Symplectic geometry is a branch of
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
and
differential topology that studies
symplectic manifolds; that is,
differentiable manifolds equipped with a
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
,
nondegenerate 2-form. Symplectic geometry has its origins in the
Hamiltonian formulation of
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
where the
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
of certain classical systems takes on the structure of a symplectic manifold.
The term "symplectic", introduced by
Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
, is a
calque
In linguistics, a calque () or loan translation is a word or phrase borrowed from another language by literal word-for-word or root-for-root translation. When used as a verb, "to calque" means to borrow a word or phrase from another language ...
of "complex"; previously, the "symplectic group" had been called the "line complex group". "Complex" comes from the Latin ''com-plexus'', meaning "braided together" (co- + plexus), while symplectic comes from the corresponding Greek ''sym-plektikos'' (συμπλεκτικός); in both cases the stem comes from the Indo-European root
*pleḱ- The name reflects the deep connections between complex and symplectic structures.
By
Darboux's Theorem, symplectic manifolds are isomorphic to the standard
symplectic vector space locally, hence only have global (topological) invariants. "Symplectic topology," which studies global properties of symplectic manifolds, is often used interchangeably with "symplectic geometry."
Introduction
A symplectic geometry is defined on a smooth even-dimensional space that is a
differentiable manifold. On this space is defined a geometric object, the
symplectic 2-form, that allows for the measurement of sizes of two-dimensional objects in the
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
. The symplectic form in symplectic geometry plays a role analogous to that of the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
in
Riemannian geometry. Where the metric tensor measures lengths and angles, the symplectic form measures oriented areas.
Symplectic geometry arose from the study of
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
and an example of a symplectic structure is the motion of an object in one dimension. To specify the trajectory of the object, one requires both the
position ''q'' and the
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
''p'', which form a point (''p'',''q'') in the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
ℝ
2. In this case, the symplectic
form
Form is the shape, visual appearance, or configuration of an object. In a wider sense, the form is the way something happens.
Form also refers to:
*Form (document), a document (printed or electronic) with spaces in which to write or enter data
* ...
is
:
and is an
area form that measures the area ''A'' of a region ''S'' in the plane through
integration:
:
The area is important because as
conservative dynamical systems evolve in time, this area is invariant.
[
Higher dimensional symplectic geometries are defined analogously. A 2''n''-dimensional symplectic geometry is formed of pairs of directions
:
in a 2''n''-dimensional manifold along with a symplectic form
:
This symplectic form yields the size of a 2''n''-dimensional region ''V'' in the space as the sum of the areas of the projections of ''V'' onto each of the planes formed by the pairs of directions][
:
]
Comparison with Riemannian geometry
Symplectic geometry has a number of similarities with and differences from Riemannian geometry, which is the study of differentiable manifolds equipped with nondegenerate, symmetric 2-tensors (called metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
s). Unlike in the Riemannian case, symplectic manifolds have no local invariants such as curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the can ...
. This is a consequence of Darboux's theorem which states that a neighborhood of any point of a 2''n''-dimensional symplectic manifold is isomorphic to the standard symplectic structure on an open set of ℝ2''n''. Another difference with Riemannian geometry is that not every differentiable manifold need admit a symplectic form; there are certain topological restrictions. For example, every symplectic manifold is even-dimensional and orientable. Additionally, if ''M'' is a closed symplectic manifold, then the 2nd de Rham cohomology group ''H''2(''M'') is nontrivial; this implies, for example, that the only ''n''-sphere that admits a symplectic form is the 2-sphere. A parallel that one can draw between the two subjects is the analogy between geodesics
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
in Riemannian geometry and pseudoholomorphic curves in symplectic geometry: Geodesics are curves of shortest length (locally), while pseudoholomorphic curves are surfaces of minimal area. Both concepts play a fundamental role in their respective disciplines.
Examples and structures
Every Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
is also a symplectic manifold. Well into the 1970s, symplectic experts were unsure whether any compact non-Kähler symplectic manifolds existed, but since then many examples have been constructed (the first was due to William Thurston); in particular, Robert Gompf has shown that every finitely presented group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
occurs as the fundamental group of some symplectic 4-manifold, in marked contrast with the Kähler case.
Most symplectic manifolds, one can say, are not Kähler; and so do not have an integrable complex structure compatible with the symplectic form. Mikhail Gromov, however, made the important observation that symplectic manifolds do admit an abundance of compatible almost complex structures, so that they satisfy all the axioms for a Kähler manifold ''except'' the requirement that the transition maps be holomorphic.
Gromov used the existence of almost complex structures on symplectic manifolds to develop a theory of pseudoholomorphic curves, which has led to a number of advancements in symplectic topology, including a class of symplectic invariants now known as Gromov–Witten invariants. Later, using the pseudoholomorphic curve technique Andreas Floer invented another important tool in symplectic geometry known as the Floer homology.[Floer, Andreas. "Morse theory for Lagrangian intersections." Journal of differential geometry 28.3 (1988): 513–547.]
See also
* Contact geometry
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distributio ...
* Geometric mechanics Geometric mechanics is a branch of mathematics applying particular geometric methods to many areas of mechanics, from mechanics of particles and rigid bodies to fluid mechanics to control theory.
Geometric mechanics applies principally to systems ...
* Moment map In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the act ...
* Poisson geometry
* Symplectic integration
* Symplectic vector space
Notes
References
*
*
*
* ''(An undergraduate level introduction.)''
*
*
* Reprinted by Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large.
The press was founded by Whitney Darrow, with the financial ...
(1997). . .
External links
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