IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i20p4400-d1265566.html
   My bibliography  Save this article

General Fractional Noether Theorem and Non-Holonomic Action Principle

Author

Listed:
  • Vasily E. Tarasov

    (Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia
    Department of Physics, 915, Moscow Aviation Institute (National Research University), Moscow 125993, Russia)

Abstract

Using general fractional calculus (GFC) of the Luchko form and non-holonomic variational equations of Sedov type, generalizations of the standard action principle and first Noether theorem are proposed and proved for non-local (general fractional) non-Lagrangian field theory. The use of the GFC allows us to take into account a wide class of nonlocalities in space and time compared to the usual fractional calculus. The use of non-holonomic variation equations allows us to consider field equations and equations of motion for a wide class of irreversible processes, dissipative and open systems, non-Lagrangian and non-Hamiltonian field theories and systems. In addition, the proposed GF action principle and the GF Noether theorem are generalized to equations containing general fractional integrals (GFI) in addition to general fractional derivatives (GFD). Examples of field equations with GFDs and GFIs are suggested. The energy–momentum tensor, orbital angular-momentum tensor and spin angular-momentum tensor are given for general fractional non-Lagrangian field theories. Examples of application of generalized first Noether’s theorem are suggested for scalar end vector fields of non-Lagrangian field theory.

Suggested Citation

  • Vasily E. Tarasov, 2023. "General Fractional Noether Theorem and Non-Holonomic Action Principle," Mathematics, MDPI, vol. 11(20), pages 1-35, October.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:20:p:4400-:d:1265566
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/20/4400/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/11/20/4400/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Vasily E. Tarasov, 2022. "General Non-Local Continuum Mechanics: Derivation of Balance Equations," Mathematics, MDPI, vol. 10(9), pages 1-43, April.
    2. Kerins, John & Boiteux, Michel, 1983. "Applications of Noether's theorem to inhomogeneous fluids," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 117(2), pages 575-592.
    3. Vasily E. Tarasov, 2021. "General Fractional Vector Calculus," Mathematics, MDPI, vol. 9(21), pages 1-87, November.
    4. Tarasov, Vasily E., 2023. "Nonlocal statistical mechanics: General fractional Liouville equations and their solutions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
    5. Mohammed Al-Refai & Yuri Luchko, 2023. "The General Fractional Integrals and Derivatives on a Finite Interval," Mathematics, MDPI, vol. 11(4), pages 1-13, February.
    6. Vasily E. Tarasov, 2022. "Nonlocal Probability Theory: General Fractional Calculus Approach," Mathematics, MDPI, vol. 10(20), pages 1-82, October.
    7. Garra, Roberto & Taverna, Giorgio S. & Torres, Delfim F.M., 2017. "Fractional Herglotz variational principles with generalized Caputo derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 94-98.
    8. Vasily E. Tarasov, 2023. "Multi-Kernel General Fractional Calculus of Arbitrary Order," Mathematics, MDPI, vol. 11(7), pages 1-32, April.
    9. Yun-Die Jia & Yi Zhang, 2021. "Fractional Birkhoffian Mechanics Based on Quasi-Fractional Dynamics Models and Its Noether Symmetry," Mathematical Problems in Engineering, Hindawi, vol. 2021, pages 1-17, April.
    10. Vasily E. Tarasov, 2021. "General Fractional Calculus: Multi-Kernel Approach," Mathematics, MDPI, vol. 9(13), pages 1-14, June.
    11. Stefan G. Samko & Rogério P. Cardoso, 2003. "Integral equations of the first kind of Sonine type," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2003, pages 1-24, January.
    12. Yuri Luchko, 2021. "General Fractional Integrals and Derivatives with the Sonine Kernels," Mathematics, MDPI, vol. 9(6), pages 1-17, March.
    13. Yuri Luchko & Masahiro Yamamoto, 2020. "The General Fractional Derivative and Related Fractional Differential Equations," Mathematics, MDPI, vol. 8(12), pages 1-20, November.
    14. Lim, S.C., 2006. "Fractional derivative quantum fields at positive temperature," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 363(2), pages 269-281.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Vasily E. Tarasov, 2023. "General Fractional Calculus in Multi-Dimensional Space: Riesz Form," Mathematics, MDPI, vol. 11(7), pages 1-20, March.
    2. Vasily E. Tarasov, 2023. "Multi-Kernel General Fractional Calculus of Arbitrary Order," Mathematics, MDPI, vol. 11(7), pages 1-32, April.
    3. Vasily E. Tarasov, 2024. "General Fractional Economic Dynamics with Memory," Mathematics, MDPI, vol. 12(15), pages 1-24, August.
    4. Mohammed Al-Refai & Yuri Luchko, 2023. "The General Fractional Integrals and Derivatives on a Finite Interval," Mathematics, MDPI, vol. 11(4), pages 1-13, February.
    5. Tarasov, Vasily E., 2023. "Nonlocal statistical mechanics: General fractional Liouville equations and their solutions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
    6. Yuri Luchko, 2022. "Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense," Mathematics, MDPI, vol. 10(6), pages 1-24, March.
    7. Maryam Alkandari & Yuri Luchko, 2024. "Operational Calculus for the 1st-Level General Fractional Derivatives and Its Applications," Mathematics, MDPI, vol. 12(17), pages 1-23, August.
    8. Vasily E. Tarasov, 2021. "General Fractional Vector Calculus," Mathematics, MDPI, vol. 9(21), pages 1-87, November.
    9. Maryam Al-Kandari & Latif A-M. Hanna & Yuri Luchko, 2022. "Operational Calculus for the General Fractional Derivatives of Arbitrary Order," Mathematics, MDPI, vol. 10(9), pages 1-17, May.
    10. Vasily E. Tarasov, 2022. "General Non-Local Continuum Mechanics: Derivation of Balance Equations," Mathematics, MDPI, vol. 10(9), pages 1-43, April.
    11. Zhang, Yi & Jia, Yun-Die, 2023. "Generalization of Mei symmetry approach to fractional Birkhoffian mechanics," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    12. Yuri Luchko, 2023. "Fractional Integrals and Derivatives: “True” versus “False”," Mathematics, MDPI, vol. 11(13), pages 1-2, July.
    13. Huang, Li-Qin & Zhang, Yi, 2024. "Herglotz-type vakonomic dynamics and Noether theory of nonholonomic systems with delayed arguments," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    14. Salahshour, S. & Ahmadian, A. & Abbasbandy, S. & Baleanu, D., 2018. "M-fractional derivative under interval uncertainty: Theory, properties and applications," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 84-93.
    15. Anatoliy Martynyuk & Gani Stamov & Ivanka Stamova & Ekaterina Gospodinova, 2023. "Formulation of Impulsive Ecological Systems Using the Conformable Calculus Approach: Qualitative Analysis," Mathematics, MDPI, vol. 11(10), pages 1-15, May.
    16. Muñoz-Vázquez, Aldo Jonathan & Martínez-Fuentes, Oscar & Fernández-Anaya, Guillermo, 2022. "Generalized PI control for robust stabilization of dynamical systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 202(C), pages 22-35.
    17. Tian, Xue & Zhang, Yi, 2019. "Noether’s theorem for fractional Herglotz variational principle in phase space," Chaos, Solitons & Fractals, Elsevier, vol. 119(C), pages 50-54.
    18. Taylor, C.M. & Widom, B., 2005. "Line tension on approach to a wetting transition," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 358(2), pages 492-504.
    19. Danyang Kang & Cuiling Liu & Xingyong Zhang, 2020. "Existence of Solutions for Kirchhoff-Type Fractional Dirichlet Problem with p -Laplacian," Mathematics, MDPI, vol. 8(1), pages 1-17, January.
    20. Ervin Kaminski Lenzi & Luiz Roberto Evangelista & Luciano Rodrigues da Silva, 2023. "Aspects of Quantum Statistical Mechanics: Fractional and Tsallis Approaches," Mathematics, MDPI, vol. 11(12), pages 1-15, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:20:p:4400-:d:1265566. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.