Minimally classifying relative equilibria

Palmore, Julian

doi:10.1007/bf01793953

Lett Math Phys

1977

DOI: 10.1007/bf01793953

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Minimally classifying relative equilibria

Julian Palmore

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Cited by 4 publications

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“…, where 2a 0 is the number of noncollinear local minima, 2a 1 the number of non-collinear saddles of index 1, etc., then R(t) − Q(t) = (1 + t)S(t), where S(t) is a polynomial with non-negative integer coefficients. 4) Palmore's "ignored estimates" are developed in [15], where a cellular decomposition of the configuration space is considered. Palmore does not explain why U should have the critical points corresponding to this cell decomposition.…”

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confidence: 99%

Open Problem 1: Are Palmore’s “Ignored Estimates” on the Number of Planar Central Configurations Correct?

Albouy

2015

Qual. Theory Dyn. Syst.

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Abstract. We wish to draw attention on estimates on the number of relative equilibria in the Newtonian n-body problem that Julian Palmore published in 1975.Julian Palmore published in [12] a first estimate on the number of planar central configurations with given number of bodies n and given positive masses m 1 , . . . , m n . If these central configurations are non-degenerate, they are at least (3n − 4)(n − 1)!/2.The word "configuration" is ambiguous. Here we consider two "configurations" which are deduced one from the other by homothety and rotation as the same configuration. But we consider that applying a reflection to a non-collinear configuration produces a distinct configuration. We also consider that the bodies are "distinguishable": configurations which differ only in the numbering of the bodies are nevertheless considered as distinct.Palmore published in [13] a second estimate for the same number, under the same non-degeneracy hypothesis. There are at least (n − 2)!(2 n−1 (n − 2) + 1) planar central configurations of n bodies. We call this estimate the "ignored" Palmore estimate, because the subsequent authors on lower bounds did not even mention it (maybe because the proof is missing). Palmore also gave detailed estimates, i.e. a lower bound on the number of central configuration with given index. We do not know if these estimates are true. They seem however compatible with all the known results.When we speak of degeneracy or index, we think of central configurations as critical points of a function. This function is the Newtonian potential U = i

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confidence: 99%

Open Problem 1: Are Palmore’s “Ignored Estimates” on the Number of Planar Central Configurations Correct?

Albouy

2015

Qual. Theory Dyn. Syst.

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Relative equilibrium solutions in the four body problem

Simó

1978

Celestial Mechanics

128

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Relative Equilibria in Continuous Stellar Dynamics

Serrano

Pino

Dolbeault

2010

Commun. Math. Phys.

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We study a three dimensional continuous model of gravitating matter rotating at constant angular velocity. In the rotating reference frame, by a finite dimensional reduction, we prove the existence of non radial stationary solutions whose supports are made of an arbitrarily large number of disjoint compact sets, in the low angular velocity and large scale limit. At first order, the solutions behave like point particles, thus making the link with the relative equilibria in N -body dynamics.

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Minimally classifying relative equilibria

Cited by 4 publications

References 5 publications

Open Problem 1: Are Palmore’s “Ignored Estimates” on the Number of Planar Central Configurations Correct?

Open Problem 1: Are Palmore’s “Ignored Estimates” on the Number of Planar Central Configurations Correct?

Relative equilibrium solutions in the four body problem

Relative Equilibria in Continuous Stellar Dynamics

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