Symplectomorphisms are the homomorphisms of symplectic manifolds.
In the context of mechanics where symplectic manifolds model phase spaces, symplectomorphisms are essentially what are called canonical transformations.
A symplectomorphism or symplectic diffeomorphism from a symplectic manifold to a symplectic manifold is a diffeomorphism preserving the symplectic form, i.e. such that
The symplectomorphisms from a symplectic manifold to itself form an infinite-dimensional Lie group that is a subgroup of the diffeomorphism group of , the symplectomorphism group:
Its Lie algebra
is that of symplectic vector fields: those vector fields such that their Lie derivative annihilates the symplectic form
The further subgroup corresponding to those symplectic vector fields which are flows of Hamiltonian vector fields coming from a smooth family of Hamiltonians
is the group of Hamiltonian symplectomorphisms or Hamiltonian diffeomorphisms.
In the generalization to n-plectic geometry there are accordingly -plectomorphisms. See at higher symplectic geometry.
Preservation of volume
Inasmuch as a symplectic manifold carries a canonical volume form , it is clear that a symplectomorphism is locally volume-preserving.
Relation to Poisson brackets
The Lie algebra given by the Poisson bracket of a symplectic manifold is that of a central extension of the group of Hamiltonian symplectomorphisms. (It integrates to the quantomorphism group.)
The central extension results form the fact that the Hamiltonian associated with every Hamiltonian vector field is well defined only up to the addition of a constant function.
If is a symplectic vector space then there is corresponding to it a Heisenberg Lie algebra. This sits inside the Poisson bracket algebra, and accordingly the Heisenberg group is a subgroup of the group of (necessarily Hamiltonian) symplectomorphisms of the symplectic vector space, regarded as a symplectic manifold.
Relation to Lagrangian correspondences
A symplectomorphisms canonically induces a Lagrangian correspondence between and , given by its graph.
Extensions under geometric quantization
higher and integrated Kostant-Souriau extensions:
(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for -principal ∞-connection)
(extension are listed for sufficiently connected )
Given a symplectic vector space regarded as a symplectic manifold, then those symplectomorphisms which are linear maps on form, under composition, the symplectic group .
The linear Hamiltonian symplectomorphisms are also known as the Hamiltonian matrices?.
A curious example: volumes of balls
The following example, due to Andreas Blass and Stephen Schanuel, is a categorified way to calculate volumes of even-dimensional balls.
In any dimension , the volume of the unit ball in (with respect to the Lebesgue measure) is
where is the Euler Gamma function. In dimension , this gives
Meanwhile, we may regard as the volume of the -dimensional complex polydisc, viz. the cartesian power of the complex 1-disc , on which the symmetric group acts by permuting coordinates. The volume of the orbit space is clearly .
Given , write coordinates in polar coordinate form , and define an -invariant map by first permuting the so that and then mapping to
Then induces a continuous well-defined map . Furthermore, when restricted to the set of for which the are all distinct, induces a smooth symplectic isomorphism mapping onto the set of for which for .
In other words, writing the symplectic 2-form
is preserved by pulling back along . Since symplectic maps are locally volume-preserving, and since and are almost all of and respectively, this gives a proof that the volume of is (alternate to standard purely computational proofs).
Lecture notes include
- Augustin Banyaga, Introduction to the geometry of hamiltonian diffeomorphisms (pdf)
The example of volumes of balls is discussed in
- Andreas Blass, Stephen Schanuel, On the volumes of balls (ps).