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.

Noether’s theorem for fractional variational problems of variable order

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Tatiana Odzijewicz ◽  
Agnieszka Malinowska ◽  
Delfim Torres
Keyword(s):  
Optimality Condition ◽  
Variational Problems ◽  
Time Variable ◽  
Variable Order ◽  
Necessary Optimality Condition ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Fractional Variational Problems ◽  
Derivatives Of

AbstractWe prove a necessary optimality condition of Euler-Lagrange type for fractional variational problems with derivatives of incommensurate variable order. This allows us to state a version of Noether’s theorem without transformation of the independent (time) variable. Considered derivatives of variable order are defined in the sense of Caputo.

scholarly journals Variational Problems with Partial Fractional Derivative: Optimal Conditions and Noether’s Theorem

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Jun Jiang ◽  
Yuqiang Feng ◽  
Shougui Li
Keyword(s):  
Fractional Derivative ◽  
Sufficient Conditions ◽  
Variational Problems ◽  
Necessary And Sufficient Conditions ◽  
Optimal Conditions ◽  
Lagrange Equations ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Necessary And Sufficient

In this paper, the necessary and sufficient conditions of optimality for variational problems with Caputo partial fractional derivative are established. Fractional Euler-Lagrange equations are obtained. The Legendre condition and Noether’s theorem are also presented.

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].

scholarly journals Noether’s theorem for non-smooth extremals of variational problems with time delay

2013 ◽  
Vol 93 (1) ◽  
pp. 153-170 ◽  
Author(s):  
Gastão S.F. Frederico ◽  
Tatiana Odzijewicz ◽  
Delfim F.M. Torres
Keyword(s):  
Time Delay ◽  
Variational Problems ◽  
Noether’S Theorem ◽  
Noether's Theorem

scholarly journals Noether's symmetry and conserved quantity for a time-delayed Hamiltonian system of Herglotz type

2018 ◽  
Vol 5 (10) ◽  
pp. 180208 ◽  
Author(s):  
Yi Zhang
Keyword(s):  
Time Delay ◽  
Phase Space ◽  
Hamiltonian System ◽  
Variational Problem ◽  
Conserved Quantity ◽  
Noether Symmetry ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Special Cases ◽  
The Relationship

The variational problem of Herglotz type and Noether's theorem for a time-delayed Hamiltonian system are studied. Firstly, the variational problem of Herglotz type with time delay in phase space is proposed, and the Hamilton canonical equations with time delay based on the Herglotz variational problem are derived. Secondly, by using the relationship between the non-isochronal variation and the isochronal variation, two basic formulae of variation of the Hamilton–Herglotz action with time delay in phase space are derived. Thirdly, the definition and criterion of the Noether symmetry for the time-delayed Hamiltonian system are established and the corresponding Noether's theorem is presented and proved. The theorem we obtained contains Noether's theorem of a time-delayed Hamiltonian system based on the classical variational problem and Noether's theorem of a Hamiltonian system based on the variational problem of Herglotz type as its special cases. At the end of the paper, an example is given to illustrate the application of the results.

scholarly journals Lanczos Approach to Noether’s Theorem

2014 ◽  
Vol 11 ◽  
pp. 1-3
Author(s):  
P. Lam-Estrada ◽  
José Luis Lopez-Bonilla ◽  
R. López-Vázquez
Keyword(s):  
Lagrange Equation ◽  
Lanczos Method ◽  
Conserved Quantity ◽  
Degree Of Freedom ◽  
Infinitesimal Transformation ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Constant Of Motion

If the action A=∫t1t2L(q,q,t)dt is invariant under the infinitesimal transformation t˜=t+ετ(q,t), q˜=qr+εζr(q,t), r-1,...,n with ε=constant≤1, then the Noether’s theorem permits to construct the corresponding conserved quantity. The Lanczos method accepts that ε=qn+1 is a new degree of freedom, thus the Euler-Lagrange equation for this new variable gives the Noether’s constant of motion.

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.

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