SYMPLECTIC MANIFOLD Meaning and
Definition
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A symplectic manifold is a geometric object in mathematics that combines both the notions of a manifold (a geometric space that locally resembles Euclidean space) and a symplectic structure (a mathematical object that encodes the notion of "area" or "volume" in a geometric space).
More precisely, a symplectic manifold is a smooth manifold equipped with a symplectic form, which is a non-degenerate, closed 2-form. Non-degeneracy means that at every point in the manifold, the 2-form assigns a non-zero value to any non-zero vector in the tangent space, while closedness means that locally the 2-form behaves as an exact differential.
The symplectic structure endows a symplectic manifold with several important geometric properties. For example, it allows for the definition of a symplectic vector field, which preserves the symplectic form when it evolves along the manifold. This concept represents the notion of a "conservative" or "incompressible" flow in physics. Moreover, the symplectic structure gives rise to the idea of a symplectic transformation, which is a diffeomorphism that preserves the symplectic form.
In symplectic geometry, the study of symplectic manifolds delves into understanding the topological, geometric, and analytic properties of these structures. It has connections to various other areas of mathematics and physics, such as Hamiltonian dynamics, algebraic geometry, and quantum mechanics, making it an essential topic in modern mathematical research.
Etymology of SYMPLECTIC MANIFOLD
The word "symplectic manifold" has its etymology rooted in mathematics and geometry. The term "symplectic" originated from the Greek word "συμπλεκτικός" (symplektikos), which means "interwoven" or "complex". The word was first introduced by Hermann Weyl, a German mathematician, in the early 20th century.
Weyl used the term "symplectic" to describe a mathematical structure that he believed was more fundamental than the concept of a metric on a manifold. This structure was later developed and formalized by other mathematicians, especially by the work of Joseph Marsden and Alan Weinstein.
The term "manifold" comes from the old English word "manigfeald", which means "manyfold" or "varied". In mathematics, a manifold is a topological space that locally resembles Euclidean space.
