Exact solution of the planar motion of three arbitrary point vortices
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31 November 2015 11:13 80 Years’ Birthday of Professor Hao Bailin 9in x 6in 2nd Reading b2111-Conte-de-Seze page 1
Exact solution of the planar motion of threearbitrary point vortices
R. Conte and L. de SezeService de physique du solide et de r´esonance magn´etique,CEN Saclay, France. DPhG/PSRM/1697/80
We give an exact quantitative solution for the motion of three vorticesof any strength, which Poincar´e showed to be integrable. The absolutemotion of one vortex is generally biperiodic: in uniformly rotating axes,the motion is periodic. There are two kinds of relative equilibrium con-figuration: two equilateral triangles and one or three colinear configu-rations, their stability conditions split the strengths space into threedomains in which the sets of trajectories are topologically distinct.According to the values of the strengths and the initial positions,all possible motions are classified. Two sets of strengths lead to genericmotions other than biperiodic. First, when the angular momentum van-ishes, besides the biperiodic regime there exists an expansion spiralmotion and even a triple collision in a finite time, but the latter motionis nongeneric. Second, when two strengths are opposite, the system alsoexhibits the elastic diffusion of a vortex doublet by the third vortex.For given values of the invariants, the volume of the phase spaceof this Hamiltonian system is proportional to the period of the reducedmotion, a well known result of the theory of adiabatic invariants. We thenformally examine the behaviour of the quantities that Onsager definedonly for a large number of interacting vortices.
Introduction
Known from a long time, the problem of the motion of a system of pointvortices in interaction presents a particular interest for the study of bidi-mensional turbulence. As predicted by Onsager (1949) the study of the ther-modynamics of a large system of vortices shows the possibility of negative R. Conte and L. de Seze temperatures and eddy viscosities (see e.g. Lundgren and Pointin 1977).The case of a small number of vortices was studied, long ago, by Lord Kelvinand Mayer (1878). They used experimental methods to obtain some resultson the stability of simple geometric configurations. Recently Novikov (1975)pointed out the interest of the three vortex system as the first elementaryinteraction process in isotropic turbulence kinetics. Using properties of thetriangle he was able to solve the problem of three identical vortices.While this work was under submission, the referees pointed out to us theexistence of a paper to appear. In this paper, Hassan Aref (March 1979),extending the work of Novikov with a symmetry-preserving presentation,qualitatively solved the relative motion of three vortices and undertooka classification of the topology of the phase space; however, he gave noindication on the nature of the absolute motion and no quantitative results,except for two special cases of great importance which he solved completely:the direct or exchange scattering when two strengths are equal and oppositeto the third one, and the self-similar motion of a triple collision or a tripleexpansion to infinity when both the inertia momentum and the angularmomentum vanish.In this paper, we present a method which quantitatively gives the abso-lute motion in all cases of strengths or initial conditions. The main idea is tointroduce reduced variables with a very simple geometrical interpretationand whose number matches the number of degrees of freedom of the sys-tem. For three vortices, this can be done by defining one complex variable ζ which characterizes the shape of the triangle. Knowing the motion of ζ ,which happens to be periodic, is then sufficient to derive the behaviour ofany physically interesting variable. We thus have obtained detailed results,which resort mainly to fluid mechanics and partly to the field of differen-tiable dynamical systems.In the first part, we describe the problem, introduce the reduced motionand discuss its advantages and inconvenients. In the second part, we give thegeneral solution: generically, i.e. for arbitrary strengths and initial positions,the absolute motion of any given vortex is the product of a periodic motionof period T by a uniform rotation or, stated in other words, the motion isperiodic when referred to uniformly rotating axes centered at the center ofvorticity; in the case of a vortex plasma (zero total strength), the uniformrotation is merely replaced by a uniform translation. Due to its frequentoccurrence in nature, we give special consideration to the case of two orthree equal strengths: in a given domain of initial conditions, vortices with Exact solution of the planar motion of three arbitrary point vortices equal strengths have the same motion, up to a shift of T or T in time anda rotation in space.The third part, following the Smale’s method of study of a differentiabledynamical system, is devoted to the finding of the bifurcation set, i.e. the setof values of the invariants for which the nature of the motion changes. Thisreduces to the finding of the relative equilibria, which are shown to be of twotypes, the same than those of the three body problem of celestial mechanics:two equilateral triangles and one or three colinear configurations. The num-ber and stability of these relative equilibria are discussed, thus leading to aseparation of the strengths space in three principal domains where the setsof orbits are topologically different; an important result is that, whateverbe the strengths domain, the phase space is always multiconnected; forstrengths on the boundaries of these domains, the motion must be studiedseparately for it presents special features.In the fourth part, we study the absolute motion whenever it differsfrom the general biperiodic case, i.e. when strengths are on the boundariesof their limiting domains or when initial positions are those of a relativeequilibrium. At a relative equilibrium the motion is a rigid body rotation.Otherwise two new generic motions are found. First, for strengths such thatthe angular momentum vanishes, the vortices can go to infinity in a spiralmotion with a fixed asymptotic shape of the triangle; there even exists aspiral motion ending in a triple collision in a finite time, but it is nongenericsince it happens only for a zero value of the inertia momentum. Second,when two strengths are opposite, besides the doubly periodic motion anelastic scattering can occur in which a vortex doublet is diffused by thethird vortex; if moreover the third vortex has the strength of one of the twoothers, a third generic motion exists which is an exchange scattering.The fifth part consists in formally examining the behaviour of the ther-modynamical quantities, of course meaningless for this integrable system,that Onsager defined for a large number of vortices; the volume of the phasespace is found to be equal to the period of the relative motion; among thefeatures which could be indicative for a large number of vortices is a possiblelack of ergodicity due to a multiple connexity of the phase space.
1. Description of the problem
We consider three point vortices M j ( j = 1 , ,
3) in a plane with strengths κ j and given initial positions. Their motion is ruled by the first order R. Conte and L. de Seze differential system: ∀ j = 1 , , πi d¯ z j d t = X ℓ =1 ℓ = j κ ℓ z j − z ℓ , (1)where z = x + iy and the bar denotes the complex conjugation. This systemis equivalent to the set of Hamilton equations for the Hamiltonian: H = − π X jj < X ℓℓ κ j κ ℓ Log | z j − z ℓ | , (2)in the conjugate variables p | κ j | x j and p | κ j | sign ( κ j ) y j , j = 1 , ,
3; there-fore H is an invariant and remains equal to the energy E . The invarianceof H under translation and rotation yields two other invariants:the impulse B = X j κ j z j = X + iY,I = X j κ j | z j | , which Poincar´e called inertia momentum.In fact we shall use, instead of I , the invariant J = X jj < X ℓℓ κ j κ ℓ | z j − z ℓ | = ( κ + κ + κ ) I − | B | , (3)which depends only on the relative positions of the vortices. There are6 − κ κ κ ( z − z )( z − z )( z − z ) = 0.Among the six Poisson brackets built from the four known integrals ofmotion H, J, X, Y , only one is nonzero: { X, Y } = X j κ j = K. Since a Hamiltonian system with 2 N variables is integrable in the senseof Liouville when it has N independent invariants in involution, the threevortex system ( N = 3) is therefore integrable (Poincar´e, 1893). The purposeof this paper is to integrate it.Let us remark that this problem is not affected by the adjunction of anexternal velocity field made of a uniform translation and a uniform rotation, Exact solution of the planar motion of three arbitrary point vortices since the new equations of motion2 πi d¯ z j d t = ′ X ℓ κ ℓ z j − z ℓ + 2 πi ¯ v + 2 πω ¯ z j , ω ∈ R , v ∈ C reduce to the original ones by the change of variables Z j = ivω + (cid:18) z j − ivω (cid:19) e − iωt . The reduced motion
The main idea is to match the number of variables and the number ofdegrees of freedom of this system, so as to keep the minimum number ofindependent variables. For this purpose, let us define a time dependentcomplex plane ζ in which two of the three point vortices remain fixed. Thiscan be achieved by the following transformation z → ζ = ( κ + κ ) z − ( κ z + κ z ) z − z , (4)where 2 and 3 number two vortices whose sum of strengths is nonzero; theaffixes of these two vortices become κ and − κ under the transformation.It is clearly seen from the definition how a reduced point is geometricallydeduced from an absolute point. We shall simply note ζ the transformedof M : ζ = ξ + iη = Kz − Bz − z ,ζ therefore represents the shape of the triangle, and the inverse transfor-mation is represented by z = z , z = ( κ K − κ ζ ) z + ( ζ − κ ) Bsζ ,z = ( − κ K − κ ζ ) z + ( ζ + κ ) Bsζ , (5)where s = κ + κ = 0.The reader will have noticed the main disadvantage of the above defini-tion of two reduced coordinates ξ, η : it does not reflect the invariance of theproblem under the permutations of the three elements ( κ j , z j ) and there-fore every result we can get may be uneasy to interpret. Nevertheless, theadvantages are numerous. First, every physical quantity can systematicallybe expressed, as we shall soon see, as a function of ζ and Kz − B only; more-over, since ζ is invariant under a change of length and Kz − B extensive in R. Conte and L. de Seze the lengths, such a physical function will quite generally be the product ofa function of ζ by a function of Kz − B and we are going to see that thisuncoupling between intensive and extensive variables will enable us to solvethe motion not only qualitatively but also quantitatively. Secondly, unlikeNovikov and Aref, we do not have to eliminate some unphysical portionsof our ζ plane (which will be seen to be the initial conditions plane) sinceevery ζ point describes a physical situation. Thirdly, this reduction leavesonly the required number of degrees of freedom.
2. The solution for the general case
To obtain the absolute motion we need only determine the evolution of ζ and z , since we have the parametric representation (5). Provided K and B do not simultaneously vanish the motions of z and ζ are ruled by:2 πi d¯ z d t = 1 Kz − B s ζ ( ζ + d )( ζ + κ )( ζ − κ ) , (6)2 πi d¯ ζ d t = − | Kz − B | s | ζ | [¯ ζ ( ζ + dζ − Q ) − sK ( ζ + d )]( ζ + κ )( ζ − κ ) , (7)where d = κ − κ and Q = κ κ + κ κ + κ κ .We can express the invariants E and J as functions of z and ζ : J = | Kz − B | κ | ζ | + κ κ Ks | ζ | , (8) e − πE = | Kz − B | Q | ζ | − Q (cid:12)(cid:12)(cid:12)(cid:12) ζ − κ s (cid:12)(cid:12)(cid:12)(cid:12) κ κ (cid:12)(cid:12)(cid:12)(cid:12) ζ + κ s (cid:12)(cid:12)(cid:12)(cid:12) κ κ , (9)expressions where we notice the factorized dependency on ζ and z .Unless Q and J simultaneously vanish, at least one of the two aboveequations expresses | Kz − B | as a function of ζ and from (7) we obtain afirst order differential system for ζ . For example if J is nonzero the elimi-nation of z between (7) and (8) gives:2 πi d¯ ζ d t = − ( κ | ζ | + κ κ K )[¯ ζ ( ζ + dζ − Q ) − sK ( ζ + d )] J ( ζ + κ )( ζ − κ ) . (10) Exact solution of the planar motion of three arbitrary point vortices There is no need to solve this system since the equation of the reducedtrajectory is given by the straightforward elimination of z between (8)and (9): κ | ζ | + κ κ KsQ (cid:12)(cid:12)(cid:12)(cid:12) ζ − κ s (cid:12)(cid:12)(cid:12)(cid:12) − κ κ Q (cid:12)(cid:12)(cid:12)(cid:12) ζ + κ s (cid:12)(cid:12)(cid:12)(cid:12) − κ κ Q = JQ e π EQ . (11)This represents in the ζ plane a set of closed orbits, indexed by the non-dimensional variable c = JQ e πEQ which is invariant under a change of lengthor a change of unit of vorticity.The set of orbits is symmetrical relative to the ξ axis and also, when κ equals κ , to the η axis. When two vortices are close to each other, theyremain as such and therefore the ζ curves are near to circles in the vicinityof − κ , κ and ∞ . Figures 2 to 3 show examples of the ζ plane.
350 351 140 252 250Kk <0 K k > K ( k + k ) < K ( k + k ) > K ( k + k ) < K k >
33 30 34030Q>0Q>0 ∆ >
352 151 150 152 BC Fig. 1. The strengths space. The numbering of regions reflects the ternary symmetry. R. Conte and L. de Seze –4 –3 –2domain 351K =–2 , K = 1 , K =4 P–1 0 1 2 3 4M T T M c i r c l e j = Fig. 2. ζ plane in the domain 351 ( Q <
0, ∆ > κ = − , κ = 1 , κ = 4. Since there is in general no stationary point on the orbit, the reducedmotion is periodic and the period is expressed by T = − πJs I | ( ζ + κ )( ζ − κ ) | ( κ | ζ | + κ κ K ) d | ζ | Im( ζ + dζ ) . (12)When J is zero we use (9) instead of (8) to obtain a similar expression,a result which shows that the condition J = 0 alone represents nothingspecial as one would believe in the Aref classification (for more details seethe third part of this paper). To have a dimensionless result we can takeas unit of time T u = π JQK which is, as we shall see later, the period ofthe absolute motion when the vortices are in a configuration of relativeequilibrium; therefore, in every domain of the multiply-connected ζ plane, TT u depends only on c . Exact solution of the planar motion of three arbitrary point vortices P P P M x T x T M –0.5–1–1.5 0 0.5 1 1.5domain 1K = –2 , K = 5 , K =9 c i r c l e j = Fig. 3. ζ plane in the domain 1 ( Q >
0, ∆ <
0) domain 1 κ = − , κ = 5 , κ = 9. The absolute motion
Using the parametric representation of the z j ’s we easily obtain the follow-ing relations: Kz − Bsζ = Kz − Bκ K − κ ζ = Kz − B − κ K − κ ζ . (13)Then, using (8) or (9), we conclude that, when the reduced motion isperiodic, the modulus of z j − BK has the period T of the reduced motion.As to the arguments of these affixes, after one period they have all been R. Conte and L. de Seze M P J = T T P P M domain 12K = –6 , K = 7 , K =11 Fig. 4. ζ plane in the domain 12 ( Q <
0, ∆ <
0) domain 12 κ = − , κ = 7 , κ = 11. increased by the same value, modulo 2 π : ∆ ϕ = (cid:20) arg (cid:18) z − BK (cid:19)(cid:21) To = I − K Re { ( ζ + dζ )(¯ ζ + d ¯ ζ − κ κ ) } | ζ | ( κ | ζ | + κ κ K )Im( ζ + dζ ) d | ζ | = ∆ ϕ, ∆ ϕ = ∆ ϕ + (cid:20) arg (cid:18) κ K − κ ζsζ (cid:19)(cid:21) To = ∆ ϕ modulo 2 π, ∆ ϕ = ∆ ϕ + (cid:20) arg (cid:18) − κ K − κ ζsζ (cid:19)(cid:21) To = ∆ ϕ modulo 2 π. (14) Exact solution of the planar motion of three arbitrary point vortices M M P P P T Q = 0 domain (12 and 1)K = –2 , K = 3 , K =6 Fig. 2. ζ plane on the line Q = 0. The nonperiodic domain is hatched Q = 0 domain(12 and 1) κ = − , κ = 3 , κ = 6. For the motion it means that, after a time interval of T , the shape andsize of the triangle are again the same, i.e. the new positions are deducedfrom their initial values by a rotation of ∆ ϕ around the center of vorticity.Depending on the domain of initial conditions, the number of turns aroundthe barycentrum may vary from one vortex to another by an integer value.Therefore, when K is nonzero, the absolute motion of any vortex is theproduct of a uniform rotation about the barycentrum and of a periodicmotion, the two periods being the same for the three vortices: ∀ j = 1 , , z j ( t ) = BK + (cid:18) z j (0) − BK (cid:19) e i ∆ ϕ tT f j (cid:18) tT (cid:19) , where f j is periodic with period 1. In other words, in uniformly rotatingaxes centered at the barycentrum, the absolute motion is periodic. Figure 8shows the absolute trajectory of M both in fixed axes and in rotating axes,for ~κ = ( − , ,
4) and ζ o = − K (the ζ orbit is the small curve surrounding P in Figure 2). R. Conte and L. de Seze P –2 2 33–3 P M P M T c i r c l e j = domain (11 and 12)K = –2 , K = 2 , K =3 Fig. 3. ζ plane on the line (11 and 12). The diffusion domain is hatched domain (11and 12) κ = − , κ = 2 , κ = 3. Some symmetries may exist in the absolute motion: for instance if the ζ axis is an axis of symmetry of the ζ orbit, then at intervals distant of T the vortices are colinear; if we choose for origin of time an instant ofcolinearity, the absolute trajectory of every vortex, when run between 0and T , possesses an axis of symmetry (see Figure 8).Let us now mention an important result concerning this dynamical sys-tem: since integral (14) is a continuous function of both the strengths ~κK and the initial conditions, the angle ∆ ϕ is generically incommensurablewith 2 π , i.e. commensurability occurs only for a set of strengths and initialconditions of zero measure. We conclude that generically, due to the incom-mensurability of ∆ ϕ with 2 π , the trajectory of a given vortex completely fillsan annulus centered at the barycentrum (see Landau and Lifchitz Figure 9Chap. III).In the case of a vortex plasma ( K = 0) and when the barycentrum is atinfinity ( B = 0), the variation of z j over one period does not depend on j Exact solution of the planar motion of three arbitrary point vortices E Ωτ E Fig. 7. Volume Ω and temperature τ versus energy in the domain 0( k j K > ~κ = (2 , , and is given by iB [¯ z j ] To = I ( ζ + dζ )(¯ ζ + d ¯ ζ − κ κ )2 κ | ζ | Im( ζ + dζ ) d | ζ | , (15)which evaluates to a real quantity. The rotation has become a uniformtranslation in the direction normal to the direction of the impulse. Thecommon mean velocity of the vortices is [ z ] /T and, for instance in thecase of an initial equilateral triangle where the absolute motion is a uniformtranslation, this velocity evaluates to iQ πB . R. Conte and L. de Seze = –2 , K = 1 , K =4 T2– Fig. 8. Absolute motion z ( t ) (left) and rotating motion z ( t ) e − i ∆ ϕ tT (right), − T 21 1 3 1 t = 0t = 00 Ht = + ∞ t = – ∞ M Fig. 9. Absolute motion of the three vortices for an elastic diffusion. ~κ = ( − , , ζ o = − K ( c = 1 . Exact solution of the planar motion of three arbitrary point vortices Let us conclude all that with a geometrical remark: the relations κ | Kz − B | + κ κ K | z − z | = sJ, and 2 π d | z − z | d t = 2 κ s Im( ζ + dζ ) | ( ζ + κ )( ζ − κ ) | , show that the extrema of | z j − BK | occur simultaneously with those of | z ℓ − z m | ( j, ℓ, m permutation of 1 , , 3) and that they are reached when thetriangle is either flat or isosceles with M j as a summit. The absolute motion for two or three equal strengths An important practical case is that of the invariance of a ζ orbit underone of the six permutations of the three elements ( κ j , z j ). For instance thepermutation (132) acts by: κ → κ , κ → κ and is seen to leave the set of ζ orbits invariant provided κ = κ . Since most of the vortices encounteredin nature have equal, or opposite, strengths, we shall consider this particularcase in some detail.Let us therefore assume κ = κ and consider a ζ orbit having the ori-gin as a center of symmetry (e.g. the circle J = 0 on the Figure 3 assumedcontinuously deformed so as to admit the origin as a center of symmetry;note that, on the same figure, the orbit surrounding T has not the requiredproperty). Having chosen such an initial condition t = 0, ζ = ζ o , during theevolution there will happen a time when ζ evaluates to − ζ o ; this time is nec-essarily equal to the half-period T / t = 0, ζ = ζ o ) and( t = T , ζ = − ζ o ) are identical from a point of view of initial conditions, weconclude to the identity of the motions for 0 < t < T and for T < t < T ; thetriangle at t = T is deduced from the triangle at t = 0 by some fixed rotationaround the barycentrum (a change of size and a translation are excludedbecause of the invariants) and of course by the exchange of 2 and 3 ( B = 0): ∀ t : z ( t ) = z (cid:18) t − T (cid:19) e iα , z ( t ) = z (cid:18) t − T (cid:19) e iα ,z ( t ) = z (cid:18) t − T (cid:19) e iα . By iterating, we see that 2 α is equal to ∆ ϕ modulo 2 π . Therefore theabsolute motions of 2 and 3 are identical, up to a rotation in space and atranslation in time. Every absolute trajectory has two independent axes ofsymmetry and therefore, an infinity: they are defined by the barycentrum R. Conte and L. de Seze and the absolute positions of the vortex when the triangle is either isoscelesor flat (i.e. when ζ crosses one of its two symmetry axes).If K is zero, i.e. ~κ = ( − , , L/ L being the translation after oneperiod; the initial conditions which have the required symmetry are definedby 0 < c < √ < c . The shortest of the three mutual distances is always the same,say M M , the ζ orbit is invariant by a permutation of 2 and 3 and theabove conclusions apply. The three vortices are colinear at intervals of T / 2. Choosing for t = 0 a colinear configuration, the triangle is isoscelesat t = T + n T , n ∈ Z and the summit of the triangle is then alwaysthe same vortex M .(2) 1 < c < √ 2, i.e. a vicinity of the equilateral configurations. The ζ orbitis invariant under any circular permutation and, using quite similararguments, we deduce for instance ( B = 0): ∀ t : z ( t ) = z (cid:18) t − T (cid:19) e iα , z ( t ) = z (cid:18) t − T (cid:19) e iα ,z ( t ) = z (cid:18) t − T (cid:19) e iα , with 3 α = ∆ ϕ modulo 2 π . This time, the three motions are identical.The vortices are never colinear. Choosing for t = 0 an isosceles con-figuration, we see that the triangle regains the same shape at t = n T ,successively with the summits 1 , , , , t = T + n T .The period T , that Novikov gave as a hyperelliptic integral, is reducibleto an ordinary elliptic integral (see Appendix I). 3. The relative equilibria and the bifurcation set In two fundamental papers linking topology to mechanics, Smale (1970)explains how to split the study of any dynamical system into two simplerproblems. He first defines the integral manifolds as the set of points in thephase space with given values of the invariants, or better as the quotient ofthat set by the symmetry group of the system. Then the first problem is to Exact solution of the planar motion of three arbitrary point vortices find the topology of the integral manifolds of the phase space and more pre-cisely to find the bifurcation set, i.e. the set of values of the invariants ( E, J )for which this topology, and therefore the nature of the motion, changes.The second problem, which has been solved above at least in the generalcase, is the study of the dynamical systems on the integral manifolds.Since we do not want to insist on the mathematics, we shall only give thebifurcation set, i.e., for every value of the strengths, we shall determine thevalues of E and J which cause a qualitative change in the absolute motion;such a research will introduce separating lines in the strengths space.Our discussion will therefore take place in two different spaces: the spaceof strengths (parameter space) and the space of invariants, spaces which weare now going to describe in more detail.Due to the homogeneity of the equations of motion, the strengths spacemay be represented by its section by a plane K = constant and, in thisplane, by a figure invariant under a rotation of π around the point κ = κ = κ . Its dimension is therefore 2 and we shall represent a point by itspolar coordinates: ρ cos θ = √ κ − κ )2 K , ρ sin θ = − κ + κ + κ K , (16) θ describing any interval of amplitude π (see Figure 1).For given values of the strengths, the space of invariants is a priori bidi-mensional, since the center of vorticity is not a relevant invariant exceptwhen K is zero. Let us compare this space to the space of initial conditions.As we have seen, an initial condition is an orbit in the ζ plane, since two setsof absolute positions z i whose ζ values belong to the same ζ orbit evolvein the same absolute motion, up to a translation in time and a translation,rotation and scale change in space. On a given orbit, c is constant, but,inversely, the equation c = cst represents a finite number (between 0 and4, see Table 1) of orbits. From this fact, we draw two conclusions: first,the space of invariants ( E, J ) is in fact of dimension one, two points beingidentified if they lead to the same value of c = JQ e π EQ ; second, an initialcondition is characterized by, and therefore equivalent to, a value of theinvariant c plus an index of region in the ζ plane (or of sheet in the phasespace). We can therefore identify the space of initial conditions to the prod-uct of the one-dimensional space of invariants by the finite set of the indicesof region. Accordingly, the most precise graphical representation will be theorbits of the ζ plane, but we shall also use for simplicity a plane ( E, J ) or N o v e m b e r : Y e a r s ’ B i r t hd a y o f P r o f e ss o r H a o B a ili n i n x i n nd R e a d i n g b - C o n t e - d e - S e z e p a g e R . C o n t e a n d L . d e S e z e Table 1. Summary of results.degeneracystrengths domain sign (Q) sign (∆) c = −∞ c = 0 c = 1 c = + ∞ topology0 + − , , , − , , , − − , , , − − , , − − , − + 0 , − + 0 1, 2 2 M E M1 5 1 − + 1 2 2, 1 M E M K = 0, κ < < κ < κ − + 0 2 2 M E M∆ = 0 (11 and 351) − − Q ( κ j + κ l ) = 0 (11 and 12) − + 2, 1 2 2, 1 E M H Xid. (10 and 11) − + 1, 2, 1 2 2 X H E Mid. (140 and 351) − − Q = 0 (1 and 12) 0 − − , , − − , , − − 2, 2 3 2 X H Xpoint C ( − , √ , − c the number of differentstates, i.e. of ζ orbits, having the same value of c ; for instance, in the domain 11, the limiting values of c are −∞ , , + ∞ (atthe points M j ), 1 (at the triangles), c , c , c (at the colinear relative equilibria), they define 6 intervals, hence a sequence of 6degeneracy numbers; when Q is zero, e πE/K is used in place of c for the classification.The column “topology” lists the sequence of the nature of the real remarkable ζ points along the real axis: M stands for apoint M j (we omit M at infinity), E, H, P for elliptic, hyperbolic, parabolic, X for an M coinciding with E or H and Q for theonly higher-order point we found. The information on the nature of the triangles is contained in the column sign ( Q ). All this issufficient to draw every ζ plane. Exact solution of the planar motion of three arbitrary point vortices an axis c with some handwritten information on it to take into account theinteger index.Let us now proceed to the determination of the bifurcation set. Weexclude the set of collisions, represented in the ζ plane by three points ofaffixes ∞ , κ , − κ where c evaluates to +0, − 0, + ∞ or −∞ depending onthe signs of J, Q and the strengths; these limiting values of c will thereforebelong to the bifurcation set.The nature of the motion (at least its topological nature since it isalways generically biperiodic) changes when there is a stationary point onthe ζ orbit; such points, where the velocity of ζ vanishes, are given by thesolutions of the complex equation¯ ζ ( ζ + dζ − Q ) − sK ( ζ + d ) = 0 , (17)equivalent to K ¯ z − ¯ Bz − z + K ¯ z − ¯ Bz − z + K ¯ z − ¯ Bz − z = 0 . These points are also the critical points of the function ( ξ, η ) → c andthey give all the relative equilibrium configurations of the three vortices,i.e. the states for which the system moves generally like a rigid body. Thesolutions of (17) are:(a) one or three colinear configurations, defined by the real zeros of P ( ζ ) ≡ ζ + dζ − ( sK + Q ) ζ − sKd (points P , P , P of Figures 2 to 3),(b) two equilateral triangles, defined by the zeros of ζ + dζ + sK − Q ,i.e. ζ = − d ± is √ (points T and T of Figures 2 to 3); these stationarytriangles, already known to Kelvin for identical vortices, therefore existfor vortices of any strengths.(c) When Q = 0, the isolated zero ζ = − d of P and every point of thecircle | ζ | = sK on which lie the summits of the equilateral trianglesand the two other real zeros of P . On this circle J is equal to zero.The corresponding absolute motions will be described later. It is inter-esting to remark that, except for some values of the strengths like forinstance Q = 0, the relative equilibria of the three vortex system are qualita-tively the same than those of the three body problem of celestial mechanics,where there are two equilateral triangles and three colinear configurations;in fact, a simple geometric reasoning shows the same qualitative compo-sition of the set of relative equilibria for every planar three body motionruled by a two body central interaction. R. Conte and L. de Seze The study of the stability of the relative equilibria is given inAppendix II and only three different behaviours are found (in celestialmechanics, only two cases arise: the three colinear configurations are alwaysunstable, the two triangles are stable for ( m + m + m ) − m m + m m + m m ) > 0, unstable otherwise), depending on the signs of twopolynoms of the strengths of even degree:∆ > Q < 0: unstable triangles, one stable aligned configuration,∆ < Q < 0: unstable triangles, two stable aligned configurations, oneunstable,∆ < Q > 0: stable triangles, three unstable aligned configurations,with ∆ = − X j = l κ j κ l − X j Q > 0, ∆ < We must add to the above mentioned lines the three lines given by κ κ κ = 0 which are forbidden. Q ( κ j + κ l ) is represented by three lines, Q = 0 by the circle ρ = 1 and ∆ = 0 by a quartic of equation: ρ + 8 ρ sin 3 θ + 18 ρ − 27 = 0 , the form of which reflects the ternary symmetry.The intersections of these lines define three distinct remarkable pointsthat we shall study explicitly:point A: ρ = 2 , θ = 3 π − , , , point B: ρ = 3 , θ = 3 π − , , , point C: ρ = √ , sin 3 θ = − √ 525 (strengths − , √ , R. Conte and L. de Seze These lines define different domains in the strengths space, which hasbeen represented on Figure 1. When we stay in one of these domains, thetopology of the bifurcation set does not change, and Table 1 summarizesthe results concerning the bifurcation set for all domains of the strengthsspace.The classification for the bifurcation set is mainly based on the signs of Q and ∆, i.e. on the number and stability of the relative equilibria; amongthe three quantities, i.e. arithmetic, geometric and harmonic means of thestrengths, whose signs were proposed by Aref as a basis for a classification,only the third one Q κ κ κ is relevant, although the factor κ κ κ prevents itfrom being invariant under a strengths reversal, an operation equivalent toa time reversal which must leave invariant any candidate to a classification;moreover, for any N , the formula P ′ κ j κ l = 0, not P κ j = 0, expresses theassociated physical property of invariance of the energy under a change oflength.In the next part, we describe the behaviour of the three vortex systemwhen it is nongeneric, i.e. when either the strengths are on the boundaries ofthe domains represented on Figure 1 or when the initial conditions are thoseof a relative equilibrium. We shall proceed by studying first the relativeequilibria associated with nonsingular strengths, then the strengths on theseparating lines of the strengths space and finally the three particular pointsA, B, C of the strengths space. 4. Absolute motions for nongeneric strengthsor initial conditions Absolute motions for the ordinary relative equilibria Ordinary means that we exclude ζ points which are the coincidence of twoelements of the bifurcation set: this is equivalent to J nonzero and ζ not amultiple zero of P .For an initial condition in the vicinity of such a ζ point, every pointobeys the general motion, except those lying on the two curves intersectingat an unstable ζ point, in which case the motion is asymptotic to the motionat the stationary point, motion which we are going to determine:(a) K = 0. From the time variation law2 π d | Kz − B | d t = − s Im( ζ + dζ ) | ( ζ + κ )( ζ − κ ) | , Exact solution of the planar motion of three arbitrary point vortices it follows that the distance from each point to the barycentrum remainsconstant. The absolute motion is therefore a solid rotation around thebarycentrum; moreover, using N X j =1 κ j z j d¯ z j d t = Q πi , whose imaginary part yields N X j =1 κ j | z j | d arg z j d t = Q π , we see that the common angular velocity of every vortex is independent oftime and remains equal to Q πI o , where I o is the inertia momentum relativeto the barycentrum I o = X κ j (cid:12)(cid:12)(cid:12)(cid:12) z j − BK (cid:12)(cid:12)(cid:12)(cid:12) = JK , and Q π is the angular momentum of the system.The period of the uniform rotation is therefore T u = 4 π JQK , a formula valid for any number of vortices when there exists a solid rotation.A particular case is Q = 0, for which the only isolated relativeequilibrium is ζ = − d , i.e. κ κ z + κ κ z + κ κ z = 0; all the vorticesremain at rest. This situation is of course unstable (free vortices cannothave a stable rest position since the complex velocity is a nonconstant mero-morphic function of the affixes), and the small motions have for pulsation ω = ± iκ κ κ πJ (b) K = 0. The three stationary points are ordinary. ζ = 0, stable: the impulse B is zero and the affixes verify: z − z κ = z − z κ = z − z κ . The inertia momentum is nonzero and is the same at every point, andthe absolute motion is a uniform solid rotation of period T = 4 π IQ , equal to that of the small motions. R. Conte and L. de Seze At the two unstable equilateral triangles, B is nonzero and the absolutemotion is a uniform translation of velocity ∀ j : d z j d t = iQ π ¯ B , orthogonal to the impulse. Absolute motions on the boundaries of the strengths domains These lines are: Q ( κ + κ )( κ + κ )( κ + κ )∆ = 0. Q = 0 . Triple collision in a finite time, expanding motion In addition to the already studied isolated stationary point, there exists acircle of stationary ζ points (see Figure 2), on which J is zero, On this circlelie the two summits of the equilateral triangles and the two other zeros of P . These four points are the points of contact of the circle J = 0 with theset of ζ trajectories whose equation is now: (cid:12)(cid:12)(cid:12)(cid:12) ζ − κ s (cid:12)(cid:12)(cid:12)(cid:12) − κ κ K (cid:12)(cid:12)(cid:12)(cid:12) ζ + κ s (cid:12)(cid:12)(cid:12)(cid:12) − κ κ K = e πEK . We shall first study the absolute motion when the ζ point lies on thecircle J = 0, then examine its neighborhood and finally deduce the bifur-cation set.Let us assume ζ on the circle J = 0. This circle is no longer a trajectoryand the ζ point stays at rest. The two conditions Q = 0, J = 0 express thatboth the energy and the inertia momentum relative to the barycentrumare invariant under a change of length, and therefore nothing prevents thevortices from going to infinity or to zero; we see below that both cases arepossible. The absolute motion is ruled by Equation (6) which implies that dρ dt and ρ dϕ dt are constant in time ( ρ , ϕ are polar coordinates of M , B is chosen zero). Since the shape of the triangle is conserved, Equation (6)integrates as in ∀ j = 1 , , z j = z j,o (cid:18) − tt c (cid:19) / − iωt c , i.e. r − tt c = ρ j ρ j,o = exp { ( ϕ j − ϕ j,o ) / ( − ωt c ) } , Exact solution of the planar motion of three arbitrary point vortices where the characteristic time t c and the initial angular velocity ω aredefined by − ω + it c = − s πK | z , | ζ + dζζ + dζ − κ κ = 13 π X ′ j,ℓ . . . κ ℓ (¯ z j ( z ℓ − z j )) o .ω never vanishes and has the sign of K . t c vanishes and changes sign when ζ is one of the four points already mentioned where the circle is tangent to theset of ζ curves. We conclude that, for a ζ point of the circle J = 0 distinctof these four points, every vortex runs a logarithmic spiral whose pole is thebarycentrum, the shape of the triangle remains constant and, depending onthe sign of t c , the triangle either expands to infinity in an infinite time orcollapses on the barycentrum in a finite time t c . At the time of this triplecollision all the denominators of the equations of motion (1) simultaneouslyvanish like ( t c − t ) / . After the collision, the system is made of a singlemotionless vortex of strength K located at what was the center of vorticity.When t c vanishes, the spiral motion degenerates into a uniform circularmotion whose period T = πω can also be written as T = 4 π K ( | z − z | + | z − z | + | z − z | ) , for the two aligned configurations and twice the same expression for theequilateral triangles.Let us now examine the motion elsewhere in the ζ plane ( J = 0). Fig-ure 2 shows that two generic situations exist depending on whether the ζ orbit intersects or not the circle J = 0 of singular points; the limiting ζ orbits, which belong to the bifurcation set, are E = 0 (tangent at T , T )and E = E ( P ). For an energy E outside the interval [ E ( P ) , E ( P ) , ζ tra-jectory stops on the circle (note that it cannot cross it) and, since J is theproduct of | z − z | by a function of ζ vanishing on the circle, every ζ pointhaving a nonzero J and the energy of a curve intersecting the circle yieldsan expanding motion in which the trajectory of every vortex is asymptoticto a logarithmic spiral going to infinity. No motion exists which is asymp-totic to the triple collision in a finite time: therefore the points of the halfcirconference (from P to T and from P to T ) where such a collisionexists are repulsive points, while the other half is made of attractive points.Finally, the bifurcation set, which we have also represented in the ( E, J )plane on Figure 11, is the union of the following lines of the ζ plane: the setof collisions ( ζ = ∞ , − κ , κ ), the circle J = 0 (not an orbit), the orbit R. Conte and L. de Seze P biperiodic expanding expanding expanding expanding P T T T Fig. 11. Bifurcation set for Q = 0, not to scale. going through P , the orbit E = 0 tangent to the circle at T , T , theorbit E = E ( P ) tangent to the circle at P and that portion of the orbit E = E ( P ) which is interior to the circle J = 0. ∆ = 0 Two of the three colinear relative equilibria coincide except at the threepoints of retrogression ρ = 3 of the quartic ∆ = 0 (strengths − , , ρ = √ − , √ , 2) where two of the strengths are opposite (see next case). Exact solution of the planar motion of three arbitrary point vortices For the absolute motion, there is nothing qualitatively changed: in thevicinity of the double point ζ o , the orbits are equivalent to the cubics ofequation c − c o + A η + A ( ξ − ξ o ) = 0 , and the period, which behaves like J R d ξη , diverges like | c − c o | − / ; theabsolute motion is still biperiodic, although the triangle spends quite along time in nearly aligned states. On the orbit c = c o passing through theunstable ζ o point, the absolute motion is asymptotic to the usual circularmotion, the time law being different: ξ − ξ o ∼ t − , η ∼ t − . Π( κ j + κ ℓ ) = 0 . Elastic diffusion At least two vortices have opposite strengths. To fix the ideas, let us assume κ + κ = 0. The case ~κ = ( − , , 1) will be described in next section. Thepoint ζ = κ , which lies on the circle J = 0, is singular in the sense thatfor every c value there exists a ζ orbit passing through κ , which forbids areduced periodic motion.Therefore two generic situations exist, as shown on Figure 3: outside ofthe striped domain which contains J = 0, the absolute motion is biperiodic.Inside this domain, for every c value there is one and only one ζ orbit andthe ζ point is attracted by κ in an infinite time: it tends to κ normallyto the real axis following a law i ( ζ − κ ) t → πsκ e − πE/Q . In the absolute space, the point M stops and the two others go toinfinity together with a motion asymptotic to a uniform translation in thedirection normal to the line joining the final position of M to the barycen-trum, exactly as if they were alone and obeyed the motion of two vorticesof opposite strengths: at the limit, the translation velocity is √− Q π e πE/Q and the mutual distance is e − πE/Q .This creation of a doublet can also be interpreted in terms of the elasticdiffusion of the doublet by the third vortex: if we take for initial condition ζ near to K with η/K positive, the vortices 1 and 2 are initially a doubletmoving towards the third vortex; then the three mutual distances becomethe same order of magnitude, i.e. there is interaction; finally, the doubletemerges and goes to infinity away from the third vortex with unchangedmutual distance and velocity, but in a different direction. This is exactlya process of elastic diffusion. Figure 9 shows typical absolute motions. We R. Conte and L. de Seze define the scattering angle ∆( φ ) by the total variation along the trajectoryof arg( z ) ( B is chosen zero) and a dimensionless impact parameter by OHOM (Figure 9) where H is the projection of the middle of M M on the linedefined by the barycentrum and the final position of M : OHOM = Re(¯ z z + z )¯ z z = Re(( ~ζ − κ )( κ − κ ζ + sζ )) − κ | ζ − κ | ∼ sκ − rκ + s − κ − κ = c + 12 , where r is the radius of curvature in κ : r = lim η ξ − κ ) = − sκ s + κ c . The impact parameter is then our dimensionless invariant c , up to a lineartransformation. As to the scattering angle, it can be computed from anintegral taken along the ζ orbit:∆ ϕ = [arg z ] + ∞ t = −∞ = I − sK Re (cid:16) ζ + dζζ + dζ − κ κ (cid:17) Re (cid:16) ζ [ ~ζ ( ζ + dζ − Q ) − sK ( ζ + d )] ζ + dζ − κ κ (cid:17) d(arg ζ ) . This integral can be carried out exactly for c = 0, when the ζ orbit is acircle:∆ ϕ = Z πo K + d K + d ) (cid:18) − Kd K + 2 Kd + d + 2 K ( K + d ) cos θ (cid:19) d θ = π if | κ | > | K | (cid:18) − Kκ (cid:19) π if | κ | < | K | . The bifurcation set for this case is made, in the ζ plane, of the follow-ing lines: the set of collisions ( ∞ , κ , − κ ), the boundary between the tworegimes, the point P and the orbit c = c ( P ). The three particular ~κ points (a) Point A (strengths − , , ζ plane looks like the one Figure 3 assumed continuously deformedso as to admit the origin as a center of symmetry: M = P , symmetric of M , and P = 0.Aref (1979) has extensively studied the motion and we shall only brieflysummarize it. The results obtained for κ + κ = 0 still apply, but a new Exact solution of the planar motion of three arbitrary point vortices physical situation arises from the existence of ζ orbits which go from ζ = ± K to ζ = ∓ K (these are all the orbits which cross the segment T T ): themotion is then an exchange scattering in which the incident pair (1 2) isdifferent from the outgoing doublet (1 3).To sum up, three generic situations exist, depending on the initial con-ditions: • | ζK ± | > 2: in uniformly rotating axes, the absolute motion is periodic. • | ζK ± | < − < c < 1: exchange scattering; the integral giving thescattering angle is to be taken from 0 to π only. • | ζK ± | < c outside [ − , c = 0, the ζ orbit is a half-circle and, computing d z d t by derivingEquation (6), we find zero, which means a uniform linear motion M , hencea scattering angle of π . The motion of ζ = Ke iθ is ruled by d θ d t = va sin θ which integrates as ζ = K ia − vt √ a + v t where we have noted a = 2 e πE/K , v = Kπa .The origin of time being chosen when M is the summit of an isosce-les rectangle triangle, the absolute motions take place on three parallelstraight lines: z = a + i v tz = a + i v t i p a + v t z = a + i v t − i p a + v t . The exchange scattering process is clearly seen on the above equations,and every other exchange scattering motion can be thought of as a contin-uous deformation of this one.The bifurcation set is made of the set of collisions and of the boundariesof the domains limiting the three generic situations; note that the circle J = 0( | ζ | = | K | ) does not belong to it. This set is simple enough to berepresented without any ambiguity by a c axis with the number and typeof solutions in each interval: R. Conte and L. de Seze − (b) Point B summit of the quartic (strengths − , , ζ = 0, thevortex of strength − K is motionless as it coincides with the barycentrum,and the two other vortices, which are symmetric with regard to the barycen-trum, obey the circular motion. The ζ orbits around this stable point areequivalent to the quartics of equation2 − / c + 1 − (cid:16) ηK (cid:17) − (cid:18) ξK (cid:19) = 0 , and they correspond to ordinary motions; however the period, which stillbehaves like J R d ξη , diverges as ( c + 2 / ) − / .(c) Point C (strengths − , √ , 2) conjunction of ∆ = 0 and Π( κ j + κ ℓ ) = 0.This point has no new properties: the previously studied singularities onlyadd without interfering. 5. Volume of the phase space The Hamiltonian system has an invariant element of volume of the phasespace equal tod v = κ κ κ d x ∧ d y ∧ d x ∧ d y ∧ d x ∧ d y . Since there exist four real invariants E, J, X, Y ( X + iY = B ), we wantthe density of states d v d E d J d X d Y after integration over two independentvariables of the phase space.By using the two successive changes of variables ( z , z , z ) → ( z − z , ζ, B ) and ( ζ, ¯ ζ, z − z , z − z ) → ( E, J, z − z , z − z ) , whose jacobians are respectively D ( z − z , ζ, B ) D ( z , z , z ) = − sKz − z , and D ( E, J ) D ( ζ, ¯ ζ ) = iκ κ κ π | z − z | Im( ζ + dζ ) | ( ζ − κ )( ζ + κ ) | , Exact solution of the planar motion of three arbitrary point vortices we obtain d v = πκ s K | ( ζ − κ ) ( ζ + κ ) | Im( ζ + dζ )d( x − x ) ∧ d( y − y ) ∧ d E ∧ d J ∧ d X ∧ d Y, and we still have to integrate over x − x and y − y . Since | z − z | movesin time according to2 π d | z − z | d t = 2 κ s Im( ζ + dζ ) | ( ζ − κ )( ζ + κ ) | , the integration is quite easy to perform and we finally getd v = πK T ( E, J )d E ∧ d J ∧ d X ∧ d Y, which shows that the density of states is the period of the reduced motion.This is a well known result of the theory of adiabatic invariants (see e.g.Landau and Lifchitz), for energy and time are conjugate variables: when J is constant and E slowly varying, then the product ET is constant sincethe volume of the phase space if conserved.To take into account the fact that the phase space is the union of discon-nected parts, we must sum the above expression over the number (between0 and 4, see Table 1) of different domains associated to given values of E and J . The resulting volume Ω( J, E ) obeys the scaling law:Ω( E, J ) = πK X domains T ( E, J ) = (cid:12)(cid:12)(cid:12)(cid:12) JQK (cid:12)(cid:12)(cid:12)(cid:12) f (cid:18) JQ e π EQ (cid:19) . Onsager (1949) defined the entropy S and the temperature τ of anassembly of a large number of interacting vortices: S = k B Log Ω , τ = d S d E . Although it makes no sense to speak of thermodynamics about an inte-grable system, there may be some interest for the understanding of thebehaviour of a large number of vortices to examine what the Onsager’stheory gives when formally applied to the three-vortex system.The main hypothesis made by Onsager is that the total amount of vol-ume R + ∞−∞ Ωd E available to the system is finite, an hypothesis equivalentto assume the system confined in a box since the phase space and theconfiguration space are the same. Due to the scaling law for Ω, the integral R Ωd E will be either finite and proportional to J or infinite, depending onthe strengths of the vortices. The first arising question is therefore: when R. Conte and L. de Seze J is kept constant, is the integral R Ω( E, J )d E = Q π R cQJ> Ω d cc finite ornot, i.e. are all the singularities of T integrable or not?The singularities of T are: the set of collisions, the unstable relativeequilibria and, since J is kept nonzero, the limit e πE/Q → κ κ κ K < j alone): then T ∼ π J ( K − κ j ) κ j (cid:18) Qcκ j ( K − κ j ) (cid:19) − Qκjκ κ κ → , and the singularity R T d E = QJ π R TJ d cc is integrable.(b) ζ tends to an unstable ζ value, whose c value is c o . The equivalenthyperbola having for equation: G ( ζ, ¯ ζ ) ≡ c − c o c o + J κ κ κ Q ( κ | ζ | + κ κ K ) × [ µ o ( ζ − ζ o ) + 2 v o | ζ − ζ o | + ¯ µ o (¯ ζ − ¯ ζ o ) ] = 0 , the period is equivalent to T ∼ cst J Z d¯ ζG ′ ζ ( ζ, ¯ ζ ) ∼ cst J Log | c − c o | , and the singularity is therefore integrable.(c) | ζ | → − κ κ Kκ (possible in every domain, except 0 and 150).Since c tends to zero with J being kept constant and nonzero, then e πE/Q tends to zero, therefore the period T = cst e − πE/Q has a noninte-grable singularity.In conclusion, the volume of the phase space, with J being constant, isfinite and proportional to J in the domains 0, 150 and K = 0 and infiniteelsewhere. The unit of time we chose, i.e. 4 π JQK , is a posteriori convenientfor it is proportional to the volume.Let us now examine the behaviour of the thermodynamical quantities,keeping in mind that any conclusion is meaningless for three vortices andcan only be indicative for a larger system. For instance in the domain 0 (i.e. κ j K > E o → R E In addition to the fact of being an exactly soluble three body problem, thethree vortex system is very interesting in connection with the theory ofturbulence. Unfortunately its number of degrees of freedom is too small toyield a chaotic behaviour (the threshold for such a behaviour is 3) and thiswas confirmed by the results: nonperiodic behaviours are obtained onlyfor very particular values of the parameters. A four vortex system (seesome preliminary results in Conte, 1979), with its 3 independent degrees offreedom and because we do not know about its integrability, is the reallyinteresting dynamical system to study in order to have some hints aboutthe integrability of the N vortex system. Acknowledgements We want to thank Y. Pomeau for many fruitful discussions which led to thediscovery of the appropriate plane. We also greatly appreciated the formalReduce-like computer language AMP (Drouffe, 1976) which helped us toestablish the numerous necessary formulae.The work is the first chapter of the unpublished Th`ese d’´Etat of thefirst author. It is an honor and a pleasure for us to dedicate it to ProfessorHao Bailin and to wish him a long life. References H. Aref (1979), Motion of three vortices, Phys. Fluids , 393–400.Bateman manuscript project, (1953), Higher transcendental functions, vol. II,chapter XIII; A. Erd´elyi Editor, Mc Graw Hill.R. Conte (1979), Th`ese d’Etat, Universit´e de Paris VI.J.M. Drouffe, AMP language, (1976), same address as authors. R. Conte and L. de Seze S.F. Edwards and J.B. Taylor (1974), Negative temperature states of two-dimensional plasmas and vortex fluids, Proc. Roy. Soc. London, A 336 ,257–271.W. Gr¨obner and Hofreiter (1965), Integraltafel, vol. 4, Springer-Verlag.L. Landau and E. Lifchitz (1960), Mechanics, Pergamon Press.T.S. Lundgren and Y.B. Pointin (1977), Statistical mechanics of two-dimensionalvortices, Journal of statistical physics, , 323–325.A.M. Mayer (1878), Floating magnets, Nature, , 258.E.A. Novikov (1975), Dynamics and statistics of a system of vortices, JETP , 937–943.L. Onsager (1949), Statistical hydrodynamics, Nuovo Cimento, suppl., 279–287.H. Poincar´e (1893), Th´eorie des tourbillons, pages 77–84, Deslis fr`eres, Paris.C.E. Seyler, Jr. (1974), Partition function for a two-dimensional plasma in therandom phase approximation, Phys. Rev. Letters , 515–517.S. Smale (1970), Topology and mechanics, Inventiones math., , 305–331 and , 45–64.W. Thomson (1878), Floating magnets (illustrating vortex-systems), Nature, , 13–14. Exact solution of the planar motion of three arbitrary point vortices Appendix ICases of integrability For practical applications, it may be of interest to find which values of the κ j ’s lead to integrable expressions for the period (12). A first case is whentwo strengths are equal: κ = κ ; then ξη can be expressed only with | ζ | ,using (11), and the period is a simple integral in the variable | ζ | , whichcan be easily integrated numerically.Another case is when, the strengths being rational, the algebraic curve(11) is of genus one or zero (the genus of an algebraic curve of degree n isequal to ( n − n − minus the number of double points). The only curve ofgenus zero is the circle J = 0 but then the period is given by another non-integrable expression. If the trajectory is of genus one, the abelian integralexpressing the period can always be reduced to an elliptic integral by abirational transformation of the coordinates (see e.g. Bateman 1953). Forsmall integer values of the strengths, there is some chance of finding curvesof genus one.Let us just mention three particular cases. ~κ = (1 , , K/ 3. This belongs to the first but not to the second case(degree 6, genus 4 in general). The period is expressed by the hyperellipticintegral in u = | ζ | κ : TT u = 12 π I − u + 3) sign( ξη ) p c ( u + 1) − u + 3) p u + 3) − c ( u − d u. Novikov gave this expression in the variable b = u +3 = κ | z − z | J whichalways remains between 0 and 2 but he did not integrate it: TT u = 34 πc I sign( ξη ) p f ( b ) p − g ( b ) d b with f ( b ) ≡ b ( b − − c , g ( b ) ≡ b ( b − ) − c . This hyperelliptic integralhappens to be reducible to an elliptic integral (Bolza, 1898, mentioned inthe tables of Gr¨obner and Hofreiter, 1965) of the variable z = g ( b )3 b , due tothe relations: ϕ ( z ) ≡ z − z + (cid:18) − c (cid:19) z + 18 c − c = f ( b )[ h ( b )] b , d z d b = 6 h ( b )9 b R. Conte and L. de Seze with h ( b ) ≡ b − b + c ; this gives for the period: TT u = 18 πc I sign( ξη )sign( h ( b )) p − zϕ ( z ) d z Let us call b < b < b the zeros of f , b < b < b those of g ( b and b are not real for 1 < c < b < b < b those of h and z < z < z those of ϕ . The correspondence is ( b , b ) → z , ( b , b ) → z , ( b , b ) → z ,which gives the following values of the period for the two domains:1 < c < K ) I = 2 Z b b d b = 6 Z z z d z,b < b < b < b < b < b < b T | T u | = 32 πc K ( k ) p ( z − z )( − z ) , k = z ( z − z )( z − z )( − z ) , z < z < < z < c : b < b < b < b < b < b < b < b < b , z < < z < z . Two equivalent expressions lead to the period, according to whether ζ turns around M or another vortex:( ζ around M ) : sign( K ) I = 4 Z b b d b = 4 Z oz d z ( ζ around M or M ) : sign( K ) I = 2 Z b b d b = 4 Z oz d zT | T u | = K ( k ) πc p z ( z − z ) , k = − z ( z − z ) z ( z − z ) ~κ = ( − , , , ) K where three generic situations exist. We assume K > ζ curves are bicircular quartics of genus one and we derive belowthe normal forms of the scattering angle and the period of the reducedmotion:∆ ϕ = I ( u + u o ) sign( ξη )2 u p ( u − u − )( u − u + ) p ( u − u o − u ) d uTT u = 12 π I ξη ) c ( u − p ( u − u − )( u − u + ) p ( u − u o − u ) d u with the notations u = | ζK | , u o = 1+ c , u ± = c − ± c √ − c . The variable b in Aref is related to u by b = − u . K, E and Π are the complete elliptic Exact solution of the planar motion of three arbitrary point vortices integrals of the first, second and third kind, the last one being defined as ∗ Π( n, k ) = Z d x (1 − nx ) p (1 − x )(1 − k x )First regime (exchange scattering). − < c < H sign( ξη )d u = 2 R u − d u .There is no discontinuity for c = 0 where ∆ ϕ evaluates to π . − < c < ϕ = − p (1 − u o )( u − − u + ) (cid:20)(cid:18) u u + (cid:19) K ( k ) + u o ( u + − u + Π( n, k ) (cid:21) with k = (1 − u − )( u o − u + )(1 − u o )( u − − u + ) , n = u + ( u − − u − − u + , < c < ϕ = 2 p ( u o − u − )( u + − (cid:20) K ( k ) + 8 c Π( n, k ) (cid:21) with k = ( u o − u + )( u − − u o − u − )( u + − , n = u o (1 − u − ) u o − u − Second regime (direct scattering). c < − < c . H sign( ξη )d u = 2 R u o d uc < − ϕ = − p (1 − u − )( u o − u + ) (cid:20)(cid:18) u u + (cid:19) K ( k ) + ( u + − u o u + Π( n, k ) (cid:21) with k = (1 − u o )( u − − u + )(1 − u − )( u o − u + ) , n = u + ( u o − u o − u + , < c : ∆ ϕ = ( u − o − u o ) K ( k ) + (1 + √ u o ) n, k )with k = ( u o − − √ u o − √ u o , n = − (1 − √ u o ) √ u o ∗ In the tables of Gradshteyn and Ryzhik 4 th edition, the definition 8.111.4 is not con-sistent with the rest of the book; many formulae concerning elliptic integrals are wrong,among them 3.132.5, 3.132.6, 3.138.8, 3.148.1. R. Conte and L. de Seze Third regime (biperiodic). 0 < c < H sign( ξη )d u = 4 R u o u + d uTT u = 2 π s u o − u − − u o − u − ) (cid:20) K ( k ) − u o − u − u o − E ( k ) (cid:21) ∆ ϕ = 4 p ( u − − u o − u − ) (cid:20) K ( k ) − c + 8 Π( n, k ) (cid:21) with k = ( u − − u o − u + )( u + − u o − u − ) , n = u + − u o u o ( u + − ~κ = ( − , , K , a case with two possible regimes (biperiodic, expanding).The ζ trajectories are the Cassini ovals, whose genus is one. The periodfor instance is given by T = (cid:18) π JK (cid:19) α π I − 27 sign( ξη )4( u − p ( u + 1) − α p α − ( u − d u with u = | ζκ | , α = | ( ζκ ) − | = 16 e πE/K . Its reduced form is notvery compact and we shall not give it here. Appendix IIStability of the relative equilibria In order to obtain the shape of the ζ trajectories in the vicinity of the rela-tive equilibria we must determine whether they are of elliptic or hyperbolicnature.The points ∞ , − κ , κ are elliptic, neighbouring orbits are circlesdescribed with a uniform circular motion of period: T = 4 π J ( K − κ j ) κ j (cid:18) Qcκ j ( K − κ j ) (cid:19) − Qκjκ κ κ We now assume that ζ o is the affix of an ordinary relative equilibrium(the case of two coincident r.e. is studied elsewhere in the paper), whichimplies J = 0 and we study the vicinity of the equilateral triangles and ofthe aligned configurations. Exact solution of the planar motion of three arbitrary point vortices By writing f ( ζ, ¯ ζ ) for the right-hand side of Equation (10), the smallmotions of a ζ point in the vicinity of a stationary point ζ o are ruled by:2 πi d¯ ζ d t = ( ζ − ζ o ) ∂f∂ζ ( ζ o , ¯ ζ o ) + ( ζ − ζ o ) ∂f∂ ¯ ζ ( ζ o , ¯ ζ o ) , or, in real matricial notation:2 π dd t (cid:18) ξη (cid:19) = M (cid:18) ξ − ξ o η − η o (cid:19) = (cid:18) α ′ + β ′ α − βα + β − α ′ + β ′ (cid:19) (cid:18) ξ − ξ o η − η o (cid:19) with α + iα ′ = ∂f∂ζ ( ζ o , ¯ ζ o ) = µ , β + iβ ′ = ∂f∂ ¯ ζ ( ζ o , ¯ ζ o ) = ν .The stability condition is: tr( M ) = 0, det( M ) > 0. We find: − ( κ | ζ | + κ κ K ) J ( ζ + κ )( ζ − κ ) = µ | ζ | + d ¯ ζ − sK = νζ + dζ − Q with the condition (17). The trace of M is therefore zero. If ζ o is elliptic,then the small motions have the period 4 π / p det( M ). We now divide thestudy according to the two types of stationary points.(a) The equilateral trianglesdet( M ) = Q J . The stability condition is Q > π J √ Q / ; comparing with theperiod of the absolute motion which will be derived later, we find: (cid:18) T r T a (cid:19) = 1 − K X j X ℓ>j ( κ j − κ ℓ ) . The overall rotation is therefore quicker than the small motions with equal-ity only for identical strengths.(b) The aligned configurationsdet M = | ν | − | µ | = (cid:18) κ ζ + κ κ KJ ( ζ + dζ − κ κ ) (cid:19) F ( ζ ) F ( ζ )with the notation F ( ζ ) = − ζ + Ks − QF ( ζ ) = 3 ζ + 2 dζ − sK − Q = P ′ ( ζ )and the condition: ζ is a real zero of P . R. Conte and L. de Seze The determinant of M changes sign when the resultant of P and F F vanishes. We find:res( P, F ) = − s Q , res( P, F ) = s ∆ . Then we obtain the nature of the aligned configurations in the parameterspace: ∆ > Q < 0) : one stable configuration∆ < Q < < Q > ..