symplectic
Contents
EnglishEdit
EtymologyEdit
A calque of complex, coined by Hermann Weyl in his 1939 book The Classical Groups: Their Invariants and Representations. From Ancient Greek συμπλεκτικός (sumplektikós), from συμ (sum) (variant of σύν (sún)), + πλεκτικός (plektikós) (from πλέκω (plékō)); modelled on complex (from Latin complexus (“braided together”), from com- (“together”) + plectere (“to weave, braid”)). In both cases the suffix is from Proto-Indo-European *plek-.
The symplectic group had previously been called the line complex group.
AdjectiveEdit
symplectic (not comparable)
- (group theory, of a group) Whose characteristic abelian subgroups are cyclic.
- (mathematics, multilinear algebra, of a bilinear form) That is alternating and nondegenerate.
- (mathematics, multilinear algebra, of a vector space) That is equipped with an alternating nondegenerate bilinear form.
- (mathematics) Of or pertaining to (the geometry of) a differentiable manifold equipped with a closed nondegenerate bilinear form.
- 1995, V. I. Arnold, Some remarks on symplectic monodromy of Milnor fibrations, Helmut Hofer, Clifford H. Taubes, Alan Weinstein, Eduard Zehnder (editors), The Floer Memorial Volume, Birkhäuser Verlag, page 99,
- There exist interesting and unexplored relations between symplectic geometry and the theory of critical points of holomorphic functions.
- 1997, C. H. Cushman-de Vries (translator), Richard H. Cushman, Gijs M. Tuynman (translation editors), Jean-Marie Souriau, Structure of Dynamical Systems: A Symplectic View of Physics, Springer Science & Business Media (Birkhäuser).
- 2003, Fabrizio Catanese, Gang Tian (editors), Symplectic 4-Manifolds and Algebraic Surfaces: Lectures given at the C.I.M.E Summer School, Springer, Lecture Notes in Mathematics No. 1938.
- 2003, Yakov Eliashberg, Boris A. Khesin, François Lalonde (editors), Symplectic and Contact Topology: Interactions and Perspectives, American Mathematical Society.
- 2003, Maung Min-Oo, The Dirac Operator in Geometry and Physics, Steen Markvorsen, Maung Min-Oo (editors), Global Riemannian Geometry: Curvature and Topology, Springer, page 72,
- In symplectic geometry, there is a notion of fibrations with a symplectic manifold F as fiber, where the structure group is the group of (exact) Hamiltonian symplectomorphisms of the fiber. These are called symplectic fibrations. If the base manifold is also symplectic, there is a weak coupling construction, originally due to Thurston, of defining a symplectic structure on the total space .
- 1995, V. I. Arnold, Some remarks on symplectic monodromy of Milnor fibrations, Helmut Hofer, Clifford H. Taubes, Alan Weinstein, Eduard Zehnder (editors), The Floer Memorial Volume, Birkhäuser Verlag, page 99,
- That moves in the same direction as a system of synchronized waves.
AntonymsEdit
Derived termsEdit
- symplectic cut
- symplectic form (symplectic bilinear form)
- symplectic invariant (structure that is invariant under a symplectic form (regarded as a mapping))
- symplectic matrix
- symplectic Clifford algebra
Related termsEdit
NounEdit
symplectic (plural symplectics)
- (mathematics) A symplectic bilinear form, manifold, geometry, etc.
- 1967, Journal of Mathematics and Mechanics, Volume 16, Issue 1, Indiana University, page 339,
- The structure of stable symplectics on finite dimensional spaces has been studied by Krein [8], Gelfand & Lidskii [9], and Moser [10] in work of considerable practical importance.
- 1967, Journal of Mathematics and Mechanics, Volume 16, Issue 1, Indiana University, page 339,
- (ichthyology) A symplectic bone.
- 1914, The Philippine Journal of Science, Volume 9, page 27,
- The symplectics (9) consist of a somewhat curved central triangular portion with the base upward, and anteriorly and posteriorly from this extends a wing-like process.
- 1965, Agra University Journal of Research: Science, Volume 14, page 71,
- The symplectics (Fig. 8, sym) are thin slender bones placed vertically in between the quadrates and the hyomandibulars.
- 1967, Tyson R. Roberts, Studies on the Osteology and Phylogeny of Characoid Fishes, page 59,
- In many teleosts, on the other hand, including the catfishes, the symplectics have been entirely lost.
- 1914, The Philippine Journal of Science, Volume 9, page 27,
ReferencesEdit
- The Classical Groups. Their Invariants and Representations, Hermann Weyl; Princeton University Press, 1939 →ISBN, footnote, p. 165
- The Symplectization of Science, Mark J. Gotay and James A. Isenberg, p. 13
Further readingEdit
- symplectic on Wikipedia.Wikipedia
- symplectic bone on Wikipedia.Wikipedia
- Symplectic cut on Wikipedia.Wikipedia
- Symplectic geometry on Wikipedia.Wikipedia
- Symplectic group on Wikipedia.Wikipedia
- Symplectic manifold on Wikipedia.Wikipedia
- Symplectic matrix on Wikipedia.Wikipedia
- Group of symplectic type on Wikipedia.Wikipedia