Symplectomorphism

Symplectomorphism

In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds.

Formal definition

Specifically, let (M1, ω1) and (M2, ω2) be symplectic manifolds. A map

f : M1M2

is a symplectomorphism if it is a diffeomorphism and the pullback of ω2 under f is equal to ω1:

f^{*}omega_2 = omega_1.,

Examples of symplectomorphisms include the canonical transformations of classical mechanics and theoretical physics, the flow associated to any Hamiltonian function, the map on cotangent bundles induced by any diffeomorphism of manifolds, and the coajoint action of an element of a Lie Group on a coadjoint orbit.

Examples

  • Translations in mathbf{R}^n are symplectomorphisms.

Flows

Any smooth function on a symplectic manifold gives rise, by definition, to a Hamiltonian vector field and the set of all such form a subalgebra of the Lie Algebra of symplectic vector fields. The integration of the flow of a symplectic vector field is a symplectomorphism. Since symplectomorphisms preserve the symplectic 2-form and hence the symplectic-volumeform, Liouville's theorem in Hamiltonian mechanics follows. Symplectomorphisms that arise from Hamiltonian vector fields are known as Hamiltonian symplectomorphisms.

Since

{H,H} = XH(H) = 0

the flow of a Hamiltonian vector field also preserves H. In physics this is interpreted as the law of conservation of energy.

If the first Betti number of a connected symplectic manifold is zero, symplectic and Hamiltonian vector fields coincide, so the notions of Hamiltonian isotopy and symplectic isotopy of symplectomorphisms coincide.

The equations for a geodesic may be formulated as a Hamiltonian flow.

The group of (Hamiltonian) symplectomorphisms

The symplectomorphisms from a manifold back onto itself form an infinite-dimensional pseudogroup. The corresponding Lie algebra consists of symplectic vector fields. The Hamiltonian symplectomorphisms form a subgroup, whose Lie algebra is given by the Hamiltonian vector fields. The latter is isomorphic to the Lie algebra of smooth functions on the manifold with respect to the Poisson bracket, modulo the constants.

Groups of Hamiltonian diffeomorphisms are simple, by a theorem of Banyaga. They have natural geometry given by the Hofer norm. The homotopy type of the symplectomorphism group for certain simple symplectic four-manifolds, such as the product of spheres, can be computed using Gromov's theory of pseudoholomorphic curves.

Comparison with Riemannian geometry

Unlike Riemannian manifolds, symplectic manifolds are not very rigid: Darboux's theorem shows that all symplectic manifolds are locally isomorphic. In contrast, isometries in Riemannian geometry must preserve the Riemann curvature tensor, which is thus a local invariant of the Riemannian manifold. Moreover, every function H on a symplectic manifold defines a Hamiltonian vector field XH, which exponentiates to a one-parameter group of Hamiltonian diffeomorphisms. It follows that the group of symplectomorphisms is always very large, and in particular, infinite-dimensional. On the other hand, the group of isometries of a Riemannian manifold is always a (finite-dimensional) Lie group. Moreover, Riemannian manifolds with large symmetry groups are very special, and a generic Riemannian manifold has no nontrivial symmetries.

Quantizations

Representations of finite-dimensional subgroups of the group of symplectomorphisms (after hbar-deformations, in general) on Hilbert spaces are called quantizations. When the Lie group is the one defined by a Hamiltonian, it is called a "quantization by energy". The corresponding operator from the Lie algebra to the Lie algebra of continuous linear operators is also sometimes called the quantization; this is a more common way of looking at it in physics. See Weyl quantization, geometric quantization, non-commutative geometry.

Arnold Conjecture

A celebrated conjecture of V. I. Arnold relates the minimum number of fixed points for a Hamiltonian symplectomorphism f on M, in case M is a closed manifold, to Morse theory. More precisely, the conjecture states that f has at least as many fixed points as the number of critical points a smooth function on M must have (understood as for a generic case, Morse functions, for which this is a definite finite number which is at least 2).

It is known that this would follow from the Arnold-Givental conjecture named after V.I Arnold and Alexander Givental, which is a statement on Lagrangian submanifolds. It is proven in many cases by the construction of symplectic Floer homology.

References

  • Dusa McDuff and D. Salamon: Introduction to Symplectic Topology (1998) Oxford Mathematical Monographs, ISBN 0-19-850451-9.
  • Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 3.2.

Symplectomorphism groups:

  • Gromov, M. Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), no. 2, 307--347.
  • Polterovich, Leonid. The geometry of the group of symplectic diffeomorphism. Basel ; Boston : Birkhauser Verlag, 2001.

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