Symplectic
Manifolds / Geometry| 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 classical and quantum mechanics, and it is fruitful in other areas
of mathematics (knot theory, number theory, etc); V Arnold has suggested the
symplectization of
most mathematical
concepts.
Presymplectic Structure on a Manifold
* Idea: A manifold 
with
a closed, but possibly degenerate, 2-form 
;
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).
@ References: Künzle JMP(72)
[spinning particle example]; 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].
Almost Symplectic Structure
$ Def: An n-manifold M has
an almost symplectic (or Hamiltonian) structure if there exists an everywhere
non-degenerate 2-form on
it; Equivalently,
if its frame bundle is reducible to an Sp(n, R)-bundle.
* Necessary conditions:
Even dimension n, orientable M;
if M is compact, H2(M, R) 
0.
* Special case: If the 2-form 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 (, )
of a differentiable manifold and
a closed non-degenerate 2-form on .
* Remark: The fact that
the form is closed guarantees the local existence of bases such that assumes
the form "dp 
dq" and
of a symplectic potential 
, =
d, 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 
is
a symplectic form on a Banach manifold P, about any x 
P
there is a local chart such that =
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 = 
i
dxi dyi.
* Non-squeezing theorem:
The symplectic camel can pass through the symplectic needle only in 2D (1985,
Gromov).
* Example: Given any
manifold M, T*M is an even-dimensional
manifold; A canonical choice is
= pi
dqi , =
d = dpi dqi
.
* Necessary conditions:
Even dimension n, orientable M;
If M is
compact, H2(M, R) 0.
* Poisson brackets: Defined
by {f, g}:= ab 
a f b g = Xfa a g = 
X_f g.
@ Darboux's theorem: Moser TAMS(65); Weinstein BAMS(69), AiM(71); Lang
72; Marsden PAMS(72).
Related Concepts and Structure > s.a. group
types [metaplectic]; Momentum Map; Polarization.
* Volume element: On
a symplectic manifold, it is given by 
Omega =
(–1)n/2 (n/n!).
* Hamiltonian vector fields:
Every function f on defines
a vector field Xf
by Xfa:= ab b f.
* Symmetry reduction: Reduction
procedures can based on the standard momentum map (symplectic or Marsden–Weinstein
reduction) or on generalized distributions (the optimal momentum map and optimal
reduction); In general yields a stratified structure.
@ Symmetry reduction: Ortega & Ratiu LMP(04)
[rev], RPMP(06)
[stratified structure];
Butterfield phy/05-in.
@ 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; Lucey & Newman JMP(88);
Slawianowski
91.
@ Infinite-dimensional: in Lang 72; Chernoff & Marsden 74; in Marsden
74; in Choquet-Bruhat et al 82, VII.A.2; Schmid 87.
@ Generating functions: Ozorio de Almeida PRS(90).
@ 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 qa/95/IJMPA.
@ Special emphasis: Vaisman 87 [characteristic classes]; McDuff & Salamon
95, Arnold JMP(00) [topology].
@ Special topics: Schleich et al FP(88)
[phase space volume]; Bates in(90) [exotic]; Xia
CMP(96) [symplectic
diffeomorphisms];
Joyce
AGAG(04)m.DG/02,
AGAG(04)m.DG/02 [Lagrangian
submanifolds]; Weinstein m.SG/02-in
[momentum maps]; Golovnev & Ushakov a0710 [non-exact
symplectic forms].
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2009