scholarly journals About the Noether’s theorem for fractional Lagrangian systems and a generalization of the classical Jost method of proof

2019 ◽  
Vol 22 (4) ◽  
pp. 871-898 ◽  
Author(s):  
Jacky Cresson ◽  
Anna Szafrańska
Keyword(s):  
Local Group ◽  
Type Theorem ◽  
Classical Case ◽  
Alternative Proof ◽  
Lagrangian Systems ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Definition Of ◽  
Variational Symmetries ◽  
Discrete Setting

Abstract Recently, the fractional Noether’s theorem derived by G. Frederico and D.F.M. Torres in [10] was proved to be wrong by R.A.C. Ferreira and A.B. Malinowska in (see [7]) using a counterexample and doubts are stated about the validity of other Noether’s type Theorem, in particular ([9], Theorem 32). However, the counterexample does not explain why and where the proof given in [10] does not work. In this paper, we make a detailed analysis of the proof proposed by G. Frederico and D.F.M. Torres in [9] which is based on a fractional generalization of a method proposed by J. Jost and X.Li-Jost in the classical case. This method is also used in [10]. We first detail this method and then its fractional version. Several points leading to difficulties are put in evidence, in particular the definition of variational symmetries and some properties of local group of transformations in the fractional case. These difficulties arise in several generalization of the Jost’s method, in particular in the discrete setting. We then derive a fractional Noether’s Theorem following this strategy, correcting the initial statement of Frederico and Torres in [9] and obtaining an alternative proof of the main result of Atanackovic and al. [3].

SUPERFIELDS AND CANONICAL METHODS IN SUPERSPACE

1986 ◽  
Vol 01 (04) ◽  
pp. 293-302 ◽  
Author(s):  
J.A. DE AZCÁRRAGA ◽  
J. LUKIERSKI ◽  
P. VINDEL
Keyword(s):  
Quantum Mechanics ◽  
Evolution Equation ◽  
Supersymmetric Quantum Mechanics ◽  
Noether’S Theorem ◽  
Equal Time ◽  
Heisenberg Picture ◽  
Noether's Theorem ◽  
Time Limits ◽  
Definition Of

We consider the “supersymmetric roots” of the Heisenberg evolution equation as describing the dynamics of superfields in superspace. We investigate the superfield commutators and their equal time limits and exhibit their noncanonical character even for free superfields. For simplicity, we concentrate on the D=1 case, i.e., the superfield formulation of supersymmetric quantum mechanics in the Heisenberg picture and, as a soluble example, the supersymmetric oscillator. Finally, we express Noether’s theorem in superspace and give the definition of the global conserved supercharges.

scholarly journals Lie algebra classification, conservation laws and invariant solutions for modification of the generalization of the Emden--Fowler equation.

2021 ◽  
Author(s):  
Yeisson Acevedo Agudelo ◽  
Gabriel Loaiza Ossa ◽  
Oscar Londoño Duque ◽  
Danilo García Hernández
Keyword(s):  
Conservation Laws ◽  
Lie Algebra ◽  
Invariant Solutions ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Generating Operators ◽  
Variational Symmetries ◽  
The Given ◽  
Lie Algebra Classification ◽  
Fowler Equation

We obtain the optimal system’s generating operators associated to a modification of the generalization of the Emden–Fowler Equation. equation. Using those operators we characterize all invariant solutions associated to a generalized. Moreover, we present the variational symmetries and the corresponding conservation laws, using Noether’s theorem and Ibragimov’s method. Finally, we classify the Lie algebra associated to the given equation.

scholarly journals Restoration of the symmetries broken by reversible growth in hyperelastic bodies

2003 ◽  
pp. 311-331 ◽  
Author(s):  
Alfio Grillo ◽  
Salvatore Federico ◽  
Gaetano Giaquinta ◽  
Walter Herzog ◽  
Rosa La
Keyword(s):  
The Body ◽  
Broken Symmetry ◽  
Reference Configuration ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Broken Symmetries ◽  
Elastic Bodies ◽  
Theoretical Investigations ◽  
Hyper Elastic ◽  
Definition Of

In this paper, we interpret the development of material in homogeneities in continuum, hyper elastic bodies in the presence of reversible growth in terms of broken symmetries [1]. By applying Noether's Theorem [1, 2, 3, 4], we find a set of equations yielding the fields necessary to compensate for the broken symmetry. As growth occurs, these fields provide for an instantaneously updated reference configuration of the body, and are responsible for the dynamical restoring of the body symmetries. In addition, we propose to use these compensating fields in order to generalize the definition of the transplant operator given in [5,6]. This work has been motivated by the current theoretical investigations on the biomechanical aspects of growth in particular cartilage.

Noether’s Symmetries and Its Inverse for Fractional Logarithmic Lagrangian Systems

2019 ◽  
Vol 7 (1) ◽  
pp. 90-98 ◽  
Author(s):  
Jun Jiang ◽  
Yuqiang Feng ◽  
Shuli Xu
Keyword(s):  
Variational Problems ◽  
Conserved Quantity ◽  
Inverse Theorem ◽  
Lagrangian Systems ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Hamilton Principle ◽  
Fractional Variational Problems

Abstract In this paper, Noether’s theorem and its inverse theorem are proved for the fractional variational problems based on logarithmic Lagrangian systems. The Hamilton principle of the systems is derived. And the definitions and the criterions of Noether’s symmetry and Noether’s quasi-symmetry of the systems based on logarithmic Lagrangians are given. The intrinsic relation between Noether’s symmetry and the conserved quantity is established. At last an example is given to illustrate the application of the results.

CURRENT OPERATORS IN THE LOWEST LANDAU LEVEL

1993 ◽  
Vol 07 (26) ◽  
pp. 4389-4401 ◽  
Author(s):  
JUAN MARTÍNEZ ◽  
MICHAEL STONE
Keyword(s):  
Landau Level ◽  
Current Operator ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Lowest Landau Level ◽  
Definition Of

We use Noether’s theorem to generate a consistent definition of the current operator for electrons restricted to the lowest Landau level. We exhibit the connection between this current and the Moyal bracket, or W∞, algebra, and use it to derive the edge-charge algebra for the ν=1/(2n+1) FQHE states.

Introduction

2017 ◽  
Author(s):  
Laurent Baulieu ◽  
John Iliopoulos ◽  
Roland Sénéor
Keyword(s):  
Conservation Laws ◽  
Lagrangian Mechanics ◽  
General Introduction ◽  
Noether’S Theorem ◽  
Noether's Theorem

General introduction with a review of the principles of Hamiltonian and Lagrangian mechanics. The connection between symmetries and conservation laws, with a presentation of Noether’s theorem, is included.

scholarly journals Stochastic fractal and Noether's theorem

2021 ◽  
Vol 103 (2) ◽  
Author(s):  
Rakibur Rahman ◽  
Fahima Nowrin ◽  
M. Shahnoor Rahman ◽  
Jonathan A. D. Wattis ◽  
Md. Kamrul Hassan
Keyword(s):  
Noether’S Theorem ◽  
Noether's Theorem

scholarly journals Conditional Rényi Entropy and the Relationships between Rényi Capacities

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 526
Author(s):  
Gautam Aishwarya ◽  
Mokshay Madiman
Keyword(s):  
Mutual Information ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Basic Properties ◽  
Reference Measure ◽  
Definition Of ◽  
Discrete Setting

The analogues of Arimoto’s definition of conditional Rényi entropy and Rényi mutual information are explored for abstract alphabets. These quantities, although dependent on the reference measure, have some useful properties similar to those known in the discrete setting. In addition to laying out some such basic properties and the relations to Rényi divergences, the relationships between the families of mutual informations defined by Sibson, Augustin-Csiszár, and Lapidoth-Pfister, as well as the corresponding capacities, are explored.

Hardy spaces on ZN

2002 ◽  
Vol 132 (1) ◽  
pp. 25-43 ◽  
Author(s):  
Santiago Boza ◽  
María J. Carro
Keyword(s):  
Maximal Function ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Classical Case ◽  
Positive Answer ◽  
Riesz Transforms ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Definition Of ◽  
Open Question

The work of Coifman and Weiss concerning Hardy spaces on spaces of homogeneous type gives, as a particular case, a definition of Hp(ZN) in terms of an atomic decomposition.Other characterizations of these spaces have been studied by other authors, but it was an open question to see if they can be defined, as it happens in the classical case, in terms of a maximal function or via the discrete Riesz transforms.In this paper, we give a positive answer to this question.

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