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.
Highlights
There are numerous applications of linear, second-order ordinary differential equations (ODEs) in applied mathematics and physics [1,2]
The E–L equation forces this variable to appear in the original equation, and our results show that in order to remove it from this equation, the Lagrangian formalism must be amended by an auxiliary condition, which is a novel phenomenon in the calculus of variations
We derived the mixed Lagrangians for Do y( x ) = 0 and for Dy( x ) = 0. For the former, we demonstrated that the mixed Lagrangians depend on an arbitrary constant and that the gauge functions can be defined for all these MLs since they are equivalent to the null Lagrangians (NLs)
Summary
There are numerous applications of linear, second-order ordinary differential equations (ODEs) in applied mathematics and physics [1,2]. The most commonly used are the ODEs whose solutions are given by the special functions (SFs) of mathematical physics defined in [3,4,5]. Let D = d2 /dx2 + B( x )d/dx + C ( x ) be a linear operator with B( x ) and C ( x ) being ordinary (with the maps B : R → R and C : R → R, with R denoting the real numbers) and smooth with at least two continuous derivatives (C 2 ) functions defined either over a restricted interval ( a, b) or an infinite interval (−∞, ∞) depending on the ODE of Qs f (see Section 3), and let Dy( x ) = 0 be a linear second-order ODE with non-constant coefficients.
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