Symplectomorphism

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

1 Formal definition
2 Flows
3 Comparison with Riemannian geometry
4 The group of (Hamiltonian) symplectomorphisms
5 Quantizations
6 Arnold conjecture
7 References

[edit] Formal definition

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

f : M₁ → M₂

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 are the canonical transformations of classical mechanics and theoretical physics. A Hamiltonian symplectomorphism is a symplectomorphism that arises as the flow of a Hamiltonian vector field, and hence from some Hamiltonian function.

[edit] Flows

The flow of a symplectic vector field on a symplectic manifold is a symplectomorphism. This follows from the closedness of the symplectic form and Cartan's formula for the Lie derivative in terms of the exterior derivative. Since Hamiltonian vector fields are symplectic, as a consequence we have Liouville's theorem: the symplectic volume is invariant under a Hamiltonian flow. Since

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

the flow of a Hamiltonian vector field also preserves H. In physics this is interpreted as the law of conservation of energy. Liouville's theorem is interpreted as the conservation of phase volume in Hamiltonian systems, which is the basis for classical statistical mechanics.

We have shown that there is a one-to-one correspondence between infinitesimal symplectomorphisms and closed one-forms on a symplectic manifold. If the first Betti number of the manifold is zero, and it is connected, the latter set is the same as the space of smooth functions modulo addition of constants.

A common introductory example of a Hamiltonian flow is the presentation of the geodesic equations formulated as a Hamiltonian flow.

[edit] Comparison with Riemannian geometry

Unlike general Riemannian manifolds, symplectic manifolds are not very rigid: they have many symplectomorphisms coming from Hamiltonian vector fields. The fundamental difference between Riemannian and symplectic geometry is that a symplectic manifold has no local invariants: according to Darboux's theorem for every point x in a symplectic manifold there is a local coordinate system with coordinates, called the canonical coordinates,

p₁,...,p_n, q¹,...,qⁿ,

such that

$\omega=\sum_n dq^i \wedge dp_i$

the canonical symplectic form.

[edit] The group of (Hamiltonian) symplectomorphisms

The group of symplectomorphisms from a manifold back onto itself forms an infinite-dimensional Lie group. In the corresponding Lie algebra, the Lie bracket is given by the Poisson bracket; the algebra is the space of smooth functions on a symplectic manifold. The article on the Poisson bracket provides an explicit demonstration of the Lie algebra.

Locally, symplectomorphisms can be generated by a generating function over a (local) Darboux coordinates. See Hamilton-Jacobi equation.

[edit] 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.

[edit] 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, which is a statement on Lagrangian submanifolds. It is proven in many cases by the construction of symplectic Floer homology.

[edit] References

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

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Categories: Symplectic topology | Hamiltonian mechanics