Are Palmore's "ignored estimates" on the number of planar central configurations correct?
aa r X i v : . [ m a t h - ph ] A ug Are Palmore’s “ignored estimates” on the number of planar centralconfigurations correct?
Alain Albouy, [email protected], CNRS, Observatoire de Paris,77, avenue Denfert-Rochereau, 75014 Paris, France
Abstract.
We wish to draw attention on estimates on the number of relativeequilibria in the Newtonian n -body problem that Julian Palmore published in1975.Julian Palmore published in [12] a first estimate on the number of planar cen-tral configurations with given number of bodies n and given positive masses m , . . . , m n . If these central configurations are non-degenerate, they are atleast (3 n − n − / n − n − ( 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 noteven mention it (maybe because the proof is missing). Palmore also gave de-tailed estimates, i.e. a lower bound on the number of central configuration withgiven index. We do not know if these estimates are true. They seem howevercompatible with all the known results.When we speak of degeneracy or index, we think of central configurations ascritical points of a function. This function is the Newtonian potential U = P i 2. But, as the Morse polynomial is obtained from the Poincar´epolynomial by adding terms of the form t k + t k +1 , we should add the samenumber of saddles of index n − n − n ! / n ! / − ( n − n − n − / Z / Z . There is a simple way to reach the formulas of [7], which isinspired by the examples given in [2]. We do not know any rigorous argumentthat would justify this simplification of [7]. Instead of the Poincar´e polynomial,we should write the fraction (1 + 2 t )(1 + 3 t ) · · · (1 + ( n − t ) / (1 − t ). TheMorse inequalities take the usual form, but we should divide the contributionof an invariant critical point by 1 − t . Here, the invariant critical points arethe configurations that are invariant by reflection, i.e., collinear. We write(1 + 2 t )(1 + 3 t ) · · · (1 + ( n − t ) / (1 − t ) = ( n ! / t n − / (1 − t ) + Q ( t ), where Q ( t ) is a polynomial. If R ( t ) = a + a t + · · · , where 2 a is the number of non-collinear local minima, 2 a 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-negativeinteger coefficients.4) Palmore’s “ignored estimates” are developed in [15], where a cellular decom-position of the configuration space is considered. Palmore does not explain why U should have the critical points corresponding to this cell decomposition. Pal-more predicts ( n − n ( n − n − U is the reflection symmetry, which is likely tobe fully taken into account by McCord. Concerning the local minima, McCordpredicts only two of them, and it is probably impossible to predict more thantwo without using further properties of U . In the first non-trivial case, n = 4,Palmore’s estimates are easily proved by using a result by McMillan and Bartky(see [10] and [19]): there is at least one local minimum, a convex quadrilateral,for each cyclic ordering of the four bodies. This gives, compared to the firstPalmore estimate, 5 more local minima, and we should consequently also add 5saddles of index 1. We get 34 central configurations in total, which is Palmore’sbound. The argument of [10] and [19] is basically that no central configurationcan cross the border from convex to non-convex. This is no longer true withfive bodies in a plane, as discovered independently in [6] and in [3]. We also seein the lists [9] or [5] of central configurations with equal masses that the indexof the convex configurations is growing when n is growing (see also [18]).5) Zhihong Xia gave in [20] the exact number of central configurations in the case m ≫ m ≫ · · · ≫ m n . The explicit formulas corresponding to his constructiondo not appear in his paper, but Moeckel and Tien computed them (see [8], p.81). Surprisingly, the recursion formulas they found are those used by Palmoreto get his “ignored estimate”. The numbers are the same, and the detailednumbers index by index are obtained by a reasonable guess: that the index of2 configuration obtained by Xia’s construction is the number of times a saddlewas chosen as the position of the next body.We give below the numbers corresponding to these estimates for n = 3, 4 and 5.We add some examples. Examples with n = 4 are from [14] and [16]. Exampleswith n = 5 are from [17], [9] and [5]. According to Carles Sim´o, assuming non-degeneracy, the number of planar central configurations with n = 5 and givenpositive masses is very likely to be always between 294 and 450.I wish to thank Alain Chenciner, Joseph Fayad, Chris McCord and Rick Moeckelfor their help. n = 3 Index 0 Index 1 Totalbouquet 1 2 3first Palmore 2 3 5McCord 2 3 5Ignored Palmore 2 3 5All examples 2 3 5 n = 4 Index 0 Index 1 Index 2 Totalbouquet 1 5 6 12first Palmore 1 11 12 24McCord 2 12 12 26Ignored Palmore 6 16 12 34Xia’s case 6 16 12 343 equal, 1 small 8 18 12 38Equal masses 6 24 20 50 n = 5 Index 0 Index 1 Index 2 Index 3 Totalbouquet 1 9 26 24 60first Palmore 1 9 62 60 132McCord 2 20 72 60 154Ignored Palmore 24 90 120 60 294Xia’s case 24 90 120 60 294Equal masses 54 120 120 60 3544 equal, 1 small 30 120 192 108 450 References [1] V.I. Arnold, The cohomology ring of dyed braids , Mat. Zametki 5 (1969),pp. 227–231[2] R. Bott, Lectures on Morse theory, old and new , Bull. Amer. Mat. 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