# Special Functions of Mathematical Physics: A Unified Lagrangian Formalism

## TL;DR

This paper develops a unified Lagrangian formalism for differential equations with special functions of mathematical physics, revealing new phenomena in the calculus of variations and deriving Lagrangians for classical equations like Airy, Bessel, Legendre, and Hermite.

## Contribution

It introduces a novel approach to derive Lagrangians for special functions, including standard and non-standard types, and explores the implications of auxiliary conditions and gauge functions.

## Key findings

- Lagrangians for classical special functions are derived.
- Non-standard Lagrangians require auxiliary conditions.
- Only Bessel equations possess gauge functions.

## 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.