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Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space 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 describing the flow of the system from the differential dH of a Hamiltonian function H. So we require a linear map , or equivalently, an element of . Letting ? denote a section of , the requirement that ? be non-degenerate ensures that for every differential dH there is a unique corresponding vector field VH 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 of ? 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 ? should be closed.
A symplectic form on a smooth manifold is a closed non-degenerate differential 2-form. 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. 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.
Symplectic vector spaces
Let be a basis for We define our symplectic form ? on this basis as follows:
Here are any local coordinates on and are fibrewise coordinates with respect to the cotangent vectors . Cotangent bundles are the natural phase spaces of classical mechanics. The point of distinguishing upper and lower indexes is driven by the case of the manifold having a metric tensor, as is the case for Riemannian manifolds. 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.
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 . Lagrangian 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 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 submanifolds
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,
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 and by Darboux's theorem the symplectic form ? can be, at least locally, written as , where d denotes the exterior derivative and ? denotes the exterior product. This form is called the Poincaré two-form or the canonical two-form. 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.
Let L be a Lagrangian submanifold of a symplectic manifold (K,?) given by an immersion (i is called a Lagrangian immersion). Let give a Lagrangian fibration of K. The composite is a Lagrangian mapping. The critical value set of ? ? i is called a caustic.
Two Lagrangian maps and are called Lagrangian equivalent if there exist diffeomorphisms?, ? and ? such that both sides of the diagram given on the right commute, and ? preserves the symplectic form. Symbolically:
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.
A polysymplectic manifold is a Legendre bundle provided with a polysymplectic tangent-valued -form; it is utilized in Hamiltonian field theory.