Pfaffians and the inverse problem for collinear central configurations
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(2020) 132:32
ORIGINAL ARTICLE
Pfaffians and the inverse problem for collinear central configurations D. L. Ferrario1 Received: 15 November 2019 / Revised: 17 June 2020 / Accepted: 27 June 2020 © Springer Nature B.V. 2020
Abstract We consider, after Albouy–Moeckel, the inverse problem for collinear central configurations: Given a collinear configuration of n bodies, find positive masses which make it central. We give some new estimates concerning the positivity of Albouy–Moeckel Pfaffians: We show that for any homogeneity α and n ≤ 6 or n ≤ 10 and α = 1 (computer assisted) the Pfaffians are positive. Moreover, for the inverse problem with positive masses, we show that for any homogeneity and n ≥ 4 there are explicit regions of the configuration space without solutions of the inverse problem. Keywords n-Body problem · Pfaffian · Central configuration · Inverse problem
1 Introduction Let n ≥ 2, and d ≥ 1. The configuration space of n points in the d-dimensional Euclidean space E = Rd is defined as Fn (E) = {q ∈ E n : q i = q j }, where q = (q 1 , q 2 , . . . , q n ) ∈ E and ∀ j, q j ∈ E. Given a positive parameter α > 0, and n positive masses m j > 0, the potential function U : Fn (E) → R is defined as mi m j . U (q) = q i − q j α 1≤i< j≤n
A central configuration is a configuration that yields a relative equilibrium solution of the Newton equations of the n-body problem with potential function U and can be shown (cf. Moulton 1910; Moeckel 1990; Albouy and Moeckel 2000; Ferrario 2015, 2017a, b) that it is a solution of the following n equations q j − qk m j mk . (1.1) λm j q j = −α q j − q k α+2 k = j
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D. L. Ferrario [email protected] University of Milano-Bicocca, Milan, Italy 0123456789().: V,-vol
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D. L. Ferrario
Such configurations have center of mass nj=1 m j q j = 0 ∈ E, and the parameter λ turns U (q) out to be equal to λ = −α n . A generic central configuration (with center of m j q j 2 j=1 n j=1 m j q j mass q 0 = not necessarily 0, where M = nj=1 m j ) satisfies the equation M q j − qk λm j (q j − q 0 ) = −α m j mk . (1.2) q j − q k α+2 k= j
Now, if for each i, j denote Q jk =
q j − qk q j − q k α+2
,
Equation (1.2) can be written as1 q j = M −1
n
mk qk −
α m k Q jk , λ
j = 1, . . . , n.
(1.3)
k= j
k=1
The inverse problem, introduced by Moulton (1910) [see also Buchanan (1909)], and considered by Albouy and Moeckel (2000), can be phrased as follows: Given the positions q j (or, equivalently, the mutual differences q i −q j ) to find the (positive) masses m j and λ < 0 such that (1.3) holds. As it is, the equation is not linear in the (n + 1)-tuple (m 1 , . . . , m n , λ), but can be transformed into the following equation mˆ k Q jk , j = 1, . . . , n, (1.4) q j = cˆ + k= j
because of the following lemma. Lemma 1.5 Given q ∈ Fn (E), there exists (m 1 , . . . , m n , λ), with m j > 0 satisfying (1.3) if and only if there exists (mˆ 1 , . . . , mˆ n , cˆ ) ∈ Rn+d such that (1.4) holds and mˆ j > 0 for each j. Proof If (1.3) h
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