View source for Symplectic

==English==

===Etymology===
A [[calque]] of {{m|en|complex}}, coined by {{w|Hermann Weyl}} in his 1939 book ''{{w|The Classical Groups|The Classical Groups: Their Invariants and Representations}}''. From {{der|en|grc|συμπλεκτικός}}, from {{m|grc|συμ}} (variant of {{m|grc|σύν}}), + {{m|grc|πλεκτικός}} (from {{m|grc|πλέκω}}); modelled on {{m|en|complex}} (from Latin {{m|la|complexus||braided together}}, from {{m|la|com-||together}} + {{m|la|plectere||to weave, braid}}).

The ''symplectic group'' has previously been called the ''line complex group''.

===Pronunciation===
* {{IPA|en|/sɪmˈplɛktɪk/}}
* {{rhymes|en|ɛktɪk|s=3}}

===Adjective===
{{en-adj|-}}

# Placed in or among, as if woven together.
# {{lb|en|group theory|of a [[group]]}} Whose [[characteristic]] [[abelian]] subgroups are [[cyclic]].
# {{lb|en|mathematics|multilinear algebra|of a [[bilinear form]]}} That is [[alternating]] and [[nondegenerate]].
# {{lb|en|mathematics|multilinear algebra|of a [[vector space]]}} That is equipped with an [[alternating]] [[nondegenerate]] [[bilinear form]].
# {{lb|en|mathematics}} Of or pertaining to (the [[geometry]] of) a [[differentiable]] [[manifold]] equipped with a [[closed]] [[nondegenerate]] [[bilinear form]].
#* {{quote-book|en|year=1995|author=V. I. Arnold|chapter=Some remarks on '''symplectic''' monodromy of Milnor fibrations|editors=Helmut Hofer; Clifford H. Taubes; Alan Weinstein; Eduard Zehnder|title=The Floer Memorial Volume|publisher=Birkhäuser Verlag|pageurl=https://books.google.com.au/books?id=aGr0h8iDeo4C&pg=PA99&dq=%22symplectic%22&hl=en&sa=X&ved=0ahUKEwits-yAwdTZAhXI64MKHX0LDl4Q6AEI8wMwUA#v=onepage&q=%22symplectic%22&f=false|page=99
|passage=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), [https://books.google.com.au/books?id=RnBqCQAAQBAJ&printsec=frontcover&dq=%22symplectic%22&hl=en&sa=X&ved=0ahUKEwits-yAwdTZAhXI64MKHX0LDl4Q6AEIZDAK#v=onepage&q=%22symplectic%22&f=false '''Symplectic''' 4-Manifolds and Algebraic Surfaces: Lectures given at the C.I.M.E Summer School], Springer, Lecture Notes in Mathematics No. 1938.
#* {{quote-text|en|year=2003|editors=Yakov Eliashberg; Boris A. Khesin; François Lalonde|title=[https://books.google.com.au/books?id=A73Jeq_ZrkEC&printsec=frontcover&dq=%22symplectic%22&hl=en&sa=X&ved=0ahUKEwits-yAwdTZAhXI64MKHX0LDl4Q6AEIiwEwEQ#v=onepage&q=%22symplectic%22&f=false '''Symplectic''' and Contact Topology: Interactions and Perspectives]|publisher=w:American Mathematical Society}}
#* {{quote-book|en|year=2003|author=Maung Min-Oo|chapter=The Dirac Operator in Geometry and Physics|editors=Steen Markvorsen; Maung Min-Oo|title=Global Riemannian Geometry: Curvature and Topology|publisher=Springer|pageurl=https://books.google.com.au/books?id=O2RVZKtD-bMC&pg=PA72&dq=%22symplectic%22&hl=en&sa=X&ved=0ahUKEwits-yAwdTZAhXI64MKHX0LDl4Q6AEIngMwQg#v=onepage&q=%22symplectic%22&f=false|page=72
|passage=In '''symplectic''' geometry, there is a notion of fibrations <math>\pi : P \rightarrow M</math> 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 <math>(M,\omega_M)</math> is also '''symplectic''', there is a weak coupling construction, originally due to Thurston, of defining a '''symplectic''' structure on the total space <math>P</math>.}}
# That moves in the same direction as a system of [[synchronized]] [[wave]]s.
# {{lb|en|petrology|mineralogy}} Of or pertaining to a [[symplectite]]; {{l|en|symplectitic}}.

====Antonyms====
* [[antiplectic]]

====Derived terms====
* {{l|en|microsymplectic}}
* {{l|en|symplectic cut}}
* {{l|en|symplectic form}} {{gloss|symplectic bilinear form}}
* {{l|en|symplectic group}}
* {{l|en|symplectic invariant}} {{gloss|structure that is invariant under a symplectic form (regarded as a mapping)}}
* {{l|en|symplectic matrix}}
* {{l|en|symplectic Clifford algebra}}

====Related terms====
* {{l|en|symplectomorphism}}

===Noun===
{{en-noun}}

# {{lb|en|mathematics}} A symplectic bilinear form, manifold, geometry, etc.
#* {{quote-journal|en|year=1967|journal={{w|Indiana University Mathematics Journal|Journal of Mathematics and Mechanics}}|volume=16|issue=1|publisher=Indiana University|pageurl=https://books.google.com.au/books?id=2_YuAAAAIAAJ&q=%22symplectics%22&dq=%22symplectics%22&hl=en&sa=X&ved=0ahUKEwi9wabd_9TZAhXCurwKHV5aCRIQ6AEImQEwGA|page=339
|passage=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.}}
# {{lb|en|ichthyology}} A [[bone]] in the [[teleostean]] [[fish]]es that forms the lower [[ossification]] of the [[suspensorium]], and which [[articulate]]s below with the [[quadrate bone]] by which it is firmly held.
#* {{quote-book|en|year=1914|title=The Philippine Journal of Science, Volume 9|pageurl=https://books.google.com.au/books?id=87RMAAAAYAAJ&q=%22symplectics%22&dq=%22symplectics%22&hl=en&sa=X&ved=0ahUKEwiXworC0NLZAhUqiVQKHXljDEAQ6AEISjAG|page=27
|passage=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.}}
#* {{quote-book|en|year=1965|title=Agra University Journal of Research: Science, Volume 14|pageurl=https://books.google.com.au/books?id=DsMWAQAAIAAJ&q=%22symplectics%22&dq=%22symplectics%22&hl=en&sa=X&ved=0ahUKEwi9wabd_9TZAhXCurwKHV5aCRIQ6AEIwgIwPQ|page=71
|passage=The '''symplectics''' (Fig. 8, sym) are thin slender bones placed vertically in between the quadrates and the hyomandibulars.}}
#* {{quote-book|en|year=1967|author=Tyson R. Roberts|title=Studies on the Osteology and Phylogeny of Characoid Fishes|pageurl=https://books.google.com.au/books?id=b4hCAAAAIAAJ&q=%22symplectics%22&dq=%22symplectics%22&hl=en&sa=X&ved=0ahUKEwi9wabd_9TZAhXCurwKHV5aCRIQ6AEIbDAO|page=59
|passage=In many teleosts, on the other hand, including the catfishes, the '''symplectics''' have been entirely lost.}}

===References===
* [http://books.google.com/books?isbn=0691057567 The Classical Groups. Their Invariants and Representations], {{w|Hermann Weyl}}; Princeton University Press, 1939 {{ISBN|978-0-691-05756-9}}, [https://books.google.com/books?id=zmzKSP2xTtYC&lpg=PP1&dq=isbn%3A0691057567&pg=PA165#v=onepage&q&f=false footnote, p. 165]
* [https://web.archive.org/web/20110613221105/http://www.pims.math.ca/~gotay/Symplectization(E).pdf The Symplectization of Science], Mark J. Gotay and {{w|James A. Isenberg}}, p. 13

===Further reading===
* {{pedia}}
* {{pedia|symplectic bone}}
* {{pedia|Symplectic cut}}
* {{pedia|Symplectic geometry}}
* {{pedia|Symplectic group}}
* {{pedia|Symplectic manifold}}
* {{pedia|Symplectic matrix}}
* {{pedia|Group of symplectic type}}

{{cln|en|terms prefixed with sym-|sort=plectic}}