Dynamics of Perturbed Relative Equilibria of Point Vortices on the Sphere or Plane

Transcription

1 J. Nonlinear Sci. Vol. 0: pp (2000) DOI: 0.007/s Springer-Verlag New York Inc. Dynamics of Perturbed Relative Equilibria of Point Vortices on the Sphere or Plane G. W. Patrick Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, S7N 5E6, Canada Received June 23, 999; revised manuscript accepted for publication December, 999 Communicated by Stephen Wiggins Summary. The system of point vortices on the sphere is a Hamiltonian system with symmetry SO(3), and there are stable relative equilibria of four point vortices, where three identical point vortices form an equilateral triangle circling a central vortex. These relative equilibria have zero (nongeneric) momentum and form a family that extends to arbitrarily small diameters. Using the energy-momentum method, I show their shape is stable while their location on the sphere is unstable, and they move, after perturbation to nonzero momentum, on the sphere as point particles move under the influence of a magnetic monopole. In the analysis the internal and external degrees of freedom are separated and the mass of these point particles determined. In addition, two identical such relative equilibria attract one another, while opposites repel, and in energetic collisions, opposites disintegrate to vortex pairs while identicals interact by exchanging a vortex. An analogous situation also occurs for the planar system with its noncompact SE(2) symmetry. Key words. Point vortex, collisions, relative equilibria, stability MSC numbers. 37J25, 70H09, 53D20 PAC numbers Jj, Cc Introduction The system of N point vortices on a sphere or plane is a Hamiltonian system with symmetry SO(3). These systems have been studied for over a century for an an excellent Supported by the Natural Sciences and Engineering Council of Canada.

2 402 G. W. Patrick review see Aref [983]. Recently there has been interest in the spherical system, beginning with Bogomolov [977], and this system, from the point of view of geometric mechanics (see Marsden and Ratiu [994]), has been taken up by Kidambi and Newton [998] and Pekarsky and Marsden [998]. For N 4 these systems are decidedly nontrivial and even chaotic (Aref and Pomphrey [982]). Steadily rotating or translating arrangements of vortices are the relative equilibria of these systems. Relative equilibria of systems of planar point vortices are well studied. An early reference is Havelock [93], who considered identical point vortices in a ring as well as concentric rings of point vortices. The situation of a ring of vortices surrounding a central vortex is considered by Mertz [978], and climatological applications of this configuration are considered by Morikawa and Swenson [97] and Bauer and Morikawa [976]. Aref [995] considers the equilibria of an infinite row of point vortices. Other kinds of relative equilibria are discussed in Campbell and Kadtke [987] and Lewis and Ratiu [996], and O Neil [987] counts relative equilibria in the general case. In many of the works, but certainly not all, when the stability of these relative equilibria is considered, it is mostly the linear stability that is discussed. The determination of the nonlinear stability of relative equilibria in general is the objective of the energymomentum method (see Marsden [992] and the references therein). By that general theory, in the case of point vortices on the sphere, a relative equilibrium at nonzero momentum (i.e., nonzero center of vorticity), if stable, will under sufficiently small perturbation maintain its shape and location on the sphere, in perpetuity. However, if the relative equilibrium has zero momentum, then location is not stable even if its shape is, as established by Patrick [992], [995], [999], Ortega and Ratiu [999], and Lerman and Singer [998]. In this article I show that there are relative equilibria of N vortices on the sphere with zero momentum. These relative equilibria consist of vortices of one strength arranged in a regular (N )-gon, the center of which is occupied by a vortex of a different strength. There is essentially one of these relative equilibria for every sufficiently small diameter, and they are formally stable for N = 3 and N = 4. For N = 4, under small perturbation to nonzero momentum, they move on the sphere, to first order in the momentum perturbation, as a free particles move under the influence of a magnetic monopole. The effect of the monopole diminishes as the relative equilibria are taken to be more localized. Since a given momentum perturbation implies a particular velocity of motion of each relative equilibrium as a whole, they have mass, and using a perturbation theory of Patrick [995], I find a formula for that mass. The mechanism for the instability in the location of these relative equilibria is the occurrence of a nonzero nilpotent part of the linearized system. Duplicated eigenvalues of the linearization are a consequence of the symmetry group of these systems. That a nontrivial Jordan block of the linearized system can be a mechanism for instability in the context of the system of point vortices in the plane has been previously noticed by Khazin [976]. The clear identification of nongeneric momentum as a particularly noteworthy special case is a gift of the geometric viewpoint that seems slighted by the existing point vortex literature. Translation of point vortices through the pairing of two equal but opposite vortices (a dipole) has long been observed, and Aref [983], on page 359, remarks that a dipole has qualities reminiscent of a classical particle. The collision of a dipole with a fixed

3 Dynamics of Perturbed Equilibria of Point Vortices 403 vortex, in the planar system, is a part of the study of the (integrable) three-vortex problem, and is taken up in Aref [978], while in Eckhardt and Aref [988] one finds a study of the collision of two dipoles, again for the planar system. The four-vortex relative equilibria considered here are reminiscent of classical particles which have mass, but their translational motion is through an entirely different mechanism than that of a dipole. When two such relative equilibria are placed on the sphere, they will move independently until they are within a distance comparable to their size, whereupon an interaction occurs. Generally, opposites repel and likes attract. One relative equilibria can collide with its opposite (obtained by reversing the first s component vortex strengths), the result being radiation in the form of four separate dipoles. When two identicals collide, one often observes them to exchange one of their vortices. Disintegrations to dipoles are reported in Aref [982] and exchange-type scattering of a dipole off a single vortex are reported in Aref [978]. In this article I numerically determine some general features of these interactions, which are the subject of ongoing investigations. Over vanishingly small regions of the sphere, the point vortex system on the sphere becomes the point vortex system on the plane, and since the above relative equilibria may be arbitrarily localized, one expects such in the planar case as well. Towards the end of this article I show this in indeed true. The planar system of N point vortices will be a worthy motivating example for extensions of the stability theory of Hamiltonian systems with symmetry to the case of nonequivariant momentum mappings and/or noncompact groups.. Context and Notations I briefly summarize here the basic elements of the system of N point vortices on the sphere and on the plane. For more details see Kidambi and Newton [998], Pekarsky and Marsden [998], and the many references therein. The system of N point vortices of strengths Ɣ i R on the sphere S 2 of radius R is the Hamiltonian system with phase space P sp (S 2 ) N and Hamiltonian H sp (x,...,x N ) Ɣ 4π R 2 n Ɣ m ln l 2 mn, () where x n represents the location of the n th vortex and l 2 mn = 2(R2 x m x n ) is the chord length between x n and x m. The symplectic structure ω sp on P sp is a direct sum of the symplectic structures on S 2 weighted by vortex strengths; specifically, where ω sp N n= m<n Ɣ n R ω S2, (2) ω S 2(x)(v, w) x (v w). (3) R2 Most significant from a geometric mechanics point of view is that this system admits the

4 404 G. W. Patrick symmetry of diagonal multiplication of SO(3) on P sp with momentum map J sp R N Ɣ n x n. (4) These are the notations of Pekarsky and Marsden [998]. In the system of N vortices in the plane the n th vortex has location z n = x n + iy n P pl (C 2 ) N, and the Hamiltonian and symplectic form are n= H pl 4π Ɣ n Ɣ m ln z n z m 2, m<n N ω pl Ɣ n ω 0, (5) n= where generally I will use the notation ω 0 (a, b) Im(a b) for complex numbers a, b C. This system admits the symmetry group SE(2) ={(e iθ, a)} of Euclidean symmetries acting diagonally on each factor C of P pl by (e iθ, a) z e iθ z + a, and a momentum mapping is N [ J pl Ɣ 2 z ] n 2 n, (6) iz n n= where se(2) is identified with se(2) ={( θ,ȧ)} = R 3 by the standard inner product of R 3. These notations are as in Lewis and Ratiu [996]. When attention is confined to a sufficiently small region of the sphere, the spherical system becomes the planar one. For example, using, on each factor, the map to pull back the sphere to the plane, one obtains φ(z) = (z, R 2 z 2 ) (7) φ ω sp = zr ωpl R 2 ωpl, φ H sp R 2 Hpl. (8) Thus the pulled back equations of motion are approximately the same as the equations of motion of the planar system, since the factors of /R 2 here multiply both the symplectic form and the Hamiltonian. As noted in Adams and Ratiu [988], the momentum mapping of the planar system is equivariant if and only if the total vortex strength n Ɣ n vanishes. Indeed, the adjoint and coadjoint actions of SE(2) are Ad (e iθ,a)( θ,ȧ) = ( θ,e iθ a i θa), (9) CoAd (e iθ,a)(µ, ν) = (µ ω 0 (e iθ ν, a), e iθ ν), and the deviation of the momentum map J pl from equivariance is the cocycle σ : SE(2) se(2) defined by σ J pl ((e iθ, a) p) Coad (e iθ,a) J pl (p) (the evaluation point p P is irrelevant). For the case at hand, the cocycle is ( ) [ ] σ(e iθ, a) = Ɣ 2 a 2 n, ia n

5 Dynamics of Perturbed Equilibria of Point Vortices 405 and the derivative of this cocycle at the identity is the skew-symmetric two form : se(2) 2 R given by ( ( θ, ȧ ), ( θ 2, ȧ 2 ) ) ( ) = Ɣ n ω 0 (ȧ, ȧ 2 ). (0) Generalities on nonequivariant momentum mappings may be found in Abraham and Marsden [978]; a prime fact is the momentum commutation identity n {J ξ, J η }=J [ξ,η] (ξ,η). () Both the planar and spherical systems have simple closed form solutions in the case N = 2: for the sphere any two vortices evolve as the action of the one parameter group with generator J sp /2πl 2 2 while for the plane they evolve as the action of the one parameter group with generator θ = (Ɣ + Ɣ 2 )/2π z z 2 2, ȧ = J pl /2π z z 2 2. Numerically, the action of these one parameter subgroups is easily computed, and since the Hamiltonians H sp and H pl are sums of pairwise interactions, the full system of N vortices may be numerically integrated in a symplectic, symmetry-preserving and momentum-preserving way using splitting methods, as in Channell and Neri [993]. After this article was completed I noticed that these explicit vortex integrators had been previously discovered in the planar case by Zhang and Qin [993]. 2. The Relative Equilibria To find the relative equilibria for the system of N vortices in the sphere, one seeks ξ e so(3) = R 3 and p e P sp such that dh sp (p e ) dj ξe (p e ) = 0. Equivalently, using the obvious extensions of H sp and J sp to (R 3 ) N, one can solve the equations m H sp m J sp ξ e = λ ( 2 x m 2 ) = λ m x m, m =,...,N, where the λ m R are Lagrange multipliers. Inserting () and (4) yields the N equations Ɣ n 2π R2 = λ m x m + 2π Rξ e, λ m λ m, m =,...,N. (2) Ɣ m l n m nm 2 General analytic solutions are not to be expected to these nontrivial nonlinear equations. I seek a manifold of solutions to (2) contained in the zero level set of J sp, which in some limit is confined to arbitrarily small regions of phase space, and which is formally stable near that limit. I try a regular polygonal configuration where the first N vortices of equal strengths surround the (possibly different strength) N th central vortex, N 3. I will denote the strength of the central vortex by Ɣ. For convenience I locate the central N th vortex at Rk where k (0, 0, ), and the first vortex at (R sin α, 0, R cos α), so that α is the opening angle the angle at the center of the sphere between any outer vortex and the central vortex. The momentum is zero if and only if Ɣ (x + x 2 + +x N ) + Ɣx N = (N )Ɣ R cos αk + ƔRk = 0,

6 406 G. W. Patrick which is equivalent to Ɣ Ɣ = (N ) cos α, (3) while equations (2) reduce to λ = =λ N and the two equations Ɣ X 0 + RƔ k = λ x + 2π Rξ e, (4) l 2 N (N )Ɣ R cos α k = λ N Rk + 2π Rξ e, (5) l 2 N where ξ e depends on Ɣ, N, α and R, and where Setting so that µ N N k= X 0 x 2 l2 2 + x 3 l x N ln, 2. (6) e 2πik/N, µ N + ν N = Ɣ λ N = R 2 sin 2 α N k= ν N N k= e 2πik/N e 2πik/N 2, e 2πik/N 2, one sees by symmetry that µ N and ν N are real, and also that µ N = 2 (N ), ν N = 2 (N 2 ). Then scaling µ N and ν N gives X 0 i = µ N + ν N, R sin α X 0 j = 0, X 0 k = ν N cos α R sin 2 α. (7) Equation (5) implies ξ e is along k and in (4) it is clear that the j component of both sides is zero, so writing (4) and (5) in components gives three linear equations in the three unknowns λ, λ N, and ξ e k. These equations can be routinely solved to obtain Ɣ (N 2)(N 6) λ = R 2 cos α sin 2, α 2(N ) ( ) (8) cos α +, (9) N 2(N ) ( N + cos α Ɣ ξ e k = 4π R 2 sin 2 α In the limit α 0, from (3), the total vortex strength Ɣ N = Ɣ + (N )Ɣ = Ɣ n ( cos α ). (20) ) (2)

7 Dynamics of Perturbed Equilibria of Point Vortices 407 vanishes while the regular polygon of outer vortices collapses upon the central vortex. Although these relative equilibria depend on α, Ɣ, and N, I will denote them below simply by p e. Formal stability means definiteness of the Hessian of the reduced Hamiltonian at the relative equilibrium, and by the energy-momentum method, as in Marsden [992], this is equivalent to definiteness of the function H sp J ξe on a subspace tangent to the momentum level set and complimentary to the tangent space to the group orbit. For N = 3, and at zero momentum, the reduced spaces are points and formal stability is immediate. I have determined the stability p e for N = 4, 5, 6 with the aid of a symbolic manipulator. For N = 4 the relative equilibria are formally stable without conditions, and in particular are formally stable arbitrarily near the limit α 0. For N = 5, but numerically now, one has formal stability if and only if α>.95.8 o and for N = 6 if and only if α > o. Presumably this pattern continues and the relative equilibria are formally stable for N > 6 if and only if α is sufficiently near π. As already mentioned, since the momenta of such relative equilibria are zero, formal stability implies stability only modulo SO(3) (Patrick [992]). Dynamically, these relative equilibria, when perturbed to nonzero momentum, approximately maintain their shape, which oscillates on a fast time scale, while they move around the sphere on a slow time scale. The left of Figure is the result of a simulation and shows a typical motion of one such relative equilibrium. For short, I will call a point of phase space that results from perturbing one of the relative equilibria p e a preq. The motion of preq may be approximated as a direct application of the perturbation theory of Patrick [995]. In summary, the symplectic SO(3) symmetry implies that the linearization of the relative equilibrium has double eigenvalues and hence can be expected to have a nonzero nilpotent part, say N α. It can be shown that the image of N α is contained in the tangent space to the group orbit and the tangent space to the momentum level set is contained in the kernel of N α. Consequently there is a unique (it can be shown to be symmetric) bilinear form (here with the same name) N α on so(3) (or, equivalently, a symmetric map N α : so(3) so(3)), such that the following diagram commutes: T pe P sp dj(p e ) N α T pe P sp ξ ξ p e so(3) N α so(3) I have calculated the N α in the case N = 3 and N = 4; they are the diagonal matrices with entries J N, J N2, J N3 where J 3 = J 32 8π R 2 cos α, (22) J 33 8π R 2 sin 4 α (2 cos3 α + 3 cos 2 α + 2 cos α ), (23)

8 408 G. W. Patrick and J 4 = J 42 3 cos 2 α + cos α + 2 2π R 2 ( cos α)(9 cos 2 α + 4 cos α + 3), (24) J 43 2π R 2 sin 4 α (3 cos3 α + 4 cos 2 α + 3 cos α 2). (25) The perturbation theory asserts that to first order the motion of the preq is that of the drift system: the reduction, by the normal form symmetry of the right-hand action of the torus generated by k so(3), of the left invariant drift Hamiltonian π π ξ e + 2 π t N α π on T SO(3) with canonical symplectic form. That the drift Hamiltonian is invariant under this toral action is predicted by the theory and is evident by the equality J N = J N2. Actually, this procedure is only valid under the condition that the frequencies of the linearization of the corresponding equilibrium on the reduced space (called the reduced frequencies) are disjoint from the rotation frequencies (called the group frequencies) of the relative equilibrium itself. Were there to be such a - resonance the preq would acquire the structure of a magnetic moment, as in Patrick [999]. For the case N = 3 this is not an issue since the reduced spaces are points and so there are no reduced frequencies. For N = 4 one verifies by calculating the linearization of the equilibrium that if the reduced frequencies are ±ω red then ξ e 2 ω 2 red = 3 (9 cos2 α + 4 cos α + 3) sin 2 α. 3 cos 2 α + 2 cos α + 3 Since the second term does not vanish for 0 <α<π, the group frequencies ±ξ e are never equal to the reduced frequencies, and the simpler perturbation theory of Patrick [995] suffices. If the location of the preq on the sphere is denoted by y, then the equations of motion for it (i.e., the equations of motion of the drift system) turn out to be m α d 2 y dt 2 = m v 2 y α R R + σ B v, v dy dt, B y, (26) R3 where σ is the momentum associated to the normal form symmetry and m α J N R. 2 Thus, the drift system is the same as that of particle of mass m α and charge σ moving on the sphere under the influence of the magnetic monopole B. Given that one is perturbing a relative equilibrium where the central vortex has been located at Rk, the initial location of the preq may be taken to be y(0) = Rk, while its initial velocity v(0) and the charge σ may be obtained from the momentum perturbation µ by the equations v(0) = y(0) µ, σ = µ k. m α R2 Here σ is both the momentum of the normal form symmetry and (minus) the z-component of the momentum perturbation. By solving Equations (26), one finds that the prediction

9 Dynamics of Perturbed Equilibria of Point Vortices 409 of the theory is that the preq will rotate about the perturbed momentum, say µ, with angular velocity J N µ. Specified momentum perturbations of the relative equilibria p e may be accomplished by moving the central vortex from its original location at Rk to (R sin δ, 0, R cos δ) while changing the angle α to α + α, as follows. The momentum of the perturbed configuration is easily verified to be 0 sin δ J sp = (n )Ɣ 0 cos(α + α) Ɣ 0 cos δ, so this is µ [ µ, 0, µ 3 ]if sin δ = µ Ɣ, (27) cos(α + α) cos α = ( µ 3 + Ɣ(cos δ )), (28) (N )Ɣ and arbitrary directions in µ may be obtained by rotating this. As these formulas are to be used to perturb a relative equilibrium, it is understood that µ and µ 3 are small; just how small depends on the validity of the approximation that is the drift system. The momentum of the relative equilibrium, being zero, does not itself provide a scale. Certainly, though, the perturbation should not significantly affect the geometry of the relative equilibrium, meaning it should not displace the vortices by amounts comparable to the diameter 2α. In using (27) and (28), one should therefore ensure δ α, α α. (29) By way of a check, and for the purpose of illustration, I have, using (27) and (28), simulated the 4-vortex preq corresponding to α = π/6 on the unit sphere for momenta µ that are small multiples of the vector [2, 0, 3], and calculated the angular velocities of the preq. The results are on the right of Figure. As already mentioned, were the drift system to be exact, the angular velocity observed for a given momentum perturbation would be linear in the momentum perturbation with slope J N. Since the drift system is only a first order approximation to the actual system, higher order terms are expected to spoil this linear relationship as the perturbation is increased. All this is consistent with Figure, where it can be seen that the drift rates fit well to a cubic polynomial with slope J 4 at zero. An essential aspect of the preq is that their location is ill-defined as a concept. Suppose for example one assigns the location Rk to a preq corresponding to the state p 0 P sp at some particular time, and sometime later the state of the system is p P sp. If there is a group element A SO(3) such that Ap 0 = p, then the location corresponding to p is RAk, unequivocally. However, because it is moving, the preq at any moment corresponds to a point of phase space with generic momentum, and for N 4 the generic reduced phase space for the vortex system has dimension 2N 4 4, so the flow on this reduced phase space is usually at least as complicated as a toral flow with two incommensurate frequencies. Thus, the reduced flow may never repeat itself and there may never be an

10 40 G. W. Patrick Fig.. Left: A 4-vortex preq with opening angle α = π/6 subjected to a large perturbation subsequently moves on the sphere in a circular path about the perturbed momentum, which points towards you. The sense of rotation is clockwise, the reverse of the right-hand sense obtained from the perturbed angular momentum. Shown are the paths of the central vortex and one outer vortex; the paths of the other two outer vortices are suppressed to avoid cluttering the picture. Upper right: the drift rate vs. µ ; the curve shown is a cubic with slope at 0 equal to J 4. Lower right: the angular velocity of a preq divided by µ, calculated over 200 consecutive intervals of its path. A such that Ap 0 = p exactly. For the simulations reported above, the location of the preq was decreed to be the average of the locations of its constituent vortices; another possibility for example is the location of its central vortex. The preq alters its shape on a fast time scale compared with its overall motion, and this gives a statistical character to the meaning of its location on the sphere. The location of the preq becomes ever more exact and its character ever less statistical as the perturbation falls to zero. This problem is illustrated in the bottom right of Figure, where the path of a preq has been regularly sampled, and plotted are the values of J 4 from angular velocities obtained from consecutive changes in the angle that the preq makes with its initial condition. Were the preq to exactly follow the drift system the plot would be constant, but instead one gets a variation about the average value of This average is just.07% off the value of J 4 =.4599 predicted by Equation (24). The issue of whether it is possible to define the location of a preq may be put more deeply as follows. Let the Marsden-Weinstein reduced space for the vortex system at its zero momentum level be P sp 0 with reduced Hamiltonian H sp. The relative equilibrium p e corresponds to the equilibrium, say p e of the reduced system. In Patrick [995] it is shown that there is a symplectomorphism defined near the group orbit SO(3) p e and onto a neighborhood of p e times the zero section of T SO(3) such that the Hamiltonian H sp becomes H sp (x,π) H sp (x) + π ξ e + 2 N απ 2 + h.o.t. Thus the reduced degrees of freedom are linked to the overall motion by the higher order terms depending on both x and π, and if these terms vanish then the location of the preq may be assigned unequivocally by following the fiber of T SO(3) to its zero section.

11 Dynamics of Perturbed Equilibria of Point Vortices 4 Conversely, the location of the preq is ill defined in as much as one is obstructed in removing these higher order terms. 3. Small Opening Angles With opening angles as large as α = π/6, such as in Figure, the relative equilibrium p e does not have the appearance of a localized particle. But one may choose α arbitrarily small, and now I discuss some aspects of small α and report some numerics for α = π/ deg. Firstly, in the α 0 limit there is an essential difference between the 3-vortex and 4-vortex preq: For the 3-vortex preq, from (22), while for the 4-vortex preq, from (24), m α = 8π + O(α 2 ), m α = 8 3 πα2 + O(α 4 ). Since the mass of the 3-vortex preq does not fall with α while that of the 4-vortex preq does, the 3-vortex preq is very heavy in comparison to the 4-vortex preq for small α. Thus, large momentum perturbations are required to move a 3-vortex preq. As I shall show, momentum perturbations must also fall with α or else they will destroy the relative equilibrium by too grossly perturbing it. Thus, for dynamical purposes the 3-vortex preq is infinitely heavy in the α = 0 limit. This is confirmed upon passage to the planar system, as will be seen in Section 4. The restrictions (29) in the α = 0 limit have a different character as they affect µ vs. µ 3. The effect on µ is clear: directly from (27) and the first of (29) µ = Ɣ sin δ O(α). However, by elementary manipulations, (28) becomes where, temporarily, α = cos (z cos α) α, z cos δ + µ 3 Ɣ. (30) By considering the function z cos (z cos α) α for small α, one verifies that cos (z cos α) α α is equivalent to 0 < z cos α or 0 < z cos 2α cos α. In any case this amounts to z < O(α 2 ), and since δ α, one gets by (30) that µ 3 O(α 2 ). The point is that for momentum perturbations µ, µ 3 to be considered small the first must fall as α while the second must fall as α 2. Since µ

12 42 G. W. Patrick Fig. 2. Left: A preq incident from the lower left approaches its anti-preq incident from the lower right; a repulsive force is evident. Right: the same preq and anti-preq collide head on and break up into dipoles. The bars at the lower left indicate the diameter of the preqs. µ 3 causes preq motion that is rotation about an axis perpendicular to the direction of the preq, the effect is that the preqs can have momentum O(α) only for motions O(α)-close to great circle paths. To obtain preq motion along smaller circular paths on the sphere, one must take µ µ 3 = O(α 2 ), which implies a much smaller O(α 2 ) momentum. Since the maximum reasonable momentum perturbations are O(α) and the 4-vortex mass is O(α 2 ), the maximum 4-vortex angular velocity on the sphere is O(/α). Thus greater velocities are available to smaller opening angles. The velocity available to a 4-vortex preq compared with its diameter is O(/α 2 ). A most interesting aspect of preqs with small opening angles is that more than one of them may be positioned on the sphere, all initially far apart (compared with their radii) from one another. Because the vortex-vortex interaction falls as the vortex-vortex distance increases, as long as the separate preqs remain separated they will not strongly interact with one another, and they will maintain their separate identities. However, as they move separately on the sphere they may closely approach one another. When two or more preqs closely approach, the above perturbation theory becomes inapplicable and they undergo an interaction, where they may move apart from one another largely unchanged, or they may be partially or completely destroyed. There is no theory of these interactions at this time, but the main features, obtained by numerical simulation and illustrated in Figures 2 and 3, are as follows: Two preqs interact when their distance falls to lengths comparable with their diameters. Two preqs with vortices having the same sign are attracted to one another while if the vortices have opposite signs they repel one another. Thus identical preqs attract while a preq and its anti-preq (obtained from the first by changing the signs of each vortex) repel. The collision of two identical preqs, because those preqs attract, tends to result in two groups of four vortices representing a state of the vortex system far away from the relative equilibria p e. In this sense the 4-vortex preqs are usually destroyed in samesense vortex preq collisions. In such an interaction, the two 4-vortex preqs usually exchange one of their outer vortices.

13 Dynamics of Perturbed Equilibria of Point Vortices 43 Fig. 3. Left: A preq incident from the lower left approaches an identical preq initially at rest in the center; and attractive force is evident. Right: A preq moving rightward and slightly upward collides with an identical preq moving leftward and slightly downward. The denser tracks at the center are the incident preqs. The preq incident from the left remains and rebounds back the way it came. The preq incident from the right has been nearly destroyed and is a four vortex motion far from the original relative equilibrium. Clearly visible in the interaction is the exchange of an outer vortex. A very energetic collision of a 4-vortex preq with its anti-preq usually results in the destruction of both preqs into vortex dipoles. The detailed results of energetic or non-elastic 4-vortex preq collisions are extremely sensitive to initial condition, while the gross aspects of the collision (e.g. whether or not the collision is elastic) are relatively more robust. Many a happy hour may be spent watching the antics of 4-vortex preqs colliding on the sphere. The nature of the 4-vortex interaction is the subject of current investigations. 4. Transcription to the Planar System As already noted (see Equations (8)), when restricted to sufficiently small regions of the sphere, the system of point vortices on the sphere reverts to the system of point vortices on the plane. This is pertinent for the study of small-opening-angle preq interactions, since the interactions occur only as the preq closely approach. Moreover, the planar case, being an example with noncompact symmetry group, is currently interesting, as the general focus of the Hamiltonian stability literature has been on the compact case, while the noncompact situation is known to have distinctive features, as shown by Leonard and Marsden [997]. Consequently, I close this article with a short transcription of the above to the planar case; the results should be viewed as the α = 0 limit of the spherical system. The transcription is easy: Through the pull-back (7) of the sphere to the plane, the relative equilibria p e with opening angle α have radii α = R sin α, and by (20) they rotate at angular velocity Ɣ θ α = 4π R 2 sin 2 α ( ) N + cos α = N Ɣ N 4π α + O( 2 α4 ), (3)

14 44 G. W. Patrick while, from (3), Ɣ Ɣ = (N ) cos α = Ɣ N + O( α2 ). Deleting the higher order terms and renaming α to α, one expects planar relative equilibria with radii α, central vortex strength Ɣ, outer vortex strength Ɣ/(N ), and se(2) generator ( θ α, 0), where θ α = N Ɣ N 4πα. 2 It is easily verified that these are indeed relative equilibria of the planar system. Moreover, these relative equilibria occur for the vortex strength parameters Ɣ n such that Ɣ n = 0, parameters for which, by (0), the momentum map J pl is equivariant. For my purpose, equivariance of the momentum map is important since, in general, equivariance guarantees, for a relative equilibrium p e with momentum µ e and generator ξ e, the momentum-commutation relation coad ξe µ e = 0, and this is crucial for the perturbation theory of Patrick [995]. On the other hand, in absence of equivariance, one has the more complicated commutation relation coad µ e = (ξ e, ) = i ξe, ξ e and the perturbation theory would have to be extended in a fundamental way to cover the nonequivariant case. That not being necessary, one calculates the nilpotent parts of the linearizations of the planar system at the relative equilibria above, to obtain, for the 3-vortex relative equilibria N α = , 2πα while for the 4-vortex relative equilibria the nilpotent part turns out to be N α = πα Going to the Lagrangian viewpoint, and comparing with the Lagrangian of a twodimensional rigid body moving in the plane, one sees that the 3-vortex preq have infinite mass while the 4-vortex preq have inertia coefficient 3 2π α2 and mass 8π 3 α References [] Abraham, R. and J. E. Marsden [978]. Foundations of Mechanics (second ed.). Addision- Wesley, Reading, MA. [2] Adams, M. and T. S. Ratiu [988]. The three point vortex problem: commutative and noncommutative integrability. In K. R. Meyer and D. G. Saari (Eds.), Hamiltonian Dynamical Systems, Volume 8 of Contemporary Math., pp AMS, Providence, RI.

15 Dynamics of Perturbed Equilibria of Point Vortices 45 [3] Aref, H. [978]. Motion of three vortices. Phys. Fluids 22, [4] Aref, H. [982]. Point vortex motions with center of symmetry. Phys. Fluids 25, [5] Aref, H. [983]. Integrable, chaotic, and turbulent vortex motion in two-dimensional flows. Annual Review of Fluid Mechanics 5, [6] Aref, H. [995]. On the equilibrium and stability of row of point vortices. J. Fluid Mech. 290, [7] Aref, H. and N. Pomphrey [982]. Integrable and chaotic motions of four vortices I. The case of identical vortices. Proc. Roy. Soc. London Ser. A 380, [8] Bauer, L. and G. K. Morikawa [976]. Stability of rectilinear geostrophic vortices in stationary equilibrium. Phys. Fluids 9, [9] Bogomolov, V. A. [977]. Dynamics of vorticity at the sphere. Fluid Dynamics 2, [0] Campbell, L. J. and J. B. Kadtke [987]. Stationary configurations of point vortices and other logarithmic objects in two dimensions. Phys. Rev. Lett. 58, [] Channell, P. J. and F. P. Neri [993]. An introduction to symplectic integrators. In J. E. Marsden, G. W. Patrick, and W. F. Shadwick (Eds.), Integration Algorithms and Classical Mechanics, Volume 0 of Fields Inst. Commun., pp [2] Eckhardt, B. and H. Aref [988]. Integrable and chaotic motions of four vortices II. Collision dynamics of vortex pairs. Philos. Trans. Roy. Soc. London Ser. A 326, [3] Havelock, T. H. [93]. The stability of motion of rectilinear vorticies in ring formation. Philos. Mag., [4] Khazin, L. G. [976]. Regular polygons of point vortices and resonance instability of steady states. Soviet Phys. Dokl. 2, [5] Kidambi, R. and P. K. Newton [998]. Motion of three point vortices on a sphere. Physica D 6, [6] Leonard, N. E. and J. E. Marsden [997]. Stability and drift of underwater vehicle dynamics: Mechanical systems with rigid motion symmetry. Physica D 05, [7] Lerman, E. and S. F. Singer [998]. Stability and persistence of relative equilibria at singular points of the momentum map. Nonlinearity, [8] Lewis, D. and T. S. Ratiu [996]. Rotating n-gon/kn-gon vortex configurations. J. Nonlin. Sci. 6, [9] Marsden, J. E. [992]. Lectures on Mechanics, Volume 74 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge. [20] Marsden, J. E. and T. S. Ratiu [994]. Introduction to Mechanics and Symmetry, Volume 7 of Texts in Applied Mathematics. Springer-Verlag, New York. [2] Mertz, G. J. [978]. Stability of body-centered polygonal configurations of ideal vortices. Phys. Fluids 2, [22] Morikawa, G. K. and E. V. Swenson [97]. Interacting motion of rectilinear geostrophic vortices. Phys. Fluids 4, [23] O Neil, K. A. [987]. Stationary configurations of point vortices. Trans. Amer. Math. Soc. 302, [24] Ortega, J.-P. and T. S. Ratiu [999]. Stability of Hamiltonian relative equilibria. Nonlinearity 2, [25] Patrick, G. W. [992]. Relative equilibria in Hamiltonian systems: The dynamic interpretation of nonlinear stability on the reduced phase space. J. Geom. Phys. 9, 9. [26] Patrick, G. W. [995]. Relative equilibria of Hamiltonian systems with symmetry: Linearization, smoothness, and drift. J. Nonlin. Sci. 5, [27] Patrick, G. W. [999]. Dynamics near relative equilibria: Nongeneric momenta at a : group-reduced resonance. Math. Z. 232, [28] Pekarsky, S. and J. E. Marsden [998]. Point vortices on the sphere: stability of relative equilibria. J. Math. Phys. 39, [29] Zhang, M.-Q. and M.-Z. Qin [993]. Explicit symplectic schemes to solve vortex systems. Comput. Math. Appl. 26, 5 56.