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March  2022, 14(1): 57-89. doi: 10.3934/jgm.2021012

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 Published  March 2022 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, 2022, 14 (1) : 57-89. 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. [2] R. Abraham and J. E. Marsden, Foundations of Mechanics, Benjamin-Cummings Publ. Co., 1978. [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. [4] T. T. Atanackovic, S. Konjik, S. 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. [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. [6] N. R. O. Bastos, R. 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. [7] H. Bateman, On Dissipative systems and Related Variational Principles, Phys. Rev., 38 (1931), 815. doi: 10.1103/PhysRev.38.815. [8] P. S. Bauer, Dissipative dynamical systems, Proc. Nat. Acad. Sci., 17 (1931), 311-314.  doi: 10.1073/pnas.17.5.311. [9] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser Boston Inc., 2001. [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. [11] L. Bourdin, J. 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. [12] L. Bourdin, J. Cresson, I. 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. [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. [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. [15] J. Cresson, Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys., 48 (2007), 033504, 34 pages. doi: 10.1063/1.2483292. [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. [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. [18] J. Cresson and F. Pierret., Continuous versus discete structures Ⅰ: Discrete embeddings and ordinary differential equations, preprint, 2014, arXiv: 1411.7117. [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. [20] K. Diethelm, The Analysis of Fractional Differential Equations, Lect. Notes in Maths. Vol. 2004, Springer, 2010. [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. [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. [23] C. R. Galley, Classical mechanics of nonconservative systems, Phys. Rev. Lett., 110 (2013), 17430. doi: 10.1103/PhysRevLett.110.174301. [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. [25] P. Inizan, Dynamique fractionnaire pour le chaos hamiltonien, Ph. D. Thesis, Observatoire de Paris, 2010. [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. [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. [28] R. Leone, On the wonderfulness of Noether's theorems, 100 years later, and Routh reduction, 2018., arXiv: 1804.01714. [29] C. Lubich, Convolution quadrature and discretized operational calculus Ⅱ, Numer. Math., 52 (1988), 413-425.  doi: 10.1007/BF01462237. [30] C. Lubich, Convolution quadrature and discretized operational calculus Ⅰ, Numer. Math., 52 (1988), 129-145.  doi: 10.1007/BF01398686. [31] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.  doi: 10.1017/S096249290100006X. [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. [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). [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. [35] F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53 (1996), 1890-1899.  doi: 10.1103/PhysRevE.53.1890. [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. [37] R. Utiyama, Invariant theoretical interpretation of interaction, Phys. Rev. Lett., 101 (1956), 1597-1607.  doi: 10.1103/PhysRev.101.1597. [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.

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. [2] R. Abraham and J. E. Marsden, Foundations of Mechanics, Benjamin-Cummings Publ. Co., 1978. [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. [4] T. T. Atanackovic, S. Konjik, S. 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. [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. [6] N. R. O. Bastos, R. 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. [7] H. Bateman, On Dissipative systems and Related Variational Principles, Phys. Rev., 38 (1931), 815. doi: 10.1103/PhysRev.38.815. [8] P. S. Bauer, Dissipative dynamical systems, Proc. Nat. Acad. Sci., 17 (1931), 311-314.  doi: 10.1073/pnas.17.5.311. [9] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser Boston Inc., 2001. [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. [11] L. Bourdin, J. 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. [12] L. Bourdin, J. Cresson, I. 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. [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. [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. [15] J. Cresson, Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys., 48 (2007), 033504, 34 pages. doi: 10.1063/1.2483292. [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. [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. [18] J. Cresson and F. Pierret., Continuous versus discete structures Ⅰ: Discrete embeddings and ordinary differential equations, preprint, 2014, arXiv: 1411.7117. [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. [20] K. Diethelm, The Analysis of Fractional Differential Equations, Lect. Notes in Maths. Vol. 2004, Springer, 2010. [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. [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. [23] C. R. Galley, Classical mechanics of nonconservative systems, Phys. Rev. Lett., 110 (2013), 17430. doi: 10.1103/PhysRevLett.110.174301. [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. [25] P. Inizan, Dynamique fractionnaire pour le chaos hamiltonien, Ph. D. Thesis, Observatoire de Paris, 2010. [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. [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. [28] R. Leone, On the wonderfulness of Noether's theorems, 100 years later, and Routh reduction, 2018., arXiv: 1804.01714. [29] C. Lubich, Convolution quadrature and discretized operational calculus Ⅱ, Numer. Math., 52 (1988), 413-425.  doi: 10.1007/BF01462237. [30] C. Lubich, Convolution quadrature and discretized operational calculus Ⅰ, Numer. Math., 52 (1988), 129-145.  doi: 10.1007/BF01398686. [31] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.  doi: 10.1017/S096249290100006X. [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. [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). [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. [35] F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53 (1996), 1890-1899.  doi: 10.1103/PhysRevE.53.1890. [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. [37] R. Utiyama, Invariant theoretical interpretation of interaction, Phys. Rev. Lett., 101 (1956), 1597-1607.  doi: 10.1103/PhysRev.101.1597. [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.
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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