Symplectic manifold

In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles 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 of the system.


Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system.[1] 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 describing the flow of the system from the differential dH of a Hamiltonian function H.[2] So we require a linear map TMTM, or equivalently, an element of TMTM. Letting ω denote a section of TMTM, the requirement that ω be non-degenerate ensures that for every differential dH there is a unique corresponding vector field VH such that dH = ω(VH, · ). Since one desires the Hamiltonian to be constant along flow lines, one should have dH(VH) = ω(VH, VH) = 0, which implies that ω is alternating and hence a 2-form. Finally, one makes the requirement that ω should not change under flow lines, i.e. that the Lie derivative of ω along VH vanishes. Applying Cartan's formula, this amounts to (here   is the interior product):


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.


A symplectic form on a smooth manifold   is a closed non-degenerate differential 2-form  .[3][4] Here, non-degenerate means that for every point  , the skew-symmetric pairing on the tangent space   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.[3][4] The closed condition means that the exterior derivative 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.

Linear symplectic manifoldEdit

There is a standard linear model, namely a symplectic vector space   Let   be a basis for   We define our symplectic form ω on this basis as follows:


In this case the symplectic form reduces to a simple quadratic form. If In denotes the n × n identity matrix then the matrix, Ω, of this quadratic form is given by the 2n × 2n block matrix:


Lagrangian and other submanifoldsEdit

There are several natural geometric notions of submanifold 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  . Langrangian submanifolds are the maximal isotropic submanifolds.

The most important case of the isotropic submanifolds is that of Lagrangian submanifolds. A Lagrangian submanifold is, by definition, an isotropic submanifold of maximal dimension, namely half the dimension of the ambient symplectic manifold. One major example is that the graph of a symplectomorphism in the product symplectic manifold (M × M, ω × −ω) is Lagrangian. Their intersections display rigidity properties not possessed by smooth manifolds; the Arnold conjecture gives the sum of the submanifold's Betti numbers as a lower bound for the number of self intersections of a smooth Lagrangian submanifold, rather than the Euler characteristic in the smooth case.


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.

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 more non-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  .

Another useful class of Lagrangian submanifolds can be found using Morse theory. Given a Morse function   and for a small enough   one can construct a Lagrangian submanifold given by the vanishing locus  . For a generic morse function we have a Lagrangian intersection given by  .

Special Lagrangian submanifoldsEdit

In the case of Kahler manifolds (or Calabi-Yau manifolds) we can make a choice   on   as a holomorphic n-form, where   is the real part and   imaginary. A Lagrangian submanifold   is called special if in addition to the above Lagrangian condition the restriction   to   is vanishing. In other words, the real part   restricted on   leads the volume form on  . The following examples are known as special Lagrangian submanifolds,

  1. complex Lagrangian submanifolds of hyperKahler manifolds,
  2. fixed points of a real structure of Calabi-Yau manifolds.

The SYZ conjecture has been proved for special Lagrangian submanifolds but in general, it is open, and brings a lot of impacts to the study of mirror symmetry. see (Hitchin 1999)

Lagrangian fibrationEdit

A Lagrangian fibration of a symplectic manifold M is a fibration where all of the fibres are Lagrangian submanifolds. Since M is even-dimensional we can take local coordinates (p1,…,pn, q1,…,qn), and by Darboux's theorem the symplectic form ω can be, at least locally, written as ω = ∑ dpk ∧ dqk, where d denotes the exterior derivative and ∧ denotes the exterior product. Using this set-up we can locally think of M as being the cotangent bundle   and the Lagrangian fibration as the trivial fibration   This is the canonical picture.

Lagrangian mappingEdit

Let L be a Lagrangian submanifold of a symplectic manifold (K,ω) given by an immersion i : LK (i is called a Lagrangian immersion). Let π : KB give a Lagrangian fibration of K. The composite (πi) : LKB is a Lagrangian mapping. The critical value set of πi is called a caustic.

Two Lagrangian maps (π1i1) : L1K1B1 and (π2i2) : L2K2B2 are called Lagrangian equivalent if there exist diffeomorphisms σ, τ and ν such that both sides of the diagram given on the right commute, and τ preserves the symplectic form.[4] Symbolically:


where τω2 denotes the pull back of ω2 by τ.

Special cases and generalizationsEdit

  • Symplectic manifolds are special cases of a Poisson manifold. The definition of a symplectic manifold requires that the symplectic form be non-degenerate everywhere, but if this condition is violated, the manifold may still be a Poisson manifold.
  • A multisymplectic manifold of degree k is a manifold equipped with a closed nondegenerate k-form.[5]
  • A polysymplectic manifold is a Legendre bundle provided with a polysymplectic tangent-valued  -form; it is utilized in Hamiltonian field theory.[6]

See alsoEdit


  1. ^ Webster, Ben. "What is a symplectic manifold, really?".
  2. ^ Cohn, Henry. "Why symplectic geometry is the natural setting for classical mechanics".
  3. ^ a b de Gosson, Maurice (2006). Symplectic Geometry and Quantum Mechanics. Basel: Birkhäuser Verlag. p. 10. ISBN 3-7643-7574-4.
  4. ^ a b c Arnold, V. I.; Varchenko, A. N.; Gusein-Zade, S. M. (1985). The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1. Birkhäuser. ISBN 0-8176-3187-9.
  5. ^ Cantrijn, F.; Ibort, L. A.; de León, M. (1999). "On the Geometry of Multisymplectic Manifolds". J. Austral. Math. Soc. Ser. A. 66 (3): 303–330. doi:10.1017/S1446788700036636.
  6. ^ Giachetta, G.; Mangiarotti, L.; Sardanashvily, G. (1999). "Covariant Hamiltonian equations for field theory". Journal of Physics. A32: 6629–6642. arXiv:hep-th/9904062. doi:10.1088/0305-4470/32/38/302.


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