On the integrability of Friedmann–Robertson–Walker models with conformally coupled massive scalar fields

Coelho, Leandro Alves; Skea, Jim; Stuchi, T. J.

doi:10.1088/1751-8113/41/7/075401

Cited by 7 publications

(24 citation statements)

References 33 publications

(104 reference statements)

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“…As the change of variables is only a scale transformation, for all ε different from zero, the original and the transformed systems (5) and (8) have essentially the same phase portrait, and additionally system (8) for ε sufficiently small is close to an integrable one First we change the Hamiltonian (7) and the equations of motion (8) to convenient polar coordinates in such a way that for ε = 0 we have a pair of harmonic oscillators. Thus we consider the change of variables…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Periodic orbits and non-integrability in a cosmological scalar field

Llibre

Vidal

2012

Journal of Mathematical Physics

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Abstract. We apply the averaging theory of first order to study the periodic orbits of Hamiltonian systems describing an universe filled with a scalar field which possesses three parameters. The main results are the following.First, we provide sufficient conditions on the parameters of these cosmological model, which guarantee that at any positive or negative Hamiltonian level, the Hamiltonian system has periodic orbits, the number of such periodic orbits and their stability change with the values of the parameters. These periodic orbits live in the whole phase space in a continuous family of periodic orbits parameterized by the Hamiltonian level.Second, under convenient assumptions we show the non-integrability of these cosmological systems in the sense of Liouville-Arnol'd, proving that cannot exist any second first integral of class C 1 .It is important to mention that the tools (i.e the averaging theory for studying the existence of periodic orbits and their kind of stability, and the multipliers of these periodic orbits for studying the integrability of the Hamiltonian system) used here for proving our results on the cosmological scalar field, can be applied to Hamiltonian systems with an arbitrary number of degrees of freedom.

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Section: Proof Of Theoremmentioning

confidence: 99%

“…Our results are on the non-integrability in the sense of Liouville-Arnold for any second first integral of class C 1 . In [6] and [8] also the problem of non-existence of any additional meromorphic first integral in the Hamiltonian system (5) was considered.…”

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confidence: 99%

Periodic orbits and non-integrability in a cosmological scalar field

Llibre

Vidal

2012

Journal of Mathematical Physics

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“…Since this work was submitted, we have been aware of several other recent works on other configurations of the Friedman-RobertsonWalker models using the Hamiltonian viewpoint and the variational approach that we use below ( [6], [9]). …”

Section: The Problemmentioning

confidence: 99%

“…This last point may be tedious. In [5,6,9], the Kovacic algorithm was used; in [6,9], additional techniques (systems with homogeneous potentials) were further used. Here, we will use a criterion (from [2,3]) which is easy to apply and generalizes to equations of higher order: if the linearized (variational) equation is irreducible and has local solutions with logarithms, then the original system was not integrable (a more general version of our criterion is theorem 3 in [3]).…”

Section: The Problemmentioning

confidence: 99%

About the non-integrability in the Friedmann-Robertson-Walker cosmological model

Boucher¹,

Weil²

2007

Braz. J. Phys.

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We study the non integrability of the Friedmann-Robertson-Walker cosmological model, in continuation of the work [5] of Coehlo, Skea and Stuchi. Using Morales-Ramis theorem ([10]) and applying a practical nonintegrability criterion deduced from it, we find that the system is not completely integrable for almost all values of the parameters λ and Λ, which was already proved by the authors of [5] applying Kovacic's algorithm. Working on a level surface H = h with h = 0 and h = − 1 4λ and using the Morales-Ramis-Simo "higher variational" theory ([11]), we prove that the hamiltonian system cannot be integrable for particular values of λ among the exceptional values and that it is completely integrable in two special cases (λ = Λ = −m 2 and λ = Λ = −m 2 3 ). We conjecture that there is no other case of complete integrability and give detailed arguments towards this.Keywords: Hamiltonian systems; Integrability; Morales-Ramis-Simo Theorem; Computer algebra I. THE PROBLEMThe Friedmann-Robertson-Walker model ([7]) is a model which explains some features of the observed universe at the present time. Although it does not describe the real universe in an essential way because of too many symmetries, it remains a fundamental model. We consider the Friedmann-Robertson-Walker (FRW) universe ([5]) where the metric takes the formwhere ν is the time; a(ν) is the scale factor, which means that if a distance is measured as d 0 today, then at any other instant, it is d 0 times a; k = 0, 1, −1 is the curvature of space-time and dΩ 2 is the distance element on a two-sphere. The dynamics is described by Raychauduri equation and Klein-Gordon equation ([7]) which derive from the Hamil-After the canonical transformation of variables p a = ip a and a = −ia (suggested from [4]), and assuming k = 1, we get the Hamiltonian:where a is the scale factor of the universe; φ is the scalar field with self-coupling constant λ and with mass m (which we will assume to be non zero); Λ is the cosmological constant. • to show that the variational equation, computed along a given particular solution of the hamiltonian system, can be seen as a direct sum of two Lamé equations (when λ = 0);• to find the result of non-integrability already proved in [5] when λ = − 2m 2 (n + 1)(n + 2), n ∈ IN or Λ = − 2m 2 (N + 1)(N + 2), N ∈ IN using a criterion of non integrability ([2, 3]) deduced from Morales-Ramis theorem ([10]);• to prove that the system is not completely integrable in the particular case

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“…These new results are not proofs of non-integrability in and of themselves, but said proofs are the logical next step and will be dealt with in further theoretical studies. The techniques used not only complement those used in [7], but add a degree of specificity to forthcoming results.Hamiltonian (1) is a special case of what is called quartic Hénon-Heiles Hamiltonian (HH4) in other communities, e.g. [8, Eq (8)] and [9].…”

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confidence: 99%

Conditions and evidence for non-integrability in the Friedmann-Robertson-Walker Hamiltonian

Simon¹

2021

JNMP

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This is an example of application of Ziglin-Morales-Ramis algebraic studies in Hamiltonian integrability, more specifically the result by Morales, Ramis and Simó on higher-order variational equations, to the well-known Friedmann-Robertson-Walker cosmological model. A previous paper by the author formalises said variational systems in such a way allowing the simple expression of notable elements of the differential Galois group needed to study integrability. Using this formalisation and an alternative method already used by other authors, we find sufficient conditions whose fulfilment for given parameters would entail very simple proofs of nonintegrability -both for the complete Hamiltonian, a goal already achieved by other means by Coelho et al, and for a special open case attracting recent attention.

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On the integrability of Friedmann–Robertson–Walker models with conformally coupled massive scalar fields

Abstract: In this work we use a recently developed nonintegrability theorem of Morales and Ramis to prove that the Friedmann Robertson Walker cosmological model with a conformally coupled massive scalar field is nonintegrable.

Cited by 7 publications

References 33 publications

Periodic orbits and non-integrability in a cosmological scalar field

Periodic orbits and non-integrability in a cosmological scalar field

About the non-integrability in the Friedmann-Robertson-Walker cosmological model

Conditions and evidence for non-integrability in the Friedmann-Robertson-Walker Hamiltonian

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