Local and nonlocal solvable structures in the reduction of ODEs

Ferraioli, Diego Catalano; Morando, Paola

doi:10.1088/1751-8113/42/3/035210

Cited by 21 publications

(13 citation statements)

References 28 publications

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“…They also admit a deep interpretation by means of nontrivial geometrical language and are related to symmetries of different nature (symmetries of integral-exponential type, hidden and potential symmetries, nonlocal symmetries, and solvable structures as well): see e.g. 2 [1,2,3,4,6,7,8,18,19,23,25,31,34]. For a very recent, fairly complete and updated survey, see [20].…”

Section: Perturbed Symmetries and λ-Constants Of Motionmentioning

confidence: 99%

Symmetries of Hamiltonian Equations and Λ-Constants of Motion

Cicogna¹

2021

JNMP

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We consider symmetries and perturbed symmetries of canonical Hamiltonian equations of motion. Specifically we consider the case in which the Hamiltonian equations exhibit a Λ-symmetry under some Lie point vector field. After a brief survey of the relationships between standard symmetries and the existence of first integrals, we recall the definition and the properties of Λ-symmetries. We show that in the presence of a Λ-symmetry for the Hamiltonian equations, one can introduce the notion of "Λ-constant of motion". The presence of a Λ-symmetry leads also to a nice and useful reduction of the form of the equations. We then consider the case in which the Hamiltonian problem is deduced from a Λ-invariant Lagrangian. We illustrate how the Lagrangian Λ-invariance is transferred into the Hamiltonian context and show that the Hamiltonian equations are Λ-symmetric. We also compare the "partial" (Lagrangian) reduction of the Euler-Lagrange equations with the reduction which can be obtained for the Hamiltonian equations. Several examples illustrate and clarify the various situations.

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Section: Perturbed Symmetries and λ-Constants Of Motionmentioning

confidence: 99%

Symmetries of Hamiltonian Equations and Λ-Constants of Motion

Cicogna¹

2021

JNMP

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“…Inspired by the idea in Ferraioli and Morando and Sherring and Prince, the problem of finding first integrals can be reduced to finding Frobenius integrable 1‐forms and the corresponding symmetries that can reduce the problems to first‐order cases. In this section, by using the geometric representation of λ ‐symmetries, we will show that λ ‐symmetries together with symmetries can be used to derive first integrals when differential equations do not possess enough symmetries.…”

Section: λ‐Symmetries and First Integralsmentioning

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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“…hold. If f 1 , f 2 satisfy (11), then [ f 1 X 1 , f 2 X 2 ] = 0. The first two relations in (12) can be proved similarly.…”

Section: Commuting Generalized C ∞ −Symmetriesmentioning

confidence: 99%

“…This concept was introduced in [8] and it is based on a prolongation formula that generalizes the usual prolongation of vector fields. C ∞ −symmetries can be used to reduce the order of ODEs as Lie point symmetries do and have been widely studied and generalized from very different points of view (see [10,11,12,13] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

$\lambda$ -Symmetries and integrability by quadratures

Muriel

Romero

Ruiz

2017

IMA Journal of Applied Mathematics

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It is investigated how two (standard or generalized) λ−symmetries of a given second-order ordinary differential equation can be used to solve the equation by quadratures. The method is based on the construction of two commuting generalized symmetries for this equation by using both λ−symmetries. The functions used in that construction are related with integrating factors of the reduced and auxiliary equations associated to the λ−symmetries. These functions can also be used to derive a Jacobi last multiplier and two integrating factors for the given equation.Some examples illustrate the method; one of them is included in the XXVII case of the Painlevé-Gambier classification. An explicit expression of its general solution in terms of two fundamental sets of solutions for two related secondorder linear equations is also obtained.

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Local and nonlocal solvable structures in the reduction of ODEs

Cited by 21 publications

References 28 publications

Symmetries of Hamiltonian Equations and Λ-Constants of Motion

Symmetries of Hamiltonian Equations and Λ-Constants of Motion

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

$\lambda$ -Symmetries and integrability by quadratures

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