Qual. Theory Dyn. Syst. (2009) 8:349–356DOI 10.1007/s12346-010-0016-7

Qualitative Theoryof Dynamical Systems

Finiteness of Kite Relative Equilibria in the Five-Vortexand Five-Body Problems

Marshall Hampton

Received: 23 December 2008 / Accepted: 15 February 2010 / Published online: 10 March 2010© Birkhäuser / Springer Basel AG 2010

Abstract We study the finiteness of planar relative equilibria of the Newtonian five-body problem and in the five-vortex problem in the case that configurations form asymmetric kite (three points on a line and two additional points placed symmetricallywith respect to that line). We can prove that the equivalence classes of such relativeequilibria are finite with some possible exceptional cases. These exceptional cases aregiven explicitly as polynomials in the masses (or vorticities in the vortex problem).These results depend on computations performed with the software Sage, Singular,Magma, and Gfan.

Keywords Celestial mechanics · n-Body problem · Relative equilibria · Vortices ·Tropical geometry

Mathematics Subject Classification (2000) 70F10 · 70F15 · 37N05 · 76B47

1 Introduction

This paper presents some results on the finiteness of relative equilibria of ‘kite’ config-urations in the planar five-vortex and Newtonian five-body problem. A five-body kiteconfiguration is planar, with three points on a line and two additional points placedsymmetrically with respect to that line (see Fig. 1).

The approach is built upon the fundamental results of Bernstein [2], which overlapwith what has come to be called tropical geometry. The use of Bernstein’s ideas to

M. Hampton (B)Department of Mathematics and Statistics, University of Minnesota, 1117 University Drive,Duluth, MN 55812, USAe-mail: mhampton@d.umn.edu

350 M. Hampton

Fig. 1 An example kiteconfiguration

computationally tackle the problem of finiteness of polynomial varieties is describedin [6]. In this work we exploit recent advances in the symbolic computation of tropicalvarieties that enable an iterative approach to proving finiteness results. The specificresults, while interesting in themselves, should be regarded as a proof of concept—it is clear that sharper and more general results are possible. General functions forperforming our pipeline of computations will be added in the near future to Sage [13];Sage is used in this work to integrate Singular [5], Magma [4], and Gfan [9], and tovisualize polytopes.

Gfan in particular deserves a special mention regarding this work. Gfan version 0.3included a ‘tropical intersection’ command, which is the key novelty in making ourcalculations more efficient than before.

The five-body problem is especially interesting for the development of finitenessmethods because of the existence of a curve of solutions for particular parametervalues [11]. I conjecture that this is the only solution curve for non-zero parametervalues. Its existence, however, can be expected to complicate the analysis, and thesesystems provide a stimulating challenge for our methods. As one measure of the rela-tive complexity of the problem we can consider the mixed volume for the NewtonianAlbouy-Chenciner equations. For three, four, and five masses, the respective mixedvolumes are 99, 33201, and 133998561 (thanks to T.Y. Li for the latter calculation).

1.1 Albouy-Chenciner Equations

Our primary equations for studying relative equilibria in the Newtonian and vortex5-body problems are what we call the Albouy-Chenciner equations. These were intro-duced in [1] and some additional explanation of the particular form we use is given in[6]. A concise form can be given by using the quantities Si j :

Si j = 1

r Di j

− 1 (i �= j), Sii = 0, (1)

Finiteness of Kite Relative Equilibria 351

n∑

k=1

mk

[Sik(r

2jk − r2

ik − r2i j ) + S jk(r

2ik − r2

jk − r2i j )

]= 0. (2)

Here ri j is the distances between points i and j . When we refer to the Albouy-Chenciner equations, we mean the polynomial system that results from clearingdenominators and eliminating the Si j from the above system. These polynomials canbe written as:

fi j = 2(−mr Di j + mi + m j )

n∏

k �=i, j

r Dik r D

jk

+n∑

k �=i, j

⎡

⎣mkr Di j

⎛

⎝n∏

l �=i, j,k

r Dil +

n∏

l �=i, j

r Djl +

n∏

l �=i, j,k

r Djl +

n∏

l �=i, j

r Dil

⎞

⎠

+ mkr D−2i j

⎛

⎝r2jk

n∏

l �=i, j

r Dil

n∏

l �=i, j,k

r Djl + r2

ik

n∏

l �=i, j

r Djl

n∏

l �=i, j,k

r Dil

−(r2+Dik + r2+D

jk )

n∏

l �=i, j,k

r Djl r

Dil

⎞

⎠

⎤

⎦

where m = ∑ni mi is assumed to be nonzero. For the Newtonian case D = 3 and for

the vortex case D = 2. We will only consider configurations of five points, so n = 5.

1.2 Geometric Constraints

The mutual distances of the kite configurations we are considering satisfy a numberof relations. From symmetry we have r15 = r14, r25 = r24, and r35 = r34, so we cansimply eliminate r15, r25, and r35. The collinearity of points 1, 2, and 3 lets us assumethat r13 = r12 + r23 without loss of generality. Additional geometric constraints canbe obtained from the condition that any four points must be coplanar. If four pointsa,b,c, and d are coplanar then the determinant of the Cayley-Menger matrix

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

0 1 1 1 1

1 0 r2ab r2

ac r2ad

1 r2ab 0 r2

bc r2bd

1 r2ac r2

bc 0 r2cd

1 r2ad r2

bd r2cd 0

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

must be equal to zero (see [1] for details). This condition with (a, b, c, d) equal to(1, 2, 4, 5), (1, 3, 4, 5), and (2, 3, 4, 5) gives us three more geometric constraints. Forexample, the first of these constraints is:

352 M. Hampton

r413 − 2r2

13r214 + r4

14 − 2r213r2

34 − 2r214r2

34 + r434 + r2

13r245 = 0.

1.3 Terminology and Description of our Methods

Here we define a few terms and give some facts that will be useful, following theconventions of [8]. For basic theorems on polytopes see [18]. Another good referencefor the connection between Gröbner bases and polytopes is [14].

The face of a polyhedron P maximizing a form ω ∈ Rn is

faceω(P) := {p ∈ P : 〈ω, p〉 = maxq∈P 〈ω, q〉}.

The normal cone of a face F of a polyhedron P is

NP (F) := {ω ∈ Rn : faceω(P) = F}.

The collection of all the normal cones of faces of P is the normal fan of P , a completepolyhedral complex.

The common refinement of two fans F1 and F2 is defined as

F1 ∧ F2 = {C1 ∩ C2}(C1,C2)∈F1×F2

The Minkowski sum of two polyhedral P, Q ∈ Rn is the polyhedron

P + Q = {p + q : (p, q) ∈ P × Q}.

The common refinement of the normal fans of polyhedra P and Q is the normalfan of P + Q. Suppose f = ∑

ci xαi is a polynomial in R = k[x1, . . . , xn] where kis a field.

The Newton polytope of f is the convex hull of its exponent vectors α.The initial form of f with respect to ω is inω( f ) = ∑

cixαi where αi is in Fω(Pf )

where Pf is the Newton polytope of f .If I is an ideal, the tropical variety of I is T (I ) := {ω ∈ R

n : inω(I ) is monomialfree}. The tropical variety of a principal ideal is a tropical hypersurface.

Every polynomial ideal I has a finite tropical basis - a set of polynomials fi ∈ Ithat generate I and such that ∩iT ( fi ) = T (I ) [8].

A finite intersection of tropical hypersurfaces ∩iT ( fi ) is a called a tropical preva-riety.

In our previous work, we computed a tropical prevariety of a set of generators { fi }by first computing the normal fan of Minkowski sum of all the Newton polytopes of thegenerators, and then discarding directions ω for which there was a monomial inω fi .This was very inefficient for our equations as there were many trivial (i.e. monomial-inducing) cones in the normal fan. In contrast, Gfan is able to directly compute thetropical prevariety efficiently using linear programming. This dramatic improvementmeans that this computation not only ceases to be a bottleneck in our calculationsbut is fast enough to allow for an iterative approach to finding a tropical basis, or at

Finiteness of Kite Relative Equilibria 353

least an improvement on the original set of generators. Gfan is also able to exploit thesymmetry of ideals, which will be very important for work on the finiteness problemfor the general five-body problems.

2 Results for the Five-Vortex Kite Problem

Our starting equations were the Albouy-Chenciner equations and geometric con-straints described in Sect. 1 with substitutions si j = r2

i j . All powers of the ri j wereeven except for the collinearity constraint r13 = r12 + r23. The simplest polynomialin the si j which contains the collinearity constraint is:

pc = s212 − 2s12s13 + s2

13 − 2s12s23 − 2s13s23 + s223 (3)

so that was included in the generators of our ideal instead of the original collinearityconstraint. This may introduce some spurious solutions but the reduction in degreeobtained by using the si j seems to make the tradeoff worthwhile.

These starting equations were not enough to easily determine finiteness conditionsso more were added. Throughout the remaining calculations we assume that the sumof the vorticities is equal to 1, which results in no loss of generality unless the sum ofthe vorticities is zero. Since the Albouy-Chenciner equations are not valid in that case,a different approach would be necessary anyway and we will not consider it here. Theequations of Celli, described in [3,7], could be used instead.

The first equations added were resultants of pc with the Albouy-Chencinerequations f12 and f13 with respect to the variable s12 (i.e. Res(pc, f12; s12) andRes(pc, f13; s12)). Adding further resultants of this type did not seem to change thetropical prevariety. At this stage it was still difficult to obtain simple conditions forfiniteness so one more equation was added.

To search for more useful equations a series of small Groebner bases were com-puted. Each of the these used the degree reverse lexicographic term order with s12 �s13 � s14 � s23 � s24 � s34 � s45 � m1 � m2 � m3 � m4. The generators weretwo of the Albouy-Chenciner polynomials fi j , the collinearity polynomial pc, and themass normalization polynomial pm = m1 + m2 + m3 + 2m4 − 1. For the pair f13and f14, one polynomial from the Gröbner basis refined the tropical prevariety enoughto get some finiteness conditions. This polynomial can be expressed in terms of theoriginal generators as c1 f13 + c2 f14 + c3 pm + c4 pc where the ci are as shown below.

c1 = 5

4m2

2 + 1

2m2m4 − 1

2m2s12 + 1

2m2s14 − 1

2m4s24 − s14s24 − m2 + s24,

c2 = 1

2m2 + s23,

c3 = −5

2m2

2s12s13s14s23s34 − m2m4s12s13s14s23s34 + m2s212s13s14s23s34

−m2s12s13s214s23s34 − m2s12s13s2

14s24s34 + m4s12s13s14s23s24s34

−1

4m2

2s212s14s34 + 1

2m2

2s12s13s14s34 − 1

4m2

2s213s14s34 + 3m2

2s12s14s23s34

354 M. Hampton

+m2m4s12s14s23s34 − m2s212s14s23s34 + 1

2m2

2s13s14s23s34

+2m2s12s13s14s23s34 + m2s12s214s23s34 − 1

4m2

2s14s223s34 + 1

4m2s2

12s14s24s34

+1

2m2s12s13s14s24s34 + 1

4m2s2

13s14s24s34 − 1

2m2s12s14s23s24s34

−m4s12s14s23s24s34 − 1

2m2s13s14s23s24s34 − 2s12s2

14s23s24s34

+1

4m2s14s2

23s24s34 − 2m2s12s14s23s34 + 2s12s14s23s24s34,

c4 = −m2m4s13s14s23 + m2m4s214s23 + 1

4m1m2

2s14s34 − m32s14s34 + 1

4m2

2m3s14s34

+1

2m2

2s12s14s34 − 1

2m2

2s214s34 − m2m4s13s23s34 + m2

2s14s23s34

−1

4m1m2s14s24s34 − 1

4m2

2s14s24s34 − 1

4m2m3s14s24s34 + m2s2

14s24s34

+m2m4s23s234 + 3

4m2

2s14s34 − m2s14s23s34 − 3

4m2s14s24s34.

We find that there are finitely many solutions unless at least one of the followingpolynomials vanishes:

1. m1 + m32. m2 + m33. m1 + m24. m2m3 − m2

4

5. m2m3 + m23 + 2m3m4 + m2

4 − m3

6. (*) m22 + m2m3 + 2m2m4 + m2

4 − m2

7. m22 + 2m2m4 + m2

4 − m2

8. m23 + 2m3m4 + m2

4 − m3

9. m22 + 2m2m3 + m2

3 + 2m2m4 + 2m3m4 + m24 − m2 − m3

The polynomial marked by (*) is compatible with Roberts’ rhombus solution curvewith m1 = m3 = m4 and m2 = −m1/2.

3 Results for the Newtonian Five-Body Problem

As in the 5-vortex case, we began with the Albouy-Chenciner equations and the geo-metric constraints for kites. However, two seperate cases were considered, dependingon whether r12 = r23 or not. For the case that these distances were not equal, a sortof slack variable z was used by including the polynomial r12 − r23 + z in the geo-metric constraints. In each case a Gröbner basis was calculated for only the geometricconstraints. We computed the tropical prevarieties for these bases plus the Albouy-Chenciner equations. In each case there were 14 generators for the ideal.

Finiteness of Kite Relative Equilibria 355

For the first case (when r12 = r23), there are finitely many solutions unless m2 =4m1 and m1 = −m4/2, or m1 = m3 = m4.

For the second case, we find that there are finitely many solutions unless at leastone of the following polynomials vanish:

1. m2 − 4m32. m2 + 4m3 − 43. (*) m2 + 4m3 − 14. 4m2

3 − m24

5. (*) m1 − 3m3 + 2m46. m1 − 3m3 + 2m4 + 37. m1 + 5m3 + 2m4 − 18. (*) 3m2

3 − 2m3m4 − m24

9. 12m23 − 8m3m4 + m2

4 − 24m3 + 8m4 + 12

The polynomials marked by (*) are compatible with Roberts’ rhombus solutioncurve with m1 = m3 = m4 and m2 = −m1/4.

4 Conclusion and Future Directions

In this paper we have proven that there are finitely many kite configurations for thefive-vortex and Newtonian five-body problem if certain explicit polynomials in theparameters (masses or vorticities) are nonzero.

There is definitely a lot of scope for improvement and systematization in thesemethods. Some possibilities include:

1. Systematic exploration of term orders. The work done here only used a fixeddegree reverse lexicographic term order when reducing polynomials and com-puting Gröbner bases. It would almost certainly be helpful to explore the effectof varying the term orders. In most circumstances it is not feasible to use everypossible term order. Some sort of dynamic optimization of which order to usemight be invented; this overlaps a great deal with the capabilities of Gfan but witha different focus.

2. Coordination with fast Gröbner basis implementations. For the purposes of prov-ing finiteness, Gröbner bases per se may be overkill. It would be useful to havean efficient algorithm for partially constructing a set of Gröbner bases, one whichwould terminate if it found a useful new leading term - i.e. a polynomial in theideal which, when added to the generators, would change the tropical prevariety.Ideally this could be targeted to a particular cone in the tropical prevariety.

3. Use of the mixed volume. Often a particular system of polynomials has a mixedvolume larger than the number of actual nonzero finite solutions. This happens forthe Albouy-Chenciner equations in the Newtonian case for n = 4 and n = 5, andpresumably for all larger n as well. In such a case the mixed volume may offer agood criterion to alternative equivalent systems - i.e. it is the author’s guess that asystem that realized its mixed volume would have better properties with respectto our finiteness methods.

356 M. Hampton

We hope these or other improvements will enable us to completely characterize theparameters for which there are finitely many solutions of the full five-body problems.

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