Search results for "Racket"
showing 10 items of 132 documents
Geometric Aspects of Mechanics
2010
In many respects, mechanics carries geometrical structures. This could be felt very clearly at various places in the first four chapters. The most important examples are the structures of the space–time continua that support the dynamics of nonrelativistic and relativistic mechanics, respectively. The formulation of Lagrangian mechanics over the space of generalized coordinates and their time derivatives, as well as of Hamilton–Jacobi canonical mechanics over the phase space, reveals strong geometrical features of these manifolds.
On the Leibniz bracket, the Schouten bracket and the Laplacian
2003
International audience; The Leibniz bracket of an operator on a (graded) algebra is defined and some of its properties are studied. A basic theorem relating the Leibniz bracket of the commutator of two operators to the Leibniz bracket of them is obtained. Under some natural conditions, the Leibniz bracket gives rise to a (graded) Lie algebra structure. In particular, those algebras generated by the Leibniz bracket of the divergence and the Laplacian operators on the exterior algebra are considered, and the expression of the Laplacian for the product of two functions is generalized for arbitrary exterior forms.
Closedness of Star Products and Cohomologies
1994
We first review the introduction of star products in connection with deformations of Poisson brackets and the various cohomologies that are related to them. Then we concentrate on what we have called “closed star products” and their relations with cyclic cohomology and index theorems. Finally we shall explain how quantum groups, especially in their recent topological form, are in essence examples of star products.
Geometric Origin of the Tennis Racket Effect
2020
The tennis racket effect is a geometric phenomenon which occurs in a free rotation of a three-dimensional rigid body. In a complex phase space, we show that this effect originates from a pole of a Riemann surface and can be viewed as a result of the Picard-Lefschetz formula. We prove that a perfect twist of the racket is achieved in the limit of an ideal asymmetric object. We give upper and lower bounds to the twist defect for any rigid body, which reveals the robustness of the effect. A similar approach describes the Dzhanibekov effect in which a wing nut, spinning around its central axis, suddenly makes a half-turn flip around a perpendicular axis and the Monster flip, an almost impossibl…
Remarks on quantum groups
1991
We give a Poisson-bracket realization of SL q (2) in the phase space ℝ2. We then discuss the physical meaning of such a realization in terms of a modified (regularized) toy model, the nonregularized version of which is due to Klauder. Some general remarks and suggestions are also presented in this Letter.
Shear bond strength of ceramic bracket bonded to different surface-treated ceramic materials
2018
Background This study evaluated the effect of ceramic surface treatments on bond strength of ceramic brackets to machine-able ceramics and ceramic veneering metal. Material and Methods Machined ceramic specimens (10x10x1.5 mm) were prepared from Empress® CAD (EP), and e.max® CAD (EM). Ceramic veneering metal specimens (PF) were fabricated from sintered d.Sign® porcelain (1.27 mm thickness) over d.Sign®10 metal (0.23 mm thickness). Each ceramic was divided into 3-groups and treated surface by Er-YAG laser (LE) or etching with 9.6% HF acid for 5 seconds (A5) or 15 seconds (A15). Resin adhesive (Transbond™-XT) was used for attaching ceramic brackets for each group (n=15) and cured with LED (Bl…
Singular quadratic Lie superalgebras
2012
In this paper, we give a generalization of results in \cite{PU07} and \cite{DPU10} by applying the tools of graded Lie algebras to quadratic Lie superalgebras. In this way, we obtain a numerical invariant of quadratic Lie superalgebras and a classification of singular quadratic Lie superalgebras, i.e. those with a nonzero invariant. Finally, we study a class of quadratic Lie superalgebras obtained by the method of generalized double extensions.
Star calculus on Jacobi manifolds
2002
Abstract We study the Gerstenhaber bracket on differential forms induced by the two main examples of Jacobi manifolds: contact manifolds and l.c.s. manifolds. Moreover, we obtain explicit expressions of the generating operators and the derivations on the algebra of multivector fields. We define star operators for contact manifolds and l.c.s. manifolds and we study some of its properties.
The first Chevalley–Eilenberg Cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations
2010
Abstract In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle V k to a decreasing family of k foliations F i on a manifold M . We have shown that there exists a ( 1 , 1 ) tensor J of V k such that J k ≠ 0 , J k + 1 = 0 and we defined by L J ( V k ) the Lie Algebra of vector fields X on V k such that, for each vector field Y on V k , [ X , J Y ] = J [ X , Y ] . In this note, we study the first Chevalley–Eilenberg Cohomology Group, i.e. the quotient space of derivations of L J ( V k ) by the subspace of inner derivations, denoted by H 1 ( L J ( V k ) ) .
Some remarks concerning Nambu mechanics
1996
The structure of Nambu-Poisson brackets is studied and we establish that any Nambu tensor is decomposable. We show that every Nambu-Poisson manifold admits a local foliation by canonical Nambu-Poisson manifolds. Finally, a cohomology for Nambu (Lie) algebras which is adapted to the study of formal deformations of Nambu structures is introduced.