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In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the

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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 include the canonical transformations of classical mechanics and theoretical physics, the flow associated to any Hamiltonian function, and the map on cotangent bundles induced by any diffeomorphism of manifolds. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable Suppose that φ: M → N is a Smooth map between smooth manifolds M and N; then there is an associated Linear map from In Hamiltonian mechanics, a canonical transformation is a change of Canonical coordinates (\mathbf{q} \mathbf{p} t \rightarrow (\mathbf{Q} \mathbf{P} t Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world In Mathematics, especially Differential geometry, the cotangent bundle of a Smooth manifold is the Vector bundle of all the Cotangent spaces

Flows

Any smooth function on a manifold gives rise to a Hamiltonian vector field, which are special cases of symplectic vector fields. In Mathematics and Physics, a Hamiltonian vector field on a Symplectic manifold is a Vector field, defined for any energy function In Physics and Mathematics, a symplectic vector field is one whose flow preserves a Symplectic form. Flows of the latter give rise to symplectomorphisms. Since symplectomorphisms preserve volume, Liouville's theorem in Hamiltonian mechanics follows. In Physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. 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. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός

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. In Algebraic topology, the Betti number of a Topological space is in intuitive terms a way of counting the maximum number of cuts that can be made without dividing

The equations for a geodesic may be formulated as a Hamiltonian flow. In Mathematics, the Geodesic equations are second-order non-linear Differential equations and are commonly presented in the form of Euler–Lagrange equations

The group of (Hamiltonian) symplectomorphisms

The symplectomorphisms from a manifold back onto itself form an infinite-dimensional pseudogroup. In Mathematics, a pseudogroup is an extension of the group concept but one that grew out of the geometric approach of Sophus Lie, rather than out of The corresponding Lie algebra consists of symplectic vector fields. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie 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. In Mathematics and Classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition

Groups of Hamiltonian diffeomorphisms are simple, by a theorem of Banyaga. In Mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected Normal subgroups Augustin Banyaga (b March 31, 1947 in Kigali, Rwanda) is a Rwandan-born American Mathematician whose research fields include 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. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical In Mathematics, 4-manifold is a 4-dimensional Topological manifold. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe Mikhail Leonidovich Gromov Russian Михаил Леонидович Громов (born December 23, 1943, also known In Mathematics, specifically in Topology and Geometry, a pseudoholomorphic curve (or J -holomorphic curve) is a smooth map from a

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 Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M Darboux's theorem is a Theorem in the mathematical field of Differential geometry and more specifically Differential forms, partially generalizing In contrast, isometries in Riemannian geometry must preserve the Riemann curvature tensor, which is thus a local invariant of the Riemannian manifold. In the Mathematical field of Differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most standard way to express Moreover, every function H on a symplectic manifold defines a Hamiltonian vector field XH, which exponentiates to a one-parameter group of Hamiltonian diffeomorphisms. In Mathematics and Physics, a Hamiltonian vector field on a Symplectic manifold is a Vector field, defined for any energy function In Mathematics, a one-parameter group or one-parameter subgroup usually means a continuous Group homomorphism φ: R 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. For the Mechanical engineering and Architecture usage see Isometric projection. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the 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. This article assumes some familiarity with Analytic geometry and the concept of a limit. 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, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie See Weyl quantization, geometric quantization, non-commutative geometry. In Mathematics and Physics, in the area of Quantum mechanics, Weyl quantization is a method for associating a "quantum mechanical" Hermitian In Mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given Classical theory. Noncommutative geometry, or NCG, is a branch of Mathematics concerned with the possible spatial interpretations of Algebraic structures for which the

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. Vladimir Igorevich Arnol'd or Arnold (Влади́мир И́горевич Арно́льд born June 12, 1937 in Odessa, Ukrainian SSR See also Classification of manifolds#Point-set In Mathematics, a closed manifold is type of Topological space, namely a compact "Morse function" redirects here In another context a "Morse function" can also mean an Anharmonic oscillator. 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). "Morse function" redirects here In another context a "Morse function" can also mean an Anharmonic oscillator.

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. Alexander Givental is a Mathematician working in the area of Symplectic topology, Singularity theory and their relations to topological string theories In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the It is proven in many cases by the construction of symplectic Floer homology. Floer homology is a mathematical tool used in the study of Symplectic geometry and low-dimensional Topology.

References

Symplectomorphism groups:

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