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Special Functions of Mathematical Physics: A Unified Lagrangian Formalism

Author

Listed:
  • Zdzislaw E. Musielak

    (Department of Physics, University of Texas at Arlington, Arlington, TX 76019, USA)

  • Niyousha Davachi

    (Department of Physics, University of Texas at Arlington, Arlington, TX 76019, USA)

  • Marialis Rosario-Franco

    (Department of Physics, University of Texas at Arlington, Arlington, TX 76019, USA)

Abstract

Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard Lagrangians leads to Lagrangians for which the Euler–Lagrange equation vanishes identically, and that only some of these Lagrangians become the null Lagrangians with the well-defined gauge functions. It is also demonstrated that the non-standard Lagrangians require that the Euler–Lagrange equations are amended by the auxiliary conditions, which is a new phenomenon in the calculus of variations. The existence of the auxiliary conditions has profound implications on the validity of the Helmholtz conditions. The obtained results are used to derive the Lagrangians for the Airy, Bessel, Legendre and Hermite equations. The presented examples clearly demonstrate that the developed Lagrangian formalism is applicable to all considered differential equations, including the Airy (and other similar) equations, and that the regular and modified Bessel equations are the only ones with the gauge functions. Possible implications of the existence of the gauge functions for these equations are discussed.

Suggested Citation

  • Zdzislaw E. Musielak & Niyousha Davachi & Marialis Rosario-Franco, 2020. "Special Functions of Mathematical Physics: A Unified Lagrangian Formalism," Mathematics, MDPI, vol. 8(3), pages 1-17, March.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:379-:d:330100
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    References listed on IDEAS

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    1. Musielak, Z.E., 2009. "General conditions for the existence of non-standard Lagrangians for dissipative dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2645-2652.
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    Cited by:

    1. Diana T. Pham & Zdzislaw E. Musielak, 2023. "Novel Roles of Standard Lagrangians in Population Dynamics Modeling and Their Ecological Implications," Mathematics, MDPI, vol. 11(17), pages 1-15, August.
    2. Diana T. Pham & Zdzislaw E. Musielak, 2023. "Non-Standard and Null Lagrangians for Nonlinear Dynamical Systems and Their Role in Population Dynamics," Mathematics, MDPI, vol. 11(12), pages 1-15, June.

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