
1, ..., *n*,
such that *ω* = ∑_{i}
d*x*^{i} ∧ d*y*_{i}.

* __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* = *p*_{i}
d*q*^{i} , Ω =
d*θ* = d*p*_{i} ∧ d*q*^{i} .

* __Necessary conditions__:
Even dimension *n*, orientable *M*;
If *M* is compact, H^{2}(*M*, \(\mathbb R\)) ≠ 0.

* __Poisson brackets__: Defined
by {*f*, *g*}:= Ω^{ab }∇_{a} *f* ∇_{b} *g*
= *X*_{f}^{a} ∇_{a}* 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 *ε*_{Ω} =
(–1)^{n/2} (Ω^{n}/*n*!).

* __Hamiltonian vector fields__:
Every function *f* on Γ defines a vector field *X*_{f}
by *X*_{f}^{a}:= Ω^{ab} ∇_{b}* 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.

@ __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.

@ __Special topics__: Vaisman 87 [characteristic classes]; Schleich et al FP(88)
[phase space volume]; Bates in(90) [exotic]; 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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