In this paper we provide a partial answer to a question recently posed by Hassan Aref et. al. in their article Vortex Crystals, namely whether there are certain singular equilibria of point vortices. We prove that there are no such equilibria in the four-vortex case

Hampton, Marshall

Peterson, Andrea

Stoller, Heather

Wang, Albert

English

Furman University

Furman UniversityElectronic Journal of Undergraduate MathematicsVolume 10, 1 – 4, 2005ON THE NONEXISTENCE OF SINGULAR EQUILIBRIA IN THEFOUR-VORTEX PROBLEMMARSHALL HAMPTON, ANDREA PETERSON, HEATHER STOLLER, AND ALBERTWANGAbstract. In this paper we provide a partial answer to a question recentlyposed by Hassan Aref et. al. in their article Vortex Crystals, namely whetherthere are certain singular equilibria of point vortices. We prove that there areno such equilibria in the four-vortex case.1. IntroductionThe starting point of our discussion is the set of point-vortex equations for Ninteracting vortices a = 1, 2, . . . , N with circulations Γa and (complex) positions zi:dzidt=12pii∑j 6=iΓjzi − zj .This system was introduced by Helmholtz [H] to model a two-dimensional slice ofcolumnar vortex filaments, with some refinements by Lord Kelvin [T] and Kirchhoff[K]. An extensive bibliography on the subject can be found in [N]. It is worthnoting that this system can be written in Hamiltonian form with HamiltonianH =∑i<j ΓiΓj log |zi − zj |, where the symplectic pairs of variables are multiplesof the real and imaginary parts of each zi.A vortex equilibrium is a configuration of vortices such that dzjdt = 0 for all j.We are concerned here with the following special type of vortex equilibrium:Definition 1.1. A singular equilibrium is an equilibrium such that L =∑i<j ΓiΓj |zi−zj |2 = 0, K =∑i<j ΓiΓj = 0, and S =∑i Γi 6= 0.It is already known that there are no singular equilibria in the three-vortexproblem [ANST], where it is also shown that a rigidly rotating configuration ofvortices has an angular speed of ω = SK4piL . Our introduction of the term singularequilibrium refers to the indeterminancy of this expression for the angular speed.2. Nonexistence of the four-vortex singular equilibriaWe will prove the following theorem:Theorem 2.1. There are no four-vortex singular equilibria.Received by the editors June 14, 2005.2000 Mathematics Subject Classification. Primary 70Fxx, 37N10, 76Bxx.Key words and phrases. Point vortices, equilibria.This paper was written in the summer of 2004 by the first-listed author who at the time ofthe work was a mathematics faculty member at the University of Minnesota, and the other threeauthors who were high school students at the time the work was conducted.12 M. HAMPTON, A. PETERSON, H. STOLLER, AND A. WANGProof. Our calculations can be greatly simplified by a few assumptions. Sincesetting z3 = 0 and z4 = 1 simply scales the relative distances between the vorticesand setting Γ4 = 1 scales the circulations, we can work under these conventionswithout loss of generality. From the point-vortex equations, O’Neil [O] gives thetwo solutions for four-vortex equilibria:z1 =2 + Γ2 ± i√3Γ22(1 + Γ2 + Γ3)andz2 =2 + Γ1 ∓ i√3Γ12(1 + Γ1 + Γ3).We can use the relationK =∑i<j ΓiΓj = 0 to eliminate Γ3 from these equations:Γ3 =Γ1 + Γ2 + Γ1 Γ21 + Γ1 + Γ2.Note that we cannot have 1+Γ1+Γ2 = 0 since then K reduces to −(1+Γ2+Γ22)which cannot be zero for real vorticities.This gives usz1 =(2 + Γ2)(1 + Γ1 + Γ2) + i√3Γ2(1 + Γ1 + Γ2)2(1 + Γ2 + Γ22)andz2 =(2 + Γ1)(1 + Γ1 + Γ2) + i√3Γ1(1 + Γ1 + Γ2)2(1 + Γ1 + Γ21)for the positions of the first two vortices in a singular equilibrium.Now we can use these expressions for z1 and z2, along with our conventionsz3 = 0 and z4 = 1 to find the squared distances d2ij = |zi − zj |2:d212 =(Γ21 + Γ1Γ2 + Γ22)(1 + Γ1 + Γ2)2)(1 + Γ1 + Γ21)(1 + Γ2 + Γ22)d213 =Γ21 + Γ1Γ2 + Γ221 + Γ2 + Γ22d214 =(1 + Γ1 + Γ2)21 + Γ2 + Γ22d223 =Γ21 + Γ1Γ2 + Γ221 + Γ1 + Γ21d224 =(1 + Γ1 + Γ2)21 + Γ1 + Γ21d234 = 1.Now we substitute these expressions in to the original equation for L.L =3(Γ21 + Γ31 + Γ41 + Γ1Γ2 + Γ21Γ2 + Γ31Γ2 + Γ41Γ2 + Γ22 + Γ1Γ22 + Γ31Γ22 + Γ41Γ22 + Γ32+Γ1Γ32 + Γ21Γ32 + Γ31Γ32 + Γ42 + Γ1Γ42 + Γ21Γ42)/((1 + Γ1 + Γ21)(1 + Γ2 + Γ22)).FOUR-VORTEX SINGULAR EQUILIBRIA 3The expression in the denominator is always positive. Now all that remains isto determine the sign of the numerator in parentheses, N(Γ1,Γ2). If it is alwayspositive on R2 − (0, 0) we will have proven our claim, namely that there are nofour-vortex stationary equilibria with L = 0. We start with a lemma:Lemma 2.2. ∂2N∂Γ21and ∂2N∂Γ22are non-negative.Proof. Since N is symmetric in Γ1 and Γ2 it suffices the prove the lemma for ∂2N∂Γ21.This is a quadratic function of Γ1, whose minimum (for a fixed Γ2) is(1 + Γ2)2(5 + 2Γ22 + 5Γ42)8(1 + Γ2 + Γ22)≥ 0.This lemma implies that ∂N∂Γ1 and∂N∂Γ2are monotone functions of Γ1 and Γ2respectively. Thus they have at most one zero for each fixed Γ2 (for ∂N∂Γ1 ) and Γ1(for ∂N∂Γ2 ).We need a further lemma to reach our goal:Lemma 2.3. For each fixed Γ2, ∂N∂Γ1 has its unique zero between Γ1 = Γ2 andΓ1 = −Γ2.Proof. We simply compute that∂N∂Γ1(Γ2,Γ2) = Γ2(3 + 6Γ2 + 8Γ22 + 10Γ32 + 9Γ92).Using Sturm’s theorem it is not hard to show that the above polynomial is alwayspositive for Γ2 > 0 and always negative for Γ2 < 0. Likewise, from the calculation∂N∂Γ1(−Γ2,Γ2) = −Γ2(1 + 2Γ2 + 2Γ32 + 3Γ92)we can find that ∂N∂Γ1 (−Γ2,Γ2) is always negative for Γ2 > 0 and positive for Γ2 < 0.Combined with the monotonicity of ∂N∂Γ1 as a function of Γ1 this completes thelemma.Since N is symmetric, ∂N∂Γ1 (Γ1,Γ2) =∂N∂Γ2(Γ2,Γ1). Lemma 2.3 then implies thatthe gradient of N can only be zero at the origin, since otherwise the two partials canonly vanish in the disjoint open cones bounded by the lines Γ1 = Γ2 and Γ1 = −Γ2.Thus the origin is the only critical point of N. It is elementary to compute that theorigin is a minimum of N , and thus the unique global minimum.3. ConclusionThe nonexistence of singlar equilibria in the three- and four-vortex problemsnaturally prompts the question of whether such an equilibrium can exist for alarger number of vortices. It seems quite possible that there is a more generalargument which would show the nonexistence of singular equilibria for any numberof vortices, but we are unaware of a strategy for conducting such a proof.4 M. HAMPTON, A. PETERSON, H. STOLLER, AND A. WANGBibliography[ANST] Aref, H., Newton, P. K., Stremler, M.A., Tokieda, T., and Vainchtein, D. L.,2003, Vortex Crystals, Advances In Applied Mechanics 39, 1-79.[G] Gro¨bli, W., 1877, Specielle Probleme u¨ber die Bewegung geradliniger parallelerWirfbelfa¨den, Vierteljahrschrift der naturforschenden Gesellschaft in Zu¨rich22, 37-81 , 129-165.[H] Helmholtz, H., 1858, Uber Integrale der hydrodynamischen Gleichungen, Welcheden Wirbelbewegungen entsprechen, Crelle’s Journal fu¨r Mathematik 55, 25-55. English translation by Tait, P.G., 1867, On the integrals of the hydro-dynamical equations which express vortex-motion, Philosophical Magazine,485-51.[K] Kirchhoff, G.R., 1876, Vorlesungen u¨ber Mathenatische Physik I, Teubner,Leipzig.[N] Newton, P. K., 2001 ,The N-Vortex Problem: Analytical Techniques, Appl.Math. Sci. Series 145, Springer-Verlag, New York, NY.[O] O’Neil, K.A., Stationary configurations of point vortices, Trans. AMS 302, 2,383-425.[S] Synge, J.L., On the motion of three vortices, Can. J. Math. 1, 257-270.[T] Thomson, Sir W. (Lord Kelvin), 1867, On vortex atoms, Proc. R. Soc. Edin-burgh 6, 94–105.Department of Mathematics,, University of Minnesota Duluth, Duluth, MN 55812E-mail address: mhampton@d.umn.eduE-mail address: andidawn@msn.comE-mail address: pwafithwack@ourtownusa.netE-mail address: albertw12345@msn.com

On the Nonexistence of Singular Equilibria in the Four-vortex Problem

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On the Nonexistence of Singular Equilibria in the Four-vortex Problem

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