• Previous Article
    Compatibility aspects of the method of phase synchronization for decoupling linear second-order differential equations
  • JGM Home
  • This Issue
  • Next Article
    Control and maintenance of fully-constrained and underconstrained rigid body motion on Lie groups and their tangent bundles
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. 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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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.

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 $
[1]

Delfim F. M. Torres. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 491-500. doi: 10.3934/cpaa.2004.3.491

[2]

Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417

[3]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether's theorem for higher-order variational problems of Herglotz type. Conference Publications, 2015, 2015 (special) : 990-999. doi: 10.3934/proc.2015.990

[4]

Colin J. Cotter, Michael John Priestley Cullen. Particle relabelling symmetries and Noether's theorem for vertical slice models. Journal of Geometric Mechanics, 2019, 11 (2) : 139-151. doi: 10.3934/jgm.2019007

[5]

Gastão S. F. Frederico, Delfim F. M. Torres. Noether's symmetry Theorem for variational and optimal control problems with time delay. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 619-630. doi: 10.3934/naco.2012.2.619

[6]

Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161

[7]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Variational problems of Herglotz type with time delay: DuBois--Reymond condition and Noether's first theorem. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4593-4610. doi: 10.3934/dcds.2015.35.4593

[8]

Bernard Dacorogna, Giovanni Pisante, Ana Margarida Ribeiro. On non quasiconvex problems of the calculus of variations. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 961-983. doi: 10.3934/dcds.2005.13.961

[9]

Daniel Faraco, Jan Kristensen. Compactness versus regularity in the calculus of variations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 473-485. doi: 10.3934/dcdsb.2012.17.473

[10]

Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067

[11]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760

[12]

John Fogarty. On Noether's bound for polynomial invariants of a finite group. Electronic Research Announcements, 2001, 7: 5-7.

[13]

Shaoming Guo. Oscillatory integrals related to Carleson's theorem: fractional monomials. Communications on Pure and Applied Analysis, 2016, 15 (3) : 929-946. doi: 10.3934/cpaa.2016.15.929

[14]

Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577

[15]

Božzidar Jovanović. Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems. Journal of Geometric Mechanics, 2018, 10 (2) : 173-187. doi: 10.3934/jgm.2018006

[16]

Viviana Alejandra Díaz, David Martín de Diego. Generalized variational calculus for continuous and discrete mechanical systems. Journal of Geometric Mechanics, 2018, 10 (4) : 373-410. doi: 10.3934/jgm.2018014

[17]

Artur M. C. Brito da Cruz, Natália Martins, Delfim F. M. Torres. Hahn's symmetric quantum variational calculus. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 77-94. doi: 10.3934/naco.2013.3.77

[18]

Nikos Katzourakis. Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems. Communications on Pure and Applied Analysis, 2015, 14 (1) : 313-327. doi: 10.3934/cpaa.2015.14.313

[19]

Nikos Katzourakis. Corrigendum to the paper: Nonuniqueness in Vector-Valued Calculus of Variations in $ L^\infty $ and some Linear Elliptic Systems. Communications on Pure and Applied Analysis, 2019, 18 (4) : 2197-2198. doi: 10.3934/cpaa.2019098

[20]

Ioan Bucataru, Matias F. Dahl. Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations. Journal of Geometric Mechanics, 2009, 1 (2) : 159-180. doi: 10.3934/jgm.2009.1.159

2021 Impact Factor: 0.737

Metrics

  • PDF downloads (407)
  • HTML views (381)
  • Cited by (0)

[Back to Top]