doi: 10.3934/jgm.2021012
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Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations

1. 

Laboratoire de Mathématiques et de leurs Applications, Université de Pau et des Pays de l'Adour, Pau, CNRS 5142, France

2. 

Departamento de Matemáticas Aplicadas a la Ingeniería Industrial, Universidad Politécnica de Madrid, José Gutiérrez Abascal 2, 28006 Madrid, Spain

3. 

Department of Mathematics, University of Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany

* Corresponding author: fernando.jimenez.alburquerque@upm.es

Received  February 2021 Early access July 2021

Fund Project: This work has been funded by the EPSRC project: "Fractional Variational Integration and Optimal Control"; ref: EP/P020402/1

We prove a Noether's theorem of the first kind for the so-called restricted fractional Euler-Lagrange equations and their discrete counterpart, introduced in [26,27], based in previous results [11,35]. Prior, we compare the restricted fractional calculus of variations to the asymmetric fractional calculus of variations, introduced in [14], and formulate the restricted calculus of variations using the discrete embedding approach [12,18]. The two theories are designed to provide a variational formulation of dissipative systems, and are based on modeling irreversbility by means of fractional derivatives. We explicit the role of time-reversed solutions and causality in the restricted fractional calculus of variations and we propose an alternative formulation. Finally, we implement our results for a particular example and provide simulations, actually showing the constant behaviour in time of the discrete conserved quantities outcoming the Noether's theorems.

Citation: Jacky Cresson, Fernando Jiménez, Sina Ober-Blöbaum. Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations. Journal of Geometric Mechanics, doi: 10.3934/jgm.2021012
References:
[1]

T. Abdeljawad and F. W. Atici, On the definition of Nabla fractional opetators, Abstract in Applied Analysis, 2012 (2012), Article ID 406757, 13pages. doi: 10.1155/2012/406757.  Google Scholar

[2]

R. Abraham and J. E. Marsden, Foundations of Mechanics, Benjamin-Cummings Publ. Co., 1978. Google Scholar

[3]

O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272 (2002), 368-379.  doi: 10.1016/S0022-247X(02)00180-4.  Google Scholar

[4]

T. T. AtanackovicS. KonjikS. Pilipovic and S. Simic., Variational problems with fractional derivatives: Invariance conditions and Nöther's theorem, Nonlinear Analysis, 71 (2009), 1504-1517.  doi: 10.1016/j.na.2008.12.043.  Google Scholar

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F. M. Atici and P. W. Eloe, Discrete fractional calculus with the nabla operator, Electronic Journal ofQualitative Theory of Differential Equations, 2009 (2009), 1-12.  doi: 10.14232/ejqtde.2009.4.3.  Google Scholar

[6]

N. R. O. BastosR. A. C. Ferreira and D. F. M. Torres, Discrete-time fractional variational problems, Signal Processing, 91 (2011), 513-524.  doi: 10.1016/j.sigpro.2010.05.001.  Google Scholar

[7]

H. Bateman, On Dissipative systems and Related Variational Principles, Phys. Rev., 38 (1931), 815. doi: 10.1103/PhysRev.38.815.  Google Scholar

[8]

P. S. Bauer, Dissipative dynamical systems, Proc. Nat. Acad. Sci., 17 (1931), 311-314.  doi: 10.1073/pnas.17.5.311.  Google Scholar

[9]

M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser Boston Inc., 2001. Google Scholar

[10]

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. Google Scholar

[11]

L. BourdinJ. Cresson and I. Greff, A continuous/discrete fractional Noether's theorem, Comun. Nonlinear Sci. Numer. Simulat., 18 (2013), 878-887.  doi: 10.1016/j.cnsns.2012.09.003.  Google Scholar

[12]

L. BourdinJ. CressonI. Greff and P. Inizan, Variational integrator for fractional Euler-Lagrange equations, Appl. Numer. Math., 71 (2013), 14-23.  doi: 10.1016/j.apnum.2013.03.003.  Google Scholar

[13]

M. C. Caputo and D. F. M. Torres, Duality for the left and right fractional derivatives, Signal Processing, 107 (2015), 265-271.  doi: 10.1016/j.sigpro.2014.09.026.  Google Scholar

[14]

J. Cresson and P. Inizan, Variational formulations of differential equations and asymmetric fractional embedding, Journal of Mathematical Analysis and Applications, 385 (2012), 975-997.  doi: 10.1016/j.jmaa.2011.07.022.  Google Scholar

[15]

J. Cresson, Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys., 48 (2007), 033504, 34 pages. doi: 10.1063/1.2483292.  Google Scholar

[16]

J. Cresson, Fractional variational embedding and Lagrangian formulations of dissiaptive partial differential equations, Fractional Calculus in Analysis, Dynamics and Optimal Control, Nova Publishers, New-York, (2013), 65–127. Google Scholar

[17]

J. Cresson, F. Jiménez and S. Ober-Blöbaum, Modeling of the convection-diffusion equation through fractional restricted calculus of variations, J. Nonlinear Sci., 31 (2021), Paper No. 46, 43 pp. doi: 10.1007/s00332-021-09700-w.  Google Scholar

[18]

J. Cresson and F. Pierret., Continuous versus discete structures Ⅰ: Discrete embeddings and ordinary differential equations, preprint, 2014, arXiv: 1411.7117. Google Scholar

[19]

J. Cresson and A. Szafraǹska, About the Noether's theorem for fractional Lagrangian systems and a generalization of the classical Jost method of proof, Fractional Calculus and Applied Analysis, 22 (2019), 871-898.  doi: 10.1515/fca-2019-0048.  Google Scholar

[20]

K. Diethelm, The Analysis of Fractional Differential Equations, Lect. Notes in Maths. Vol. 2004, Springer, 2010. Google Scholar

[21]

R. A. C. Ferreira and A. B. Malinowska, A counterexample to Frederico and Torres's fractional Noether-type theorem, J. Math. Anal. Appl., 429 (2015), 1370-1373.  doi: 10.1016/j.jmaa.2015.03.060.  Google Scholar

[22]

G. S. F. Frederico and 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.  doi: 10.1016/j.jmaa.2007.01.013.  Google Scholar

[23]

C. R. Galley, Classical mechanics of nonconservative systems, Phys. Rev. Lett., 110 (2013), 17430. doi: 10.1103/PhysRevLett.110.174301.  Google Scholar

[24]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, Vol. 31, second edition, 2006. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[25]

P. Inizan, Dynamique fractionnaire pour le chaos hamiltonien, Ph. D. Thesis, Observatoire de Paris, 2010. Google Scholar

[26]

F. Jiménez and S. Ober-Blöbaum, A fractional variational approach for modelling dissipative mechanical systems: continuous and discrete settings, IFAC-PapersOnLine, 51 (2018), 50-55.   Google Scholar

[27]

F. Jiménez and S. Ober-Blöbaum, Fractional damping through restricted calculus of variations, J. Nonlinear Sci., 31 (2021), Paper No. 46, 43 pp, arXiv: 1905.05608. doi: 10.1007/s00332-021-09700-w.  Google Scholar

[28]

R. Leone, On the wonderfulness of Noether's theorems, 100 years later, and Routh reduction, 2018., arXiv: 1804.01714. Google Scholar

[29]

C. Lubich, Convolution quadrature and discretized operational calculus Ⅱ, Numer. Math., 52 (1988), 413-425.  doi: 10.1007/BF01462237.  Google Scholar

[30]

C. Lubich, Convolution quadrature and discretized operational calculus Ⅰ, Numer. Math., 52 (1988), 129-145.  doi: 10.1007/BF01398686.  Google Scholar

[31]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.  Google Scholar

[32]

J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomial, Comm. Math. Phys., 139 (1991), 217-243.  doi: 10.1007/BF02352494.  Google Scholar

[33]

E. Noether, Invariante Variationsprobleme, Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math-phys. Klasse 2,235, the first english translation is due to M. A. Tavel [Transport Theor. Stat. 1,186 (1971)], (1918). Google Scholar

[34]

P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 2nd edition, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[35]

F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53 (1996), 1890-1899.  doi: 10.1103/PhysRevE.53.1890.  Google Scholar

[36]

S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Beach, Yverdon, 1993. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[37]

R. Utiyama, Invariant theoretical interpretation of interaction, Phys. Rev. Lett., 101 (1956), 1597-1607.  doi: 10.1103/PhysRev.101.1597.  Google Scholar

[38]

C. N. Yang and R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. Lett., 96 (1954), 191-195.  doi: 10.1103/PhysRev.96.191.  Google Scholar

show all references

References:
[1]

T. Abdeljawad and F. W. Atici, On the definition of Nabla fractional opetators, Abstract in Applied Analysis, 2012 (2012), Article ID 406757, 13pages. doi: 10.1155/2012/406757.  Google Scholar

[2]

R. Abraham and J. E. Marsden, Foundations of Mechanics, Benjamin-Cummings Publ. Co., 1978. Google Scholar

[3]

O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272 (2002), 368-379.  doi: 10.1016/S0022-247X(02)00180-4.  Google Scholar

[4]

T. T. AtanackovicS. KonjikS. Pilipovic and S. Simic., Variational problems with fractional derivatives: Invariance conditions and Nöther's theorem, Nonlinear Analysis, 71 (2009), 1504-1517.  doi: 10.1016/j.na.2008.12.043.  Google Scholar

[5]

F. M. Atici and P. W. Eloe, Discrete fractional calculus with the nabla operator, Electronic Journal ofQualitative Theory of Differential Equations, 2009 (2009), 1-12.  doi: 10.14232/ejqtde.2009.4.3.  Google Scholar

[6]

N. R. O. BastosR. A. C. Ferreira and D. F. M. Torres, Discrete-time fractional variational problems, Signal Processing, 91 (2011), 513-524.  doi: 10.1016/j.sigpro.2010.05.001.  Google Scholar

[7]

H. Bateman, On Dissipative systems and Related Variational Principles, Phys. Rev., 38 (1931), 815. doi: 10.1103/PhysRev.38.815.  Google Scholar

[8]

P. S. Bauer, Dissipative dynamical systems, Proc. Nat. Acad. Sci., 17 (1931), 311-314.  doi: 10.1073/pnas.17.5.311.  Google Scholar

[9]

M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser Boston Inc., 2001. Google Scholar

[10]

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. Google Scholar

[11]

L. BourdinJ. Cresson and I. Greff, A continuous/discrete fractional Noether's theorem, Comun. Nonlinear Sci. Numer. Simulat., 18 (2013), 878-887.  doi: 10.1016/j.cnsns.2012.09.003.  Google Scholar

[12]

L. BourdinJ. CressonI. Greff and P. Inizan, Variational integrator for fractional Euler-Lagrange equations, Appl. Numer. Math., 71 (2013), 14-23.  doi: 10.1016/j.apnum.2013.03.003.  Google Scholar

[13]

M. C. Caputo and D. F. M. Torres, Duality for the left and right fractional derivatives, Signal Processing, 107 (2015), 265-271.  doi: 10.1016/j.sigpro.2014.09.026.  Google Scholar

[14]

J. Cresson and P. Inizan, Variational formulations of differential equations and asymmetric fractional embedding, Journal of Mathematical Analysis and Applications, 385 (2012), 975-997.  doi: 10.1016/j.jmaa.2011.07.022.  Google Scholar

[15]

J. Cresson, Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys., 48 (2007), 033504, 34 pages. doi: 10.1063/1.2483292.  Google Scholar

[16]

J. Cresson, Fractional variational embedding and Lagrangian formulations of dissiaptive partial differential equations, Fractional Calculus in Analysis, Dynamics and Optimal Control, Nova Publishers, New-York, (2013), 65–127. Google Scholar

[17]

J. Cresson, F. Jiménez and S. Ober-Blöbaum, Modeling of the convection-diffusion equation through fractional restricted calculus of variations, J. Nonlinear Sci., 31 (2021), Paper No. 46, 43 pp. doi: 10.1007/s00332-021-09700-w.  Google Scholar

[18]

J. Cresson and F. Pierret., Continuous versus discete structures Ⅰ: Discrete embeddings and ordinary differential equations, preprint, 2014, arXiv: 1411.7117. Google Scholar

[19]

J. Cresson and A. Szafraǹska, About the Noether's theorem for fractional Lagrangian systems and a generalization of the classical Jost method of proof, Fractional Calculus and Applied Analysis, 22 (2019), 871-898.  doi: 10.1515/fca-2019-0048.  Google Scholar

[20]

K. Diethelm, The Analysis of Fractional Differential Equations, Lect. Notes in Maths. Vol. 2004, Springer, 2010. Google Scholar

[21]

R. A. C. Ferreira and A. B. Malinowska, A counterexample to Frederico and Torres's fractional Noether-type theorem, J. Math. Anal. Appl., 429 (2015), 1370-1373.  doi: 10.1016/j.jmaa.2015.03.060.  Google Scholar

[22]

G. S. F. Frederico and 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.  doi: 10.1016/j.jmaa.2007.01.013.  Google Scholar

[23]

C. R. Galley, Classical mechanics of nonconservative systems, Phys. Rev. Lett., 110 (2013), 17430. doi: 10.1103/PhysRevLett.110.174301.  Google Scholar

[24]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, Vol. 31, second edition, 2006. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[25]

P. Inizan, Dynamique fractionnaire pour le chaos hamiltonien, Ph. D. Thesis, Observatoire de Paris, 2010. Google Scholar

[26]

F. Jiménez and S. Ober-Blöbaum, A fractional variational approach for modelling dissipative mechanical systems: continuous and discrete settings, IFAC-PapersOnLine, 51 (2018), 50-55.   Google Scholar

[27]

F. Jiménez and S. Ober-Blöbaum, Fractional damping through restricted calculus of variations, J. Nonlinear Sci., 31 (2021), Paper No. 46, 43 pp, arXiv: 1905.05608. doi: 10.1007/s00332-021-09700-w.  Google Scholar

[28]

R. Leone, On the wonderfulness of Noether's theorems, 100 years later, and Routh reduction, 2018., arXiv: 1804.01714. Google Scholar

[29]

C. Lubich, Convolution quadrature and discretized operational calculus Ⅱ, Numer. Math., 52 (1988), 413-425.  doi: 10.1007/BF01462237.  Google Scholar

[30]

C. Lubich, Convolution quadrature and discretized operational calculus Ⅰ, Numer. Math., 52 (1988), 129-145.  doi: 10.1007/BF01398686.  Google Scholar

[31]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.  Google Scholar

[32]

J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomial, Comm. Math. Phys., 139 (1991), 217-243.  doi: 10.1007/BF02352494.  Google Scholar

[33]

E. Noether, Invariante Variationsprobleme, Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math-phys. Klasse 2,235, the first english translation is due to M. A. Tavel [Transport Theor. Stat. 1,186 (1971)], (1918). Google Scholar

[34]

P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 2nd edition, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[35]

F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53 (1996), 1890-1899.  doi: 10.1103/PhysRevE.53.1890.  Google Scholar

[36]

S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Beach, Yverdon, 1993. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[37]

R. Utiyama, Invariant theoretical interpretation of interaction, Phys. Rev. Lett., 101 (1956), 1597-1607.  doi: 10.1103/PhysRev.101.1597.  Google Scholar

[38]

C. N. Yang and R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. Lett., 96 (1954), 191-195.  doi: 10.1103/PhysRev.96.191.  Google Scholar

Figure 1.  Top-Left: $ \rho = 0 $; this is the classical dynamics of the non-fractional harmonic oscillator (without dissipation). Top-Right: $ \rho = 0.2 $, $ \alpha = 0.2 $. Bottom-Left: $ \rho = 0.2 $, $ \alpha = 0.4 $. Bottom-Right: $ \rho = 0.2 $, $ \alpha = 0.5 $; this is the case of the usual linearly damped harmonic oscillator. Naturally, we observe that the dissipation increases as $ \alpha $ increases for equivalent $ \rho $
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