In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds. Specifically, let (M₁, ω₁) and (M₂, ω₂) by 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: Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... This article discusses the pullback in differential geometry. ...

Symplectomorphisms are usually called canonical transformations by physicists.

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. As a direct consequence we have Liouville's theorem: the symplectic volume is invariant under a Hamiltionan flow. Since In mathematics and physics, the symplectic vector field, also known as the Hamiltonian vector field, is a vector field induced on a symplectic manifold by an energy function or Hamiltonian. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was a French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ... In mathematics, a Lie derivative is a derivation on the algebra of smooth functions over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented by vector fields, as... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...

{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. Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...

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. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is greater in absolute value than zero yet smaller than any positive real number. ... In algebraic topology, the Betti numbers of a topological space X are a sequence b0, b1, ... of topological invariants. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...

Unlike Riemannian manifolds, symplectic manifolds are extremely non-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, In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ...

p₁,...,p_n, q¹,...,qⁿ,

such that

Finite-dimensional subgroups of the group of symplectomorphisms are Lie groups. Representations of these Lie groups (after -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. In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...

Locally, symplectomorphisms can be generated by a generating function over a (local) Darboux coordinates. See Hamilton-Jacobi equation. The Hamilton-Jacobi-Bellman (HJB) equation is a partial differential equation which is central to optimal control theory. ...

References

• Dusa McDuff and D. Salamon: Introduction to Symplectic Topology (1998) Oxford Mathematical Monographs, ISBN 0-198-50451-9.
• Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 3.2.