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A. V. BORISOV, I. S. MAMAEV, A. A. KILIN
Institute of Computer Science 1, Universitetskaya str. 426034, Izhevsk, Russia E-mail: b orisov@rcd.ru E-mail: mamaev@rcd.ru E-mail: aka@rcd.ru

ABSOLUTE AND RELATIVE CHOREOGRAPHIES IN THE PROBLEM OF POINT VORTICES MOVING ON A PLANE
Received April 05, 2004

DOI: 10.1070/RD2004v009n02ABEH000269

We obtained new periodic solutions in the problems of three and four point vortices moving on a plane. In the case of three vortices, the system is reduced to a Hamiltonian system with one degree of freedom, and it is integrable. In the case of four vortices, the order is reduced to two degrees of freedom, and the system is not integrable. We present relative and absolute choreographies of three and four vortices of the same intensity which are periodic motions of vortices in some rotating and fixed frame of reference, where all the vortices move along the same closed curve. Similar choreographies have been recently obtained by C. Moore, A. Chenciner, and C. Simo for the n-body problem in celestial mechanics [6, 7, 17]. Nevertheless, the choreographies that appear in vortex dynamics have a number of distinct features.

Equations of motion and first integrals. Let us review briefly basic forms of equations and first integrals in the dynamics of p oint vortices on a plane (for a detailed discussion, see [4, 14, 16], where, in addition, hydro dynamical assumptions required for the validity of these equations are sp ecified). For n p oint vortices with Cartesian co ordinates (x i , yi ) and intensities i , the equations of motion can b e written in Hamiltonian form, i x i = , yi where the Hamiltonian is =- 1 4


i yi = - , xi

n i
i j ln |r i - r j |2 ,

Here, the Poisson bracket is:
N

{f , g } =

i=1

1 f g - f g . i x i y i y i x i

Equations (1) are invariant under the action of the Eucledian group E (2), so (in addition to the Hamiltonian) they also have three first integrals
n n n

Q=
i=1

i xi ,

P=
i=1

i yi ,

Mathematics Sub ject Classification 76B47, 37J35, 70E40

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1

i

n,

(1)

r i = (xi , yi ).

(2)

(3)

I=
i=1

2 i (x2 + yi ), i

(4)

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A. V. BORISOV, I. S. MAMAEV, A. A. KILIN

which, however, are not involutive,
N

{Q, P } =

i ,
i=1

{P , I } = -2Q,

{Q, I } = 2P .

(5)

Using these three integrals, it is p ossible to construct two involutive integrals, Q 2 + P 2 and I . Hence, according to the general theory [10, 13], we can reduce the numb er of degrees of freedom by two. Thus, in the particular case of three vortices, the system can b e reduced to one degree of freedom and b ecomes integrable (Gr¨ obli, Kirchhoff, Poincar´ [4, 14, 16, 18], while the problem of four vortices e) is reduced to a system with two degrees of freedom. Generally, the last problem is not integrable [20]. Effective reduction in the system of four vortices with intensities of the same sign was done by K. M. Khanin in [9]. In that work two pairs of vortices were considered and for each pair action-angle variables were selected. The system of four vortices was then obtained as a p erturbation of the two two-vortex systems. He proves (using the metho ds of KAM-theory) the existence of quasip erio dic solutions. As a small parameter, he takes a value inverse to the distance b etween two pairs of vortices. Reduction by one degree of freedom using the translational invariants P and Q was done by Lim in [11]. He intro duced barycentric Jacobi co ordinates (centered, in this case, at the center of vorticity), which have well-known analogs in the classical n-b o dy problem in celestial mechanics [5]. Note that even this (partial) reduction made it p ossible to apply some metho ds of KAM-theory to the problem of p oint vortices' motion [11, 12]. When n vortices on a plane have equal intensities, one of the more formal metho ds of order reduction, which was describ ed in [2] (see also [3, 4]), proved to b e most suitable. This metho d is based on using mutual variables representation of the equations of motion. As mutual variables, the squares of distances b etween the pairs of vortices and oriented areas of triangles were taken: Mij = (xi - xj )2 + (yi - yj )2 , Mutual commutation of such variables (which Here, the reduction pro cedure (more exactly, algebraic problem of intro duction of symplectic (For identical vortices the metho d of reduction in [1].)
ij k

= (r j - r k ) (r k - r i ).

(6)

are intro duced by E. Laura) leads to some Lie algebra. the last canonical step of it) is equivalent to a purely co ordinates on the orbits of corresp onding Lie algebras. based on the Fourier transformatio was intro duced also

Reduction for three and four vortices of equal intensity. Without losing in generality, we put i = j = 1, P = Q = 0, then the moment integral I (4) can b e written as 1 I=n
n

Mij ,
i
(7)

where n is the numb er of vortices. Using the complex representation of the vortices' co ordinates, zk = xk + iyk , we obtain: 1 zk = n
n

Mk j e
j =k

i

kj

,

(8)

where kj is the angle b etween the vector from the j -th vortex to the k -th vortex and the p ositive direction of O x-axis. The following prop ositions, defining the dynamics of a reduced system of vortices, can b e obtained by direct computation: Prop osition 1. For three vortices of equal intensity, evolution of the mutual distances (6) (assuming I = const) is described by a Hamiltonian system with one degree of freedom. In canonical
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variables (g , G), the system looks as


g= , G where M - 2 12
12

G = - , g

= - 1 ln M12 M13 M23 , 4
I - G G cos g , M 2
23

= 4 I -G , M
2 I - G G cos g . 2

13

= 8G - I + 2 12

Prop osition 2. For four vortices of equal intensity, evolution of the mutual distances is described by a Hamiltonian system with two degrees of freedom. In canonical variables (g , G, h, H ), the system looks as g= , G


G = - , g

h= , H

H = - , h

= - 1 ln M12 M13 M14 M23 M24 M34 , 4

where M M M
12 13 14

=I -G+2 =I +G+2 =H +2

(I - H )(I - G) cos h, (I - H )G cos(h + g ),

M M M

34 24 23

= I -G-2 =H -2

(I - H )(I - G) cos h, (I - H )G cos(h + g ),

= I +G-H -2

(H - G)G cos g ,

(I - G)G cos g .

Remark. These canonical variables are of natural geometrical origin related to representation of the equations of motion on a Lie algebra [2, 3].

Absolute motion: quadratures and geometric interpretation. According to (8), when Mij (t) are known, one needs to know the angles ij (t) to determine the vortices' co ordinates. It is obvious that only one of the angles is indep endent (in this case, we take 12 to b e indep endent), the remaining angles are computed with the cosine theorem: ij + Evolution of 4 4
12 12 12 ik

= arccos

M

jk

2

- Mij - M 2Mij M
ik

ik

,

i = j, k = i.

is obtained by quadrature [14]: (for three vortices), (12) M
-1 24

=M =

-1 -1 -1 -1 -1 13 + M23 + M12 (6 - M23 M13 - M13 M23 ) - - - - M131 + M141 + M231 + M241 + - - - - + M121 (8 - M131 M23 - M141 M24 - M231 M13 -

M14 )

In the case of p erio dic solutions of the reduced system (9), (10), there is an interesting geometric interpretation of the absolute motion. Prop osition 3. 1 there exists a center of vorticity in 2 the rotational Let (t) be a periodic solution (of period T frame of reference, uniformly rotating with which each vortex moves along some closed p velocity a is given by (to within 2 q , p, q
T
T

) of the reduced system, then some angular velocity a about the curve i (t); Z): (13)

a = 1 T
0

12 (t)dt.

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(9) = 4 I -G
2

-

(10)

(11)

(for four vortices).

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Fig. 1. The phase portrait of the reduced system in the case of three vortices. Relative choreographies corresponding to different phase tra jectories on the portrait p 3 if the velocities a and o = 2 are commensurable (i. e. a = q , p, q Z), then the vortices T o in the fixed frame also move along some closed curves ; 4 if any of the curves i (t) can be superimposed by rotation about the center of vorticity by an angle, commensurable with 2 , then there exists a (rotating) frame of reference such that the corresponding vortices move along the same curve.

Proof. We expand the p erio dic function of p erio d T in the right-hand side of (12) in a convergent Fourier series 2 n i t a(n) e T . (14) 4 12 = Integrating (14) while b earing (11) in mind, we come to a conclusion that the angles in the following way: ij (t) = ij t + gij (t), where ij = a
(0)


n

ij

dep end on t (15)

qij + 2 pij , qij , pij Z, while gij (t) = gij (t + T ) are T -p erio dic functions of time. T Substituting this into (8), we see that lo cations of the vortices on a plane is given as follows:

zk (t) =

Q + iP + uk (t)e i

it

,

uk (t) = uk (t + T1 ) C,

=a

(0)

.

(16)

Hence, in the frame of reference, rotating about the center of vorticity with angular velocity , al l the vortices move along closed analytical curves, given by functions u k (t) C. The pro of of 2 , 3 , 4 with (14)­(16) is evident.
Remark. Proposition 3 is generalized without change to the case of arbitrarily many vortices, n, provided that (t) is a periodic solution of the reduced system with 2n - 2 degrees of freedom. (For a more detailed discussion of reduction, see, for example, [4].)

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Analytical choreographies. Now we show that the four- and three-vortex problems can have remarkable p erio dic solutions, when all the vortices follow each other along the same curve; such solutions are referred to as simple (or connected) choreographies. To underline the difference b etween choreographies in a fixed frame of reference and choreographies in a rotating one, these choreographies are called absolute and relative [7]. Theorem 1 ([4]). If in the problem of three vortices of equal intensity, the constants of the integrals of motion, I and , satisfy the inequality


- ln 3 < 4 3

+ ln I < ln 2,

(17)

then this motion is a simple relative choreography (see Fig. 1). Proof. Since in the case of three vortices, the reduced system (9) has one degree of freedom, all M ij (t) are p erio dic functions of the same p erio d T . It can b e easily shown that under the restriction (17), the orientation of the vortex triangle is unchanged, and there are times t 1 and t2 such that M23 (t1 ) = = M13 (t2 ). (If (17) is not met, then t k such that M ij (k ) < Mik (t) and Mij (k ) < Mj k (t).) Moreover, since with fixed H , I all Mij are expressed through one of them (for example, M 12 ), the following relations also hold true: M13 (t1 ) = M12 (t2 ), M12 (t1 ) = M23 (t2 ).

Since the equations are invariant under cyclic p ermutation of the vortices (as implied by equality of the intensities), we find that t1 - t2 = T n, n Z. 3 Since the evolution equation for Mij is of the first order, we conclude that M12 (t) = M or M12 (t) = M
23 23

t+ T 3 t + 2T 3

=M

13

t + 2T 3 t+ T 3

= f (t),

=M

13

= f (t),

where f (t) is some T -p erio dic function. Substituting this into (8), we obtain zk (t) = u t + k-1 T 3 e
it

,

k = 1, 2, 3,

where u(t) is a T -p erio dic complex-valued function, determining the same curve, along which the vortices move, in the frame of reference rotating with angular velo city . The phase p ortrait of the reduced system of the 3-vortex problem and the corresp onding relative choreographies are given in Fig. 1. The 4-vortex problem has an exceptional solution, given by quadratures -- Goryachev's solution, where the vortices form a parallelogram at each instant of time [8]. As in the case of the 3-vortex problem, it is quite easy to show that Theorem 2 ([4]). If in the 4-vortex problem, the vortices (of equal intensity ) form a central ly symmetric configuration (a paral lelogram ), while the constants and I satisfy the inequality


- ln 2 < 2 3

+ ln I < - ln 144 , 5

then the motion is a simple relative choreography. The corresp onding relative choreographies are shown in Fig. 2.
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Fig. 2. Relative choreographies in the 4-vortex problem. Remark. The physical meaning of the inequality is as follows: when I is fixed, the type of the motion in the 3-vortex problem and in the case of Goryachev's solution is changed at energy values corresponding to the Thomson and collinear configurations.

New p erio dic solution in the 4-vortex problem. Now we show that, aside from the choreography just describ ed, the 4-vortex problem has at least one more choreography (different from Goryachev's solution). Consider the vicinity of Thomson's solution, i. e. the motion, where the vortices are lo cated at the vertices of a square and rotate uniformly ab out the vorticity center [19]. It is obvious that in the case of the reduced system with two degrees of freedom (10), Thomson's solution is represented by a fixed p oint (more precisely, by six p oints corresp onding to various arrangements of vortices at square's vertices). We consider one of the arrangenents (the rest are completely identical) with co ordinates G = 0, H = 1 , h = . 2 2 Let us find the normal form of the reduced system's (10) Hamiltonian near this p oint. To do that,

Fig. 3. Relative choreographies corresponding to the new periodic solution in the three-body problem.

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ABSOLUTE AND RELATIVE CHOREOGRAPHIES

we start with a canonical change of variables: G= x2 + X 2 , 2 g = arctg x , X H = 1 +8 2
-1/4

Y,

h= +8 2

1/4

y;

Expanding the Hamiltonian in a series up to and including quadratic terms, we find + r, 3(x2 + X 2 ) + 2 2(y 2 + Y 2 ) ,
2


= ln 2 +

(18)

2


=1 4

where the expansion of r starts with the third-order terms. Thus, the Hamiltonian 2 defines an integrable system with two incommensurable frequencies and has precisely two non-degenerate p erio dic solutions on each energy level 2 = h2 = const. These solutions are given by x = X = 0, y = Y = 0,


Fig. 4. Relative choreographies corresponding to the new periodic solution in the three-body problem in the frame of reference different from that used in Fig. 3.

According to the Lyapunov theorem [15], these solutions are preserved under p erturbations, hence, the complete system (18) in the vicinity of the fixed p oint also has two non-degenerate p erio dic solutions on each energy level. It is easy to show that one the solutions (corresp onding to (19)) is identical to Goryachev's solution -- during their motion the vortices are lo cated at the vertices of a parallelogram. At the same time, the other solution (20) do es not have such a simple geometric interpretation. Since equations (1) are invariant under a cyclic p ermutation of the vortices c (z1 , z2 , z3 ) = = (z3 , z1 , z2 ) and the eigenvalues of the Hamiltonian 2 are different, it is easy to show that b oth p erio dic solutions are also invariant under c . Thus, according to Prop osition 3, in the appropriate frame of reference, all the vortices move along the same curve, i. e. b oth solutions corresp ond to simple relative choreographies. Figure 3 shows the relative choreographies that corresp ond to the new p erio dic solution of the reduced system (10).
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y 2 + Y 2 = h2 , 2 x2 + X 2 = 4 h2 . 3

(19) (20)

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Relative and absolute choreographies. Generally, for each p erio dic solution (of p erio d T ) of the reduced system (9), (10), it is p ossible to sp ecify a countable set of rotating frames of reference, where the vortices move along closed curves. Indeed, the tra jectories will remain closed in a frame rotating with velo city p (21) a = a + q 2 , p, q Z T Nevertheless, the change (21) with arbitrary p and q do es not preserve the connectedness of the tra jectories, i. e. in the general case, after moving to the frame of reference, rotating with velo city a , a simple relative choreography decomp oses into separate closed curves, along which the vortices move. To preserve the connectedness, the following criterion should b e met. Prop osition 4. Let a periodic solution (of period T ) of the reduced system describe a connected relative choreography, the rotational velocity of the frame of reference being a , while the period of the motion of the vortices along the corresponding common curve being equal to mT . Then, if mp = k nq , (22)

where n is the number of vortices, and k Z is an arbitrary integer, the transformation (21) results in a connected choreography. Proof. For the solution in question, the absolute co ordinates of the vortices can b e presented in the form: zk (t) = u t + k-1 n mT e
ia t

,

(23)

where u(t) = u(t + mT ) is a p erio dic complex-valued function (of p erio d mT ). Solving for a from (21) and substituting into (23), we get zk (t) = u t + k-1 n mT e
-i 2 p t T q eia t

= uk (t)e

ia t

.

(24)

If all the vortices move along the same curve, then their co ordinates in the rotating (with velo city a ) frame of reference are equal to uk (t) and satisfy u account, we obtain (22).
k +1

(t) = u

k

t + mT , whence, taking (24) into n

The relation (22) is a sufficient but not necessary condition for the choreography to b e connected. If the curve along which the vortices move has additional symmetries, then, b esides the p, q that meet the condition (22), there are more velo cities of the form (21), which result in connected choreographies (see b elow, for Goryachev's solution). It is interesting to note that using the transformation (21), some choreographies can b e "disentangled" -- for example, Fig. 4 shows choreographies, corresp onding to the p erio dic solution (20), in the frame of reference, different from that used in Fig. 3, their rotational velo cities differ by - = 4 2 . 3T As it was shown ab ove (see Prop osition 3), if, for a relative choreography, the p erio d T of the reduced system's solution is commensurable with the rotational p erio d T a = 2 of the frame of

a

reference, then in a fixed frame of reference, all the vortices move along closed (and, usually, different) curves. Let us consider in greater detail the existence of absolute choreographies in the three- and fourvortex problems. According to what was said ab ove, any relative choreography, corresp onding to the
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Fig. 5. The rotational angular velocities, 1 and 4 , of relative choreographies are presented as functions of energy. The points where the graphs intersect the Ox axis correspond to the absolute choreographies of the (0) three vortices shown in the bottom (Figs. b and c). The heavy lines denote the basic angular velocity 1 , corresponding to the simplest relative choreography (shown in Fig. a), and the frequency of the periodic solution of the reduced system (9), 0 = 2 . T

(1)

(3)

Fig. 6. The rotational angular velocities, 2 and 6 , of relative choreographies are presented as functions of energy. The points where the graphs intersect the Ox axis correspond to the absolute choreographies of the (0) four vortices shown in the bottom (Figs. b and c). The heavy lines denote the basic angular velocity 2 , corresponding to the simplest relative choreography (shown in Fig. a), and the frequency of the reduced system (10), 0 = 2 . T

(1)

(2)

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A. V. BORISOV, I. S. MAMAEV, A. A. KILIN

p erio dic solution (of p erio d T ) of the reduced system (see (9), (10)), closes in time mT , m N. During (k ) this time interval, the vortices pass one and the same relative configuration m times. Let m b e the rotational velo cities of the frames of reference related to these choreographies. As it was shown ab ove, all the solutions of the reduced three-vortex system, for a fixed D and ET < E < EC (where ET and EC are the energies corresp onding to the Thomson and collinear configurations), describ e connected relative choreographies. Moreover, there is a frame of reference, where the choreography closes in (the shortest p ossible) time T (see Fig. 5); the corresp onding angular (0) velo city is denoted as 1 , its graph is shown in Fig. 5. The angular velo cities of the other relative connected choreographies are now given by
(k ) m

(E ) =

(0) 1

(E ) + 3k 0 (E ), where m N, k Z; m
2 . The velo city T (E )
(k ) m

(25) corresp onds to the

here 3k and m are coprime numb ers, and 0 (E ) =

choreography that closes in time mT . Obviously, the absolute choreographies are defined by the solutions of the equation
(k ) m

(E ) = 0,
(k )

(26)

where k , m are fixed, while E is unknown. Figure 5 shows the graphs of velo cities m (E ) together with the solutions of equation (26), as well as the corresp onding absolute choreographies (the simplest choreographies in a three-vortex system). Generally, there is a countable set of absolute choreographies with different m, k . Indeed, consider the function fa (E ) =
(0) 1

(E ) + a0 (E ),

a R,

for which the equality fa (EC ) = C < 0 holds (see Fig. 5). Since 0 (E ) > 0, there exists a numb er a such that when a > a , the function fa (E ) has at least one zero in the interval [E T , EC ]. It is clear that the interval [a , +) contains an infinite numb er of rationals of the form a = 3k , where 3k m and m are coprimes. The dotted line in Fig. 5 shows the curve f a (E ) and the relative choreography's rotational angular velo city a , calculated from (13). This choreography is the simplest disconnected choreography and is remarkable for the fact that the values at the ends of the interval, a (ET ) and a (EC ), are equal to the rotational angular velo cities of the Thomson and collinear configurations. For Goryachev's solution in the four-vortex system, the reasoning is similar but slightly mo dified. First of all, one can show that the simplest connected choreography closes in time 2T , while the (k ) vortices in this case pass one and the same relative configuration twice (i. e. velo cities 1 corresp ond to disconnected choreographies). The graph of one of the corresp onding angular velo cities, which (0) we denote as 2 , is given in Fig. 6. In this case, due to the symmetry of the curve related to the (0) choreography 2 , the frequencies corresp onding to connected choreographies must satisfy the relation different from (25),
(k ) 2m

(E ) =

(0) 2

k (E ) + m 0 (E ), where m is o dd, k Z;
T (E )

(27)

here k and m are coprimes, and 0 (E ) = 2 , where T is the p erio d of Goryachev's solution to the reduced system (10). This choreography closes in time 2mT . As ab ove, the equation (k ) 2m (E ) = 0 defines the absolute choreographies. Similar to the three-vortex problem, one can show that there is a countable set of absolute choreographies, describ ed by Goryachev's solution with different m and k .
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Stability. As a conclusion, let us discuss the stability of the sp ecified p erio dic solutions. Since the three-vortex problem is integrable, all the solutions of the reduced system (9) are p erio dic and stable. Yet it is easy to show that any (absolute or relative) choreography in this problem is neutrally stable under p erturbations of the vortices' p ositions in the absolute space. In the four-vortex problem, due to its non-integrability, the relative choreographies can b e (exp onentially) unstable. Yet, if a p erio dic solution of the reduced system (10) is stable, the corresp onding choreographies are also neutrally stable in the absolute space. The numerical analysis of the multipliers of the p erio dic solutions that describ e choreographies presented in Fig. 6 b, c shows that they are (exp onentially) unstable. We gratefully acknowledge supp ort of the Russian Foundation for Basic Research (grant 04­05­64367), the Foundation for Leading Scientific Scho ols (grant 136.2003.1) and CRDF (grant RU-M1-2583-MO-04).


References
[1] H. Aref, N. Pomphrey. Integrable and chaotic motions of four vortices. I ­ The case of identical vortices. Proc. Roy. Soc. London Ser. A. 1982. V. 380. P. 359­387 . [2] A. V. Bolsinov, A. V. Borisov, I. S. Mamaev. Lie algebras in vortex dynamics and celestial mechanics -- IV. 1. P. 23­50. Reg. & Chaot. Dyn. 1999. V. 4. [3] A. V. Borisov, I. S. Mamaev. Poisson structures and Lie algebras in Hamiltonian mechanics. Izhevsk: Udmurt University Press. 1999. (In Russian) [4] A. V. Borisov, I. S. Mamaev. Mathematical methods in the dynamics of vortex structures. In: Fundamental and applied problems of the vortex theory, eds. A. V. Borisov, I. S. Mamaev, M. A. Sokolovsky. Moscow­Izhevsk: ICS. 2003. P. 17­178. (In Russian) [5] C. V. L. Charlier. Die Mechanik des Himmels. Berlin: W. de Gruyter. 1927. [6] A. Chenciner, R. Montgomery. A remarkable periodic solution of the three-body problem in the case of equal masses. Annals of Mathematics. 2000. V. 152. 3. P. 881­901 . [7] A. Chenciner, J. Gerver, R. Montgomery, C. Simo. Simple choreographies of N bodies: a preliminary study. In: Geometry, Mechanics, and Dynamics. Volume in honor of the 60th birthday of J. E Marsden, eds. P. Newton, P. Holmes, A. Weinstein. Springer. 2001. [8] D. N. Goryachev. On certain cases of motion of rectilinear parallel vortices. Moscow University Printing House. 1898. (In Russian) [9] K. M. Khanin. Quasi-periodic motions of vortex systems. Physica D. 1982. V. 4. P. 261­269.
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