On the connections between symmetries and conservation rules of dynamical systems

Cicogna, Giampaolo

doi:10.1002/mma.2584

Cited by 4 publications

(2 citation statements)

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“…The relation between twisted symmetries (in particular, λ-symmetries), first integrals and integrating factors [4,5,12,13,28,36,48,49], as well as that between twisted symmetries (in particular, λ-symmetries) and Jacobi Last Multiplier [51,53] has been studied by several authors .…”

Section: Other Topicsmentioning

confidence: 99%

Simple and collective twisted symmetries

Gaeta¹

2021

JNMP

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After the introduction of λ-symmetries by Muriel and Romero, several other types of so called "twisted symmetries" have been considered in the literature (their name refers to the fact they are defined through a deformation of the familiar prolongation operation); they are as useful as standard symmetries for what concerns symmetry reduction of ODEs or determination of special (invariant) solutions for PDEs and have thus attracted attention. The geometrical relation of twisted symmetries to standard ones has already been noted: for some type of twisted symmetries (in particular, λ and µ-symmetries), this amounts to a certain kind of gauge transformation.In a previous review paper [23] we have surveyed the first part of the developments of this theory; in the present paper we review recent developments. In particular, we provide a unifying geometrical description of the different types of twisted symmetries; this is based on the classical Frobenius reduction applied to distribution generated by Lie-point (local) symmetries.

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Section: Other Topicsmentioning

confidence: 99%

Simple and collective twisted symmetries

Gaeta¹

2021

JNMP

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“…Especially, several relations among λ ‐symmetries and first integrals for ordinary differential equations are studied in previous studies and references therein. For applications of λ ‐symmetries to derive first integrals or conservation laws of dynamical systems or Hamiltonian systems, one can see previous studies for example. By a geometrical characterization of λ ‐prolongation of vector fields, λ ‐symmetries have been extended to partial differential equations, leading to μ ‐symmetries that can also be used to derive conservation laws of the equations …”

Section: Introductionmentioning

confidence: 99%

A relationship between λ‐symmetries and first integrals for ordinary differential equations

Zhang

2019

Math Methods in App Sciences

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In this paper, we provide some geometric properties of -symmetries of ordinary differential equations using vector fields and differential forms. According to the corresponding geometric representation of -symmetries, we conclude that first integrals can also be derived if the equations do not possess enough symmetries.We also investigate the properties of -symmetries in the sense of the deformed Lie derivative and differential operator. We show that -symmetries have the exact analogous properties as standard symmetries if we take into consideration the deformed cases. KEYWORDSdeformed property, first integrals, Frobenius integrable, symmetries, -symmetries MSC CLASSIFICATION 34A05; 34C14 INTRODUCTIONClassical symmetry groups play an important role in analyzing differential equations. They have been widely used to reduce the order of ordinary differential equations and to reduce the number of independent variables of partial differential equations. However, many of the differential equations we study, especially nonlinear differential equations, do not in general possess enough symmetries. It is therefore essential to devise some kinds of proper extensions of classical Lie method in order to handle more situations. In the last few years, various extensions have been done, and some new classes of symmetries have been introduced. Among these extensions, -symmetries have been introduced by Muriel and Romero 1 based on a new method of prolonging vector fields known as -prolongation, which strictly include symmetries.Although -symmetries are not symmetries in the proper sense, as they do not map solutions into solutions, they can be used to perform reduction procedure via exactly the same method used for standard symmetries. Several usual methods of reduction for ODEs that do not come from symmetries are derived from the existence of -symmetries; one can see Muriel and Romero 1 and relevant references for further details. They can also be viewed as a class of twisted symmetries, and the recently developments can be reviewed in Gaeta. 2 Especially, several relations among -symmetries and first integrals for ordinary differential equations are studied in previous studies 3-11 and references therein. For applications of -symmetries to derive first integrals or conservation laws of dynamical systems or Hamiltonian systems, one can see previous studies 12-14 for example. By a geometrical characterization of -prolongation of vector fields, -symmetries have been extended to partial differential equations, leading to -symmetries that can also be used to derive conservation laws of the equations. 15,16 The investigation of -symmetries has practical advantages. First of all, the determining equations for symmetries may not have nontrivial solutions but the corresponding equations for -symmetries may have. In addition, -symmetries can be algorithmically obtained by using the Jacobi last multipliers. 17 Secondly, there exists an unknown function in the determining equations for -symmetries, so comparing with classical Lie method,...

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