The projective Cartan-Klein geometry of the Helmholtz conditions

In this paper, we review classical and recent results on the Lagrangian description of dissipative systems. After having recalled Rayleigh extension of Lagrangian formalism to equations of motion with dissipative forces, we describe Helmholtz conditions, which represent necessary and sufficient conditions for the existence of a Lagrangian function for a system of differential equations. These conditions are presented in different formalisms, some of them published in the last decades. In particular, we state the necessary and sufficient conditions in terms of multiplier factors, discussing the conditions for the existence of equivalent Lagrangians for the same system of differential equations. Some examples are discussed, to show the application of the techniques described in the theorems stated in this paper.

Download Full-text

A geometrical version of the Helmholtz conditions in time-dependent Lagrangian dynamics

Journal of Physics A General Physics ◽

10.1088/0305-4470/17/7/011 ◽

1984 ◽

Vol 17 (7) ◽

pp. 1437-1447 ◽

Cited By ~ 69

Author(s):

M Crampin ◽

G E Prince ◽

G Thompson

Keyword(s):

Time Dependent ◽

Lagrangian Dynamics ◽

Helmholtz Conditions ◽

Geometrical Version

Download Full-text

Special Functions of Mathematical Physics: A Unified Lagrangian Formalism

Mathematics ◽

10.3390/math8030379 ◽

2020 ◽

Vol 8 (3) ◽

pp. 379

Author(s):

Zdzislaw E. Musielak ◽

Niyousha Davachi ◽

Marialis Rosario-Franco

Keyword(s):

Differential Equations ◽

Calculus Of Variations ◽

Special Functions ◽

Lagrange Equation ◽

Mathematical Physics ◽

Lagrangian Formalism ◽

Lagrange Equations ◽

Gauge Functions ◽

Helmholtz Conditions

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.

Download Full-text