Symplectic

Definitions

  • Webster's Revised Unabridged Dictionary
    • a Symplectic (Anat) Plaiting or joining together; -- said of a bone next above the quadrate in the mandibular suspensorium of many fishes, which unites together the other bones of the suspensorium.
    • ***
Century Dictionary and Cyclopedia
    • symplectic Placed in or among, or put between, as if ingrained or woven in: specifically noting a bone of the lower jaw of fishes interposed between others.
    • n symplectic A bone of the lower jaw or mandibular arch of some vertebrates, as fishes, between the hyomandibular bone above and the quadrate bone below, forming an inferior ossification of the suspensorium of the lower jaw, articulated or ankylosed with the quadrate or its representative. Also called mesotympanic. See cuts under palatoquadrate and teleost.
    • ***
Chambers's Twentieth Century Dictionary
    • adj Symplectic sim-plek′tik placed in or among, as if woven together
    • n Symplectic a bone in the Teleostean fishes which forms the lower ossification of the suspensorium, and which articulates below with the quadrate bone by which it is firmly held
    • ***

Etymology

Webster's Revised Unabridged Dictionary
Gr. plaiting together, fr. to plait together

Usage


In science:

M , ω ) to be any compact symplectic manifold with integral symplectic form, i.e.
Universality and scaling of zeros on symplectic manifolds
Let M be a connected, compact, symplectic C∞ manifold of dimension 2n, with symplectic form ω .
Hamiltonian symplectomorphisms and the Berry phase
In this framework, in fact, the symplectic structure of the field theory is uniquely defined by the bulk term, generated by the so called reduced symplectic current.
The First Law of Isolated Horizons via Noether Theorem
Definition 1.1. A symplectic foliation α with normal line bund le L on a symplectic manifold (M , ω) is a non-zero element of C∞ (T ∗ CM ⊗C L) which satisfies the integrability condition (1).
Codimension one symplectic foliations
We have a symplectic foliation α′ k with Kαk = Bk a symplectic smooth submanifold of codimension 4.
Codimension one symplectic foliations
We give a method to derive a symplecticity preserving reduced map from a nonlinear symplectic map of three or more sites, which is non-integrable even in the time-continuous limit.
Random Wandering Around Homoclinic-like Manifolds in Symplectic Map Chain
We present a regularization procedure to preserve the symplectic structure of the RG map near a fully elliptic fixed point of a chain of weakly nonlinear symplectic maps.
Random Wandering Around Homoclinic-like Manifolds in Symplectic Map Chain
Corollary 2.7. — There exists a symplectic fourfold X such that the intersection form on symplectic curves D ⊂ X is not hyperbolic.
Simple examples of symplectic fourfolds with exotic properties
Hamiltonian formalism is based on symplectic structures; a special but relevant class of symplectic manifolds is provided by Kahler manifolds.
Hyperhamiltonian dynamics
This can be equivalently seen as a hypersymplectic manifold (M , g , ωα ) with ωα the symplectic forms associated to Yα via g ; in the following we will refer to the hyperkahler structure even when we will focus on the symplectic aspect.
Hyperhamiltonian dynamics
This map is an isomorphism in certain cases , for instance if M is a compact symplectic manifold and the G action is Hamiltonian (preserves the symplectic form).
Quantum Field Theory and Representation Theory: A Sketch
In that case AP is a (infinite dimensional) symplectic manifold with a line bundle L (the determinant bundle for the Dirac operator) whose curvature is the symplectic form.
Quantum Field Theory and Representation Theory: A Sketch
T ∗G we consider the canonical symplectic 2-form ΩG = −δλG then T ∗G is a symplectic groupoid over A∗G (see ).
Jacobi groupoids and generalized Lie bialgebroids
Let G ⇉ M be a symplectic groupoid with exact symplectic 2-form Ω = −δθ.
Jacobi groupoids and generalized Lie bialgebroids
Our first task, then, was to find a different definition which gives complex submanifolds in the complex case and symplectic submanifolds in the symplectic case.
Submanifolds of generalized complex manifolds
***