In
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, a subject of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a symplectic manifold is a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
,
, equipped with a
closed nondegenerate differential 2-form , called the symplectic form. The study of symplectic manifolds is called
symplectic geometry
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...
or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of
classical mechanics
Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...
and
analytical mechanics
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related formulations of classical mechanics. Analytical mechanics uses '' scalar'' properties of motion representing the sy ...
as the
cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This m ...
s of manifolds. For example, in the
Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the
phase space
The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
of the system.
Motivation
Symplectic manifolds arise from
classical mechanics
Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...
; in particular, they are a generalization of the
phase space
The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
of a closed system.
In the same way the
Hamilton equations allow one to derive the time evolution of a system from a set of
differential equations, the symplectic form should allow one to obtain a
vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
describing the flow of the system from the differential
of a Hamiltonian function
.
So we require a linear map
from the
tangent manifold to the
cotangent manifold , or equivalently, an element of
. Letting
denote a
section
Section, Sectioning, or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of
, the requirement that
be
non-degenerate
In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'') given by is not an isomorphism. An equivalent definition when ' ...
ensures that for every differential
there is a unique corresponding vector field
such that
. Since one desires the Hamiltonian to be constant along flow lines, one should have
, which implies that
is
alternating and hence a 2-form. Finally, one makes the requirement that
should not change under flow lines, i.e. that the
Lie derivative
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
of
along
vanishes. Applying
Cartan's formula, this amounts to (here
is the
interior product
In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, contraction, or inner derivation) is a degree −1 (anti)derivation on the exterio ...
):
:
so that, on repeating this argument for different smooth functions
such that the corresponding
span the tangent space at each point the argument is applied at, we see that the requirement for the vanishing Lie derivative along flows of
corresponding to arbitrary smooth
is equivalent to the requirement that ''ω'' should be
closed.
Definition
A symplectic form on a smooth
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
is a closed non-degenerate differential
2-form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
.
[
] Here, non-degenerate means that for every point
, the skew-symmetric pairing on the
tangent space
In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
defined by
is non-degenerate. That is to say, if there exists an
such that
for all
, then
. Since in odd dimensions,
skew-symmetric matrices are always singular, the requirement that
be nondegenerate implies that
has an even dimension.
The closed condition means that the
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
of
vanishes. A symplectic manifold is a pair
where
is a smooth manifold and
is a symplectic form. Assigning a symplectic form to
is referred to as giving
a symplectic structure.
Examples
Symplectic vector spaces
Let
be a basis for
We define our symplectic form
on this basis as follows:
:
In this case the symplectic form reduces to a simple
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
. If
denotes the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...
then the matrix,
, of this quadratic form is given by the
block matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices.
Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix w ...
:
:
Cotangent bundles
Let
be a smooth manifold of dimension
. Then the total space of the
cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This m ...
has a natural symplectic form, called the Poincaré two-form or the
canonical symplectic form
In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T^Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus p ...
:
Here
are any local coordinates on
and
are fibrewise coordinates with respect to the cotangent vectors
. Cotangent bundles are the natural
phase space
The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
s of classical mechanics. The point of distinguishing upper and lower indexes is driven by the case of the manifold having a
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 allows ...
, as is the case for
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s. Upper and lower indexes transform contra and covariantly under a change of coordinate frames. The phrase "fibrewise coordinates with respect to the cotangent vectors" is meant to convey that the momenta
are "
soldered" to the velocities
. The soldering is an expression of the idea that velocity and momentum are colinear, in that both move in the same direction, and differ by a scale factor.
Kähler manifolds
A
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 Arnol ...
is a symplectic manifold equipped with a compatible integrable complex structure. They form a particular class of
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
s. A large class of examples come from complex
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
. Any smooth complex
projective variety
In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, th ...
has a symplectic form which is the restriction of the
Fubini—Study form on the
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
.
Almost-complex manifolds
Riemannian manifolds
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the n-sphere, hyperbolic space, and smooth surfaces in ...
with an
-compatible
almost complex structure are termed
almost-complex manifolds. They generalize Kähler manifolds, in that they need not be
integrable. That is, they do not necessarily arise from a complex structure on the manifold.
Lagrangian and other submanifolds
There are several natural geometric notions of
submanifold
In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly ...
of a symplectic manifold
:
* Symplectic submanifolds of
(potentially of any even dimension) are submanifolds
such that
is a symplectic form on
.
* Isotropic submanifolds are submanifolds where the symplectic form restricts to zero, i.e. each tangent space is an
isotropic subspace of the ambient manifold's tangent space. Similarly, if each tangent subspace to a submanifold is co-isotropic (the dual of an isotropic subspace), the submanifold is called co-isotropic.
* Lagrangian submanifolds of a symplectic manifold
are submanifolds where the restriction of the symplectic form
to
is vanishing, i.e.
and
. Lagrangian submanifolds are the maximal isotropic submanifolds.
One major example is that the graph of a
symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the ...
in the product symplectic manifold is Lagrangian. Their intersections display rigidity properties not possessed by smooth manifolds; the
Arnold conjecture
The Arnold conjecture, named after mathematician Vladimir Arnold, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry.
Strong Arnold conjecture
Let (M, \omega) be a closed (compact without boundary) ...
gives the sum of the submanifold's
Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s as a lower bound for the number of self intersections of a smooth Lagrangian submanifold, rather than the
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
in the smooth case.
Examples
Let
have global coordinates labelled
. Then, we can equip
with the canonical symplectic form
:
There is a standard Lagrangian submanifold given by
. The form
vanishes on
because given any pair of tangent vectors
we have that
To elucidate, consider the case
. Then,
and
. Notice that when we expand this out
:
both terms we have a
factor, which is 0, by definition.
Example: Cotangent bundle
The cotangent bundle of a manifold is locally modeled on a space similar to the first example. It can be shown that we can glue these affine symplectic forms hence this bundle forms a symplectic manifold. A less trivial example of a Lagrangian submanifold is the zero section of the cotangent bundle of a manifold. For example, let
:
Then, we can present
as
:
where we are treating the symbols
as coordinates of
. We can consider the subset where the coordinates
and
, giving us the zero section. This example can be repeated for any manifold defined by the vanishing locus of smooth functions
and their differentials
.
Example: Parametric submanifold
Consider the canonical space
with coordinates
. A parametric submanifold
of
is one that is parameterized by coordinates
such that
:
This manifold is a Lagrangian submanifold if the
Lagrange bracket