A note on the construction of the Ermakov$ndash$Lewis invariant

Moyo, Sibusiso

doi:10.1088/0305-4470/35/25/312

Cited by 40 publications

(65 citation statements)

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“…A sample of results can be found by referring at Leach [72], Leach and Gorringe [73], Govinder and Leach [74], Maharaj et al [75], and Moyo and Leach [76]. The (explicit) point symmetry group classification problem for third-order equations, including linearization, has also been studied (see [25,44,[77][78][79][80][81].…”

Section: Discussionmentioning

confidence: 99%

Symmetry group classification of ordinary differential equations: Survey of some results

Mahomed

2007

Math Methods in App Sciences

111

110

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SUMMARYAfter the initial seminal works of Sophus Lie on ordinary differential equations, several important results on point symmetry group analysis of ordinary differential equations have been obtained. In this review, we present the salient features of point symmetry group classification of scalar ordinary differential equations: linear nth-order, second-order equations as well as related results. The main focus here is the contributions of Peter Leach, in this area, in whose honour this paper is written on the occasion of his 65th birthday celebrations.

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Section: Discussionmentioning

confidence: 99%

Symmetry group classification of ordinary differential equations: Survey of some results

Mahomed

2007

Math Methods in App Sciences

111

110

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“…The purpose of this comment is to point out that the main results presented in a recently published paper [1], are not new. At the end of the introduction of this paper, the authors claims that ".…”

Section: +Imentioning

confidence: 85%

“…We further notice that the Lagrangian formulation for Ermakov systems in the referred publication [1] is by no means new. This can be seen by comparing the potential functions from equations (11) in the work by Haas and Goedert with the potential given by equation (3.18) in the work by Moyo and Leach.…”

Section: +Imentioning

confidence: 98%

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Comment on A note on the construction of the Ermakov Lewis invariant

Haas

Goedert

2002

J. Phys. A: Math. Gen.

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We show that the basic results on the paper referred in the title [J. Phys. A: Math. Gen. 35 (2002) [5333][5334][5335][5336][5337][5338][5339][5340][5341][5342][5343][5344][5345], concerning the derivation of the Ermakov invariant from Noether symmetry methods, are not new.PACS numbers: 02.30.Hg, 02.90.+p, 03.20.+i The purpose of this comment is to point out that the main results presented in a recently published paper [1], are not new. At the end of the introduction of this paper, the authors claims that ". . . this is the first time the Noether symmetries are being considered to discuss the source of the Ermakov-Lewis 1 invariant." Unfortunately, the authors miss the reference Dynamical symmetries and the Ermakov invariant, by Haas and Goedert [2]. In this paper, the Ermakov invariant is, apparently for the first time, deduced as a consequence of a dynamical Noether symmetry. To make this point clear, it is enough to compare equation (29) in the paper by Haas and Goedert with Proposition 1 and equation (4.17) in the paper by Moyo and Leach. Both equations present the dynamical symmetry associated to the Ermakov invariant, for the case of Lagrangian Ermakov systems, as the result from a straightforward application of the converse of Noether's theorem.We further notice that the Lagrangian formulation for Ermakov systems in the referred publication [1] is by no means new. This can be seen by comparing the potential functions from equations (11) in the work by Haas and Goedert with the potential given by equation (3.18) in the work by Moyo and Leach. This later work also ignores the Hamiltonian descriptions for Ermakov systems developed earlier in [3], for the case of frequency functions depending on time only, and in [4], for the case of frequency functions depending also on dynamical variables.As a final remark, the work by Moyo and Leach also ignores the papers [5]- [8], dedicated to the analysis of uncoupled Ermakov systems in the light of Noether's theorem. These works, however, do not deal with truly twodimensional, coupled Ermakov systems, as in the case of references [1,2]. Rather, these papers [5]-[8] deal with Ermakov systems in which one of the equations plays the principal role, while the other, decoupled from the first, is treated as an auxiliary equation. In these cases, the Lagrangian description is effectively one-dimensional and the Ermakov invariant cannot be obtained as a result from an associated dynamical Noether symmetry. AcknowledgementsThis work has been supported by the Brazilian agency Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).

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“…More recent literature [30,33,5] links the solution (6) to that of the linear third-order differential equation…”

Section: Introductionmentioning

confidence: 99%

Ermakov's Superintegrable Toy and Nonlocal Symmetries

Leach

2005

SIGMA

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Abstract. We investigate the symmetry properties of a pair of Ermakov equations. The system is superintegrable and yet possesses only three Lie point symmetries with the algebra sl(2, R). The number of point symmetries is insufficient and the algebra unsuitable for the complete specification of the system. We use the method of reduction of order to reduce the nonlinear fourth-order system to a third-order system comprising a linear second-order equation and a conservation law. We obtain the representation of the complete symmetry group from this system. Four of the required symmetries are nonlocal and the algebra is the direct sum of a one-dimensional Abelian algebra with the semidirect sum of a two-dimensional solvable algebra with a two-dimensional Abelian algebra. The problem illustrates the difficulties which can arise in very elementary systems. Our treatment demonstrates the existence of possible routes to overcome these problems in a systematic fashion.

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A note on the construction of the Ermakov$ndash$Lewis invariant

Cited by 40 publications

References 23 publications

Symmetry group classification of ordinary differential equations: Survey of some results

Symmetry group classification of ordinary differential equations: Survey of some results

Comment on A note on the construction of the Ermakov Lewis invariant

Ermakov's Superintegrable Toy and Nonlocal Symmetries

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