An algorithm to relate general solutions of different bidimensional problems

Hojman, Sergio A.; Chayet, Sergio; Núñez, Darío; Roque, Marco A.

doi:10.1063/1.529255

Cited by 17 publications

(24 citation statements)

References 14 publications

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“…In fact, as noted by Needham [26,27] two years before Bohlin's paper, Kasner [20] had established a more general duality law relating pairs of power law potentials (but the result has only been published in 1913). This relation has been rediscovered and generalized by Arnold and Vassiliev [1,2] in 1989 and quite simultaneously by Hojman et al [17]. In fact the generalization of Kasner's result had already been obtained by Collas [5] and implicitly enters into the frame of the coupling constant metamorphosis of Hietarinta et al [16,31,37].…”

Section: Introductionmentioning

confidence: 83%

Complex Representation of Planar Motions and Conserved Quantities of the Kepler and Hooke Problems

Bérard¹,

Mohrbach²

2021

JNMP

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Section: Introductionmentioning

confidence: 83%

Complex Representation of Planar Motions and Conserved Quantities of the Kepler and Hooke Problems

Bérard¹,

Mohrbach²

2021

JNMP

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“…In two dimensions, the ArnoldVassiliev duality generalizes the well-known correspondence relating the harmonic oscillator and the Kepler problem (Levi-Civita 1906;Stiefel and Scheifele 1971). It originates from the pioneering works of Bohlin and Kasner (Bohlin 1911;Kasner 1913) and has been rediscovered and generalized by Arnold and Vassiliev and others (Arnold 1990;Arnold and Vassiliev 1989;Collas 1981;Hojman et al 1991) at the end of the eighties.…”

Section: Introductionmentioning

confidence: 94%

Duality properties of Gorringe–Leach equations

Bérard

Mohrbach

2008

Celest Mech Dyn Astr

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In the category of motions preserving the angular momentum direction, Gorringe and Leach exhibited two classes of differential equations having elliptical orbits. After enlarging slightly these classes, we show that they are related by a duality correspondence of the Arnold-Vassiliev type. The specific associated conserved quantities (LaplaceRunge-Lenz vector and Fradkin-Jauch-Hill tensor) are then dual reflections of each other.

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“…In this article, we explore the analogy of this system with a relativistic particle [6], which implies, as we will show, a geometric unification of gravity and quintessence fields. With this purpose in mind, we examine a Lagrangian L that gives rise to the dynamical Equations (8) and (9). This Lagrangian is:…”

Section: Lagrangian For Frwq Systemmentioning

confidence: 99%

Quantum Cosmologies under Geometrical Unification of Gravity and Dark Energy

2019

Self Cite

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A Friedmann–Robertson–Walker Universe was studied with a dark energy component represented by a quintessence field. The Lagrangian for this system, hereafter called the Friedmann–Robertson–Walker–quintessence (FRWq) system, was presented. It was shown that the classical Lagrangian reproduces the usual two (second order) dynamical equations for the radius of the Universe and for the quintessence scalar field, as well as a (first order) constraint equation. Our approach naturally unified gravity and dark energy, as it was obtained that the Lagrangian and the equations of motion are those of a relativistic particle moving on a two-dimensional, conformally flat spacetime. The conformal metric factor was related to the dark energy scalar field potential. We proceeded to quantize the system in three different schemes. First, we assumed the Universe was a spinless particle (as it is common in literature), obtaining a quantum theory for a Universe described by the Klein–Gordon equation. Second, we pushed the quantization scheme further, assuming the Universe as a Dirac particle, and therefore constructing its corresponding Dirac and Majorana theories. With the different theories, we calculated the expected values for the scale factor of the Universe. They depend on the type of quantization scheme used. The differences between the Dirac and Majorana schemes are highlighted here. The implications of the different quantization procedures are discussed. Finally, the possible consequences for a multiverse theory of the Dirac and Majorana quantized Universe are briefly considered.

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An algorithm to relate general solutions of different bidimensional problems

Cited by 17 publications

References 14 publications

Complex Representation of Planar Motions and Conserved Quantities of the Kepler and Hooke Problems

Complex Representation of Planar Motions and Conserved Quantities of the Kepler and Hooke Problems

Duality properties of Gorringe–Leach equations

Quantum Cosmologies under Geometrical Unification of Gravity and Dark Energy

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