Infinitesimal Deformations of ECD Fields

2020 ◽  
pp. 147-154
Author(s):  
Daniel Canarutto
Keyword(s):  
Lie Derivative ◽  
Dirac Fields ◽  
Lie Derivatives ◽  
Infinitesimal Deformations ◽  
Standard Notion ◽  
Derivatives Of

The standard notion of Lie derivative is extended in order to include Lie derivatives of spinors, soldering forms, spinor connections and spacetime connections. These extensions are all linked together, and provide a natural framework for discussing infinitesimal deformations of Einstein-Cartan-Dirac fields in the tetrad-affine setting.

scholarly journals Spinor Lie derivatives and Fermion stress–energies

2016 ◽  
Vol 472 (2186) ◽  
pp. 20150757
Author(s):  
A. D. Helfer
Keyword(s):  
Discrete Symmetries ◽  
Vector Fields ◽  
Lie Derivative ◽  
General Covariance ◽  
Arbitrary Vector ◽  
Lie Derivatives ◽  
Spinor Fields ◽  
Stress Energy ◽  
Derivatives Of ◽  
Apparent Conflict

Stress–energies for Fermi fields are derived from the principle of general covariance. This is done by developing a notion of Lie derivatives of spinors along arbitrary vector fields. A substantial theory of such derivatives was first introduced by Kosmann; here, I show how an apparent conflict in the literature on this is due to a difference in the definitions of spinors, and that tracking the Lie derivative of the Infeld–van der Waerden symbol, as well as the spinor fields under consideration, gives a fuller picture of the geometry and leads to the Fermion stress–energy. The differences in the definitions of spinors do not affect the results here, but could matter in certain quantum-gravity programs and for spinor transformations under discrete symmetries.

scholarly journals Lie derivatives of sectorform fields

1988 ◽  
Vol 55 (1) ◽  
pp. 71-78 ◽  
Author(s):  
Ivan Kolář
Keyword(s):  
Lie Derivatives ◽  
Derivatives Of

A practical form of Lagrange–Hamilton theory for ideal fluids and plasmas

2003 ◽  
Vol 69 (3) ◽  
pp. 211-252 ◽  
Author(s):  
JONAS LARSSON
Keyword(s):  
Differential Geometry ◽  
Classical Mechanics ◽  
Vector Analysis ◽  
Lie Derivative ◽  
Explicit Result ◽  
Lie Derivatives ◽  
Vector Calculus ◽  
Starting Point ◽  
Ideal Fluids ◽  
Standard Vector

Lagrangian and Hamiltonian formalisms for ideal fluids and plasmas have, during the last few decades, developed much in theory and applications. The recent formulation of ideal fluid/plasma dynamics in terms of the Euler–Poincaré equations makes a self-contained, but mathematically elementary, form of Lagrange–Hamilton theory possible. The starting point is Hamilton's principle. The main goal is to present Lagrange–Hamilton theory in a way that simplifies its applications within usual fluid and plasma theory so that we can use standard vector analysis and standard Eulerian fluid variables. The formalisms of differential geometry, Lie group theory and dual spaces are avoided and so is the use of Lagrangian fluid variables or Clebsch potentials. In the formal ‘axiomatic’ setting of Lagrange–Hamilton theory the concepts of Lie algebra and Hilbert space are used, but only in an elementary way. The formalism is manifestly non-canonical, but the analogy with usual classical mechanics is striking. The Lie derivative is a most convenient tool when the abstract Lagrange–Hamilton formalism is applied to concrete fluid/plasma models. This directional/dynamical derivative is usually defined within differential geometry. However, following the goals of this paper, we choose to define Lie derivatives within standard vector analysis instead (in terms of the directional field and the div, grad and curl operators). Basic identities for the Lie derivatives, necessary for using them effectively in vector calculus and Lagrange–Hamilton theory, are included. Various dynamical invariants, valid for classes of fluid and plasma models (including both compressible and incompressible ideal magnetohydrodynamics), are given simple and straightforward derivations thanks to the Lie derivative calculus. We also consider non-canonical Poisson brackets and derive, in particular, an explicit result for incompressible and inhomogeneous flows.

scholarly journals Infinitesimal deformations of basic tensor in generalized Riemannian space

Filomat ◽  
2007 ◽  
Vol 21 (2) ◽  
pp. 235-242 ◽  
Author(s):  
Ljubica Velimirovic ◽  
S.M. Mincic ◽  
M.S. Stankovic
Keyword(s):  
Riemannian Space ◽  
Lie Derivatives ◽  
Generalized Riemannian Space ◽  
Infinitesimal Deformations ◽  
Basic Facts

At the beginning of the present work the basic facts on generalized Riemannian space (GRn) in the sense of Eisenhart's definition [Eis] and also on infinitesimal deformations of a space are given. We study the Lie derivatives and infinitesimal deformations of basic covariant and contravariant tensor at GRn.

scholarly journals Lie Derivatives of the Shape Operator of a Real Hypersurface in a Complex Projective Space

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Juan de Dios Pérez ◽  
David Pérez-López
Keyword(s):  
Differential Operator ◽  
Projective Space ◽  
Tensor Field ◽  
Complex Projective Space ◽  
Shape Operator ◽  
Symmetric Tensor ◽  
Real Hypersurfaces ◽  
Lie Derivative ◽  
Complex Projective ◽  
Derivatives Of

AbstractWe consider real hypersurfaces M in complex projective space equipped with both the Levi-Civita and generalized Tanaka–Webster connections. Associated with the generalized Tanaka–Webster connection we can define a differential operator of first order. For any nonnull real number k and any symmetric tensor field of type (1,1) B on M, we can define a tensor field of type (1,2) on M, $$B^{(k)}_T$$ B T ( k ) , related to Lie derivative and such a differential operator. We study symmetry and skew symmetry of the tensor $$A^{(k)}_T$$ A T ( k ) associated with the shape operator A of M.

scholarly journals Deforming 𝔥-trivial the Lie algebra Vect(S1) inside the Lie algebra of pseudodifferential operators Ψ𝒟𝒪

2017 ◽  
Vol 14 (06) ◽  
pp. 1750082 ◽  
Author(s):  
Imed Basdouri ◽  
Issam Bartouli ◽  
Jean Lerbet
Keyword(s):  
Lie Algebra ◽  
Cotangent Bundle ◽  
Vector Fields ◽  
Pseudodifferential Operators ◽  
Sufficient Conditions ◽  
Lie Derivative ◽  
Smooth Vector ◽  
Algebra Of Functions ◽  
Infinitesimal Deformations ◽  
Necessary And Sufficient

In this paper, we consider the action of Vect(S1) by Lie derivative on the spaces of pseudodifferential operators [Formula: see text]. We study the [Formula: see text]-trivial deformations of the standard embedding of the Lie algebra Vect(S1) of smooth vector fields on the circle, into the Lie algebra of functions on the cotangent bundle [Formula: see text]. We classify the deformations of this action that become trivial once restricted to [Formula: see text], where [Formula: see text] or [Formula: see text]. Necessary and sufficient conditions for integrability of infinitesimal deformations are given.

scholarly journals Mean-field sparse Jurdjevic–Quinn control

2017 ◽  
Vol 27 (07) ◽  
pp. 1223-1253 ◽  
Author(s):  
Marco Caponigro ◽  
Benedetto Piccoli ◽  
Francesco Rossi ◽  
Emmanuel Trélat
Keyword(s):  
Lyapunov Function ◽  
Control Function ◽  
Kinetic Equations ◽  
Mean Field ◽  
Transport Equations ◽  
Control Law ◽  
Time Varying ◽  
Nonlinear Transport ◽  
Lie Derivatives ◽  
Derivatives Of

We consider nonlinear transport equations with non-local velocity describing the time-evolution of a measure. Such equations often appear when considering the mean-field limit of finite-dimensional systems modeling collective dynamics. We address the problem of controlling these equations by means of a time-varying bounded control action localized on a time-varying control subset of small Lebesgue measure. We first define dissipativity for nonlinear transport equations in terms of Lie derivatives of a Lyapunov function depending on the measure. Then, assuming that the uncontrolled system is dissipative, we provide an explicit construction of a control law steering the system to an invariant sublevel of the Lyapunov function. The control function and the control domain are designed in terms of the Lie derivatives of the Lyapunov function. In this sense the construction can be seen as an infinite-dimensional analogue of the well-known Jurdjevic–Quinn procedure. Moreover, the control law presents sparsity properties in the sense that the support of the control is small. Finally, we show that our result applies to a large class of kinetic equations modeling multi-agent dynamics.

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