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Open AccessArticle

Special Functions of Mathematical Physics: A Unified Lagrangian Formalism

Department of Physics, University of Texas at Arlington, Arlington, TX 76019, USA
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Mathematics 2020, 8(3), 379; https://doi.org/10.3390/math8030379
Received: 24 February 2020 / Revised: 5 March 2020 / Accepted: 5 March 2020 / Published: 9 March 2020
(This article belongs to the Special Issue Special Functions and Applications)
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. View Full-Text
Keywords: calculus of variations; lagrangians; ordinary differential equations; special functions calculus of variations; lagrangians; ordinary differential equations; special functions
MDPI and ACS Style

Musielak, Z.E.; Davachi, N.; Rosario-Franco, M. Special Functions of Mathematical Physics: A Unified Lagrangian Formalism. Mathematics 2020, 8, 379.

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