Zum Hauptinhalt springen
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

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

  • EMAIL logo und
Veröffentlicht/Copyright: 23. Oktober 2019
Veröffentlichen auch Sie bei De Gruyter Brill
Informationen für Autor*innen Erkunden Sie dieses Fachgebiet
Fractional Calculus and Applied Analysis
Aus der Zeitschrift Fractional Calculus and Applied Analysis Band 22 Heft 4

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].

MSC 2010: Primary 26A33; Secondary 34A08; 70H03
Key Words and Phrases: Euler-Lagrange equations; Noether’s theorem; fractional calculus; symmetries

Appendix A. Note on numerical solving of the Euler-Lagrange equation

In order to obtain approximate solution for the Euler-Lagrange equation (4.39) we convert this equation into the integral form. First let us formulate useful composition rules between fractional operators according to Definitions 2.1 and 2.2 (see [5]):

Lemma A.1

Letα ∈ (0, 1) andx ∈ AC([a, b], ℝn), then the following relations

  • Ia+α∘cDa+αx=x−x(a),

  • Ia+α∘Da+αx=x

are satisfied almost everywhere.

In the case of the right operators the counterparts of this rules are also valid. According to the definitions of fractional integrals (2.3) and (2.4) we can conclude that for every constant C ∈ ℝ we have

Ia+αC=(t−a)αΓ(1+α)C,Ib−αC=(b−t)αΓ(1+α)C.(1.8)

Now, the integral form of the Euler-Lagrange equation (4.34):

x(t)+Ia+α∘Ib−αx(t)−t−abαIa+α∘Ib−αx(t)t=b=1−t−abαx(a)+t−abαx(b)(1.9)

can be easily derived based on (1.8), the relations between derivatives (2.7) and the composition rules defined in Lemma A.1. Note that, if we put α = 1 in (1.9), we obtain the integral form of the equation ẍ = x.

For the purpose of discretization of the integral equation (1.9) we define the equidistant partition on [a, b] : h = (b – a)/N, tk = a + kh, for k = 0, …, N, N ∈ ℕ. On the subinterval [ti, ti+1] we substitute the function f by the arithmetic average of values f(ti) and f(ti+1). We derive the approximations of the integrals:

Ia+αf(t)|t=tk=1Γ(α)∑i=0k−1∫titi+1f(s)(tk−s)1−αds≈1Γ(α)∑i=0k−1∫titi+11(tk−s)1−αf(ti)+f(ti+1)2ds=:hIa+αf(tk)(1.10)

for k = 1, …, N, and

Ib−αf(t)|t=tk=1Γ(α)∑i=kN−1∫titi+1f(s)(s−tk)1−αds≈1Γ(α)∑i=kN−1∫titi+11(s−tk)1−αf(ti)+f(ti+1)2ds=:hIb−αf(tk)(1.11)

for k = 0, …, N – 1, where the sub-integrals can be directly calculated. Then we obtain the following algebraic system of equations

X0=x(a),Xk+hIa+α∘hIb−αXk−tk−abαhIa+α∘hIb−αXN=1−tk−abαX0+tk−abαXN,XN=x(b)(1.12)

which gives an approximate solution of the Euler-Lagrange equation (4.39).

References

[1] O.P. Agrawal, Formulation of Euler–Lagrangee quations for fractional variational problems. J. Math. Anal. Appl. 272, No 1 (2002), 368–379.10.1016/S0022-247X(02)00180-4Suche in Google Scholar

[2] B. Anerot, J. Cresson, F. Pierret, About the time-scale Noether’s theorem. arXiv:1609.02698, 12 pp., 2016.Suche in Google Scholar

[3] T.M. Atanackovic, S. Konjik, S. Pilipovic, S. Simic, Variational problems with fractional derivatives: invariance conditions and Nöther’s theorem. Nonlinear Analysis71 (2009), 1504–1517.10.1016/j.na.2008.12.043Suche in Google Scholar

[4] Z. Bartosiewicz and D.F.M. Torres, Noether’s theorem on time scales. J. of Math. Anal. Appl. 342, No 2 (2008), 1220–1226.10.1016/j.jmaa.2008.01.018Suche in Google Scholar

[5] L. Bourdin, Contributions au calcul des variations et au principe du maximum de Pontryagin en calculs time scale et fractionnaire, Ph.D. Thesis, University of Pau and Pays de l’Adour, 2013.Suche in Google Scholar

[6] J. Cresson, A. Szafrańska, Comments on various extensions of the Riemann-Liouville fractional derivatives: About the Leibniz and chain rule properties. Commun. in Nonlin. Sci. and Numer. Simul., Available online 24 July 2019, # 104903; 10.1016/j.cnsns.2019.104903.Suche in Google Scholar

[7] R.A.C. Ferreira, A.B. Malinowska, A counterexample to Frederico and Torres’s fractional Noether-type theorem. J. Math. Anal. Appl. 429, No 2 (2015), 1370–1373.10.1016/j.jmaa.2015.03.060Suche in Google Scholar

[8] G.S.F. Frederico, M.J. Lazo, Fractional Noether’s theorem with classical and Caputo derivatives: constants of motion for non-conservative systems. Nonlinear Dynam. 85, No 2 (2016), 839—851.10.1007/s11071-016-2727-zSuche in Google Scholar

[9] G.S.F. Frederico, D.F.M. Torres, A formulation of Noether’s theorem for fractional problems of the calculus of variations. J. Math. Anal. Appl. 334 (2007), 834–846.10.1016/j.jmaa.2007.01.013Suche in Google Scholar

[10] G.S.F. Frederico, D.F.M. Torres, Fractional Noether’s theorem in the Riesz-Caputo sense. Appl. Math. and Comput. 217, No 3 (2010), 1023–1033.10.1016/j.amc.2010.01.100Suche in Google Scholar

[11] L. Bourdin, J. Cresson, I. Greff, A continuous/discrete fractional Noether theorem, Commun. Nonl. Sci. Numer. Simul. 18, No 4 (2013), 878–887.10.1016/j.cnsns.2012.09.003Suche in Google Scholar

[12] J. Jost and X. Li-Jost, Calculus of Variations. Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1998.Suche in Google Scholar

[13] P.J. Olver, Applications of Lie Groups to Differential Equations. 2d Ed., Graduate Textes in Mathematics, Springer-Verlag, 1993.10.1007/978-1-4612-4350-2Suche in Google Scholar

Received: 2018-08-12
Published Online: 2019-10-23
Published in Print: 2019-08-27

© 2019 Diogenes Co., Sofia

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–Volume 22–4–2019)
  4. Research Paper
  5. Fractional equations via convergence of forms
  6. About the Noether’s theorem for fractional Lagrangian systems and a generalization of the classical Jost method of proof
  7. Survey Paper
  8. The vertical slice transform on the unit sphere
  9. Research Paper
  10. Well-posedness of time-fractional advection-diffusion-reaction equations
  11. Existence of solutions for nonlinear fractional order p-Laplacian differential equations via critical point theory
  12. Exact asymptotic formulas for the heat kernels of space and time-fractional equations
  13. Estimates of damped fractional wave equations
  14. A modified time-fractional diffusion equation and its finite difference method: Regularity and error analysis
  15. On the fractional diffusion-advection-reaction equation in ℝ
  16. Controllability of nonlinear stochastic fractional higher order dynamical systems
  17. Approximate controllability and existence of mild solutions for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 < α < 2
  18. Analysis of fractional order error models in adaptive systems: Mixed order cases
  19. Fractional calculus of variations: a novel way to look at it
  20. Pricing of perpetual American put option with sub-mixed fractional Brownian motion
https://doi.org/10.1515/fca-2019-0048
Schlagwörter für diesen Artikel
Euler-Lagrange equations; Noether’s theorem; fractional calculus; symmetries

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–Volume 22–4–2019)
  4. Research Paper
  5. Fractional equations via convergence of forms
  6. About the Noether’s theorem for fractional Lagrangian systems and a generalization of the classical Jost method of proof
  7. Survey Paper
  8. The vertical slice transform on the unit sphere
  9. Research Paper
  10. Well-posedness of time-fractional advection-diffusion-reaction equations
  11. Existence of solutions for nonlinear fractional order p-Laplacian differential equations via critical point theory
  12. Exact asymptotic formulas for the heat kernels of space and time-fractional equations
  13. Estimates of damped fractional wave equations
  14. A modified time-fractional diffusion equation and its finite difference method: Regularity and error analysis
  15. On the fractional diffusion-advection-reaction equation in ℝ
  16. Controllability of nonlinear stochastic fractional higher order dynamical systems
  17. Approximate controllability and existence of mild solutions for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 < α < 2
  18. Analysis of fractional order error models in adaptive systems: Mixed order cases
  19. Fractional calculus of variations: a novel way to look at it
  20. Pricing of perpetual American put option with sub-mixed fractional Brownian motion
Heruntergeladen am 16.4.2026 von https://www.degruyterbrill.com/document/doi/10.1515/fca-2019-0048/html
Button zum nach oben scrollen