A note on algebraic potentials and Morales–Ramis theory

Combot, Thierry

doi:10.1007/s10569-013-9470-2

Cited by 22 publications

(33 citation statements)

References 13 publications

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“…However they are algebraic over C(q, p). For a study of the integrability of systems with algebraic Hamiltonians with the methods used in this paper, certain mathematical constructions must be used, see paper of Combot (2013). Generally, one can extend the system introducing new variables in such a way that the new system is still Hamiltonian with a rational Hamilton function.…”

Section: Resultsmentioning

confidence: 99%

“…The first who pointed out that this lack of the respect for the basic assumption of the theory can lead to erroneous conclusions was Combot (2013).…”

Section: Discussion and Commentsmentioning

confidence: 99%

“…For proving non-integrability we take a particular solution which lies on the zero level of Casimir F(q, r ). More advanced justification of this approach is given in Combot (2013). Just to simplify the exposition we prove Theorem 2.1 assuming that Proposition 2.3 is valid.…”

Section: Proofs Of Integrability Theorems For Anisotropic Kepler Probmentioning

confidence: 99%

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Anisotropic Kepler and anisotropic two fixed centres problems

Maciejewski

Przybylska

Szumiński

2016

Celest Mech Dyn Astr

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In this paper we show that the anisotropic Kepler problem is dynamically equivalent to a system of two point masses which move in perpendicular lines (or planes) and interact according to Newton's law of universal gravitation. Moreover, we prove that generalised version of anisotropic Kepler problem as well as anisotropic two centres problem are non-integrable. This was achieved thanks to investigation of differential Galois groups of variational equations along certain particular solutions. Properties of these groups yield very strong necessary integrability conditions.

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Section: Resultsmentioning

confidence: 99%

“…The first who pointed out that this lack of the respect for the basic assumption of the theory can lead to erroneous conclusions was Combot (2013).…”

Section: Discussion and Commentsmentioning

confidence: 99%

See 1 more Smart Citation

Anisotropic Kepler and anisotropic two fixed centres problems

Maciejewski

Przybylska

Szumiński

2016

Celest Mech Dyn Astr

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“…References will be found in [Au] and the more recent [Com1]. See also [Com2] which helps justifying the use of the Morales-Ramis theorem in most of the previous papers.…”

Section: Complexifying Poincaré's Methodsmentioning

confidence: 99%

Poincaré and the Three-Body Problem

Chenciner

2014

Henri Poincaré, 1912–2012

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Abstract. The Three-Body Problem has been a recurrent theme of Poincaré's thought. Having understood very early the need for a qualitative study of "nonintegrable" differential equations, he developed the necessary fundamental tools: analysis, of course, but also topology, geometry, probability. One century later, mathematicians working on the Three-Body Problem still draw inspiration from his works, in particular in the three volumes of Les méthodes nouvelles de la mécanique céleste published respectively in 1892, 1893, 1899.

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“…: three body problem (Boucher and Weil 2003;Maciejewski and Przybylska 2011;see also Tsygvintsev 2001see also Tsygvintsev , 2003see also Tsygvintsev , 2007, satellite in geo-magnetic field (Boucher 2006;Maciejewski and Przybylska 2003), two-body problems in constant curvature spaces (Maciejewski and Przybylska 2003), some n-body problems (Combot 2012;Simon 2007;Tosel 1998Tosel , 1999), Sitnikov's three-body problem (Morales Ruiz 1999), Hill's problem (Morales-Ruiz et al 2005), generalized two-fixed centres problem whose interaction potential is V = ar 2n (Maciejewski and Przybylska 2004), generalized anisotropic Kepler problem (Arribas et al 2003) and many others. However, as is explained in (Combot 2013), application of this theory directly to the system (1.25) is invalid as the Hamiltonian function (1.24) is not single-valued.…”

Section: Problemmentioning

confidence: 99%

Non-integrability of the dumbbell and point mass problem

Maciejewski

Przybylska

Simpson

et al. 2013

Celest Mech Dyn Astr

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This paper discusses a constrained gravitational three-body problem with two of the point masses separated by a massless inflexible rod to form a dumbbell. This problem is a simplification of a problem of a symmetric rigid body and a point mass, and has numerous applications in Celestial Mechanics and Astrodynamics. The non-integrability of this system is proven. This was achieved thanks to an analysis of variational equations along a certain particular solution and an investigation of their differential Galois group. Nowadays this approach is the most effective tool for study integrability of Hamiltonian and non-Hamiltonian systems.Keywords Three-body problem · Morales-Ramis theory · Differential Galois theory · Non-integrability Equations of motion, symmetries and reductionConsidered is the gravitational three-body problem with a single constraint. Three point masses, m 1 , m 2 and m 3 move in a plane under mutual gravitational interaction. Masses m 2 and m 3 are connected by a massless inflexible rod of length l > 0 to form a dumbbell. A pictorial description of the problem is given in Fig. 1. Various celestial objects are considered to possess such bimodal mass distribution because do not have enough gravitational force to form their shape into a spherical object, e.g. some asteroids such as (51) Nemausa and (216) Kleopatra; meteorites, especially large irons or nucleus of some comets e.g. Comet Borrelly (for detailed references see Povenmire 2002).The dumbbell satellite has attracted the attention of scientists since the middle of 20th century because it is suitable for an investigation of the general properties of the rigid body motion in a gravity field and provides important

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A note on algebraic potentials and Morales–Ramis theory

Cited by 22 publications

References 13 publications

Anisotropic Kepler and anisotropic two fixed centres problems

Anisotropic Kepler and anisotropic two fixed centres problems

Poincaré and the Three-Body Problem

Non-integrability of the dumbbell and point mass problem

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