In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the symplectic structure of phase space, and is called a canonical transformation.
The infinitesimal version of symplectomorphisms gives the symplectic vector fields.
These vector fields build a Lie subalgebra of
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 coadjoint action of an element of a Lie group on a coadjoint orbit.
Any smooth function on a symplectic manifold gives rise, by definition, to a Hamiltonian vector field and the set of all such vector fields 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 {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 symplectomorphisms from a manifold back onto itself form an infinite-dimensional pseudogroup.
The corresponding Lie algebra consists of symplectic 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.
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.
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.
Moreover, Riemannian manifolds with large symmetry groups are very special, and a generic Riemannian manifold has no nontrivial symmetries.
Representations of finite-dimensional subgroups of the group of symplectomorphisms (after ħ-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.
A celebrated conjecture of Vladimir Arnold relates the minimum number of fixed points for a Hamiltonian symplectomorphism
is a compact symplectic manifold, to Morse theory (see [3]).
The most important development in symplectic geometry triggered by this famous conjecture is the birth of Floer homology (see [6]), named after Andreas Floer.