Helmholtz conditions, covariance, and invariance identities

1985 ◽  
Vol 24 (4) ◽  
pp. 377-390 ◽  
Author(s):  
Giacomo Caviglia
Keyword(s):  
Helmholtz Conditions

Lagrangian descriptions of dissipative systems: a review

2020 ◽  
pp. 108128652097183
Author(s):  
Alberto Maria Bersani ◽  
Paolo Caressa
Keyword(s):  
Differential Equations ◽  
Equations Of Motion ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Dissipative Systems ◽  
Lagrangian Formalism ◽  
Lagrangian Description ◽  
System Of Differential Equations ◽  
Helmholtz Conditions ◽  
Necessary And Sufficient

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.

scholarly journals Special Functions of Mathematical Physics: A Unified Lagrangian Formalism

Mathematics ◽  
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.

scholarly journals The projective Cartan-Klein geometry of the Helmholtz conditions

2018 ◽  
Vol 10 (1) ◽  
pp. 69-92
Author(s):  
Carlos Durán ◽  
◽  
Diego Otero
Keyword(s):  
Helmholtz Conditions ◽  
Klein Geometry
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