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ISSN 1560-3547, Regular and Chaotic Dynamics, 2009, Vol. 14, No. 2, pp. 179-217. c Pleiades Publishing, Ltd., 2009.

The Hamiltonian Dynamics of Self-gravitating Liquid and Gas Ellipsoids
A. V. Borisov* , A. A. Kilin** , and I. S. Mamaev*
Institute of Computer Science, Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
Received August 3, 2008; accepted Decemb er 1, 2008
**

Abstract--The dynamics of self-gravitating liquid and gas ellipsoids is considered. A literary survey and authors' original results obtained using modern techniques of nonlinear dynamics are presented. Strict Lagrangian and Hamiltonian formulations of the equations of motion are given; in particular, a Hamiltonian formalism based on Lie algebras is described. Problems related to nonintegrability and chaos are formulated and analyzed. All the known integrability cases are classified, and the most natural hypotheses on the nonintegrability of the equations of motion in the general case are presented. The results of numerical simulations are described. They, on the one hand, demonstrate a chaotic behavior of the system and, on the other hand, can in many cases serve as a numerical proof of the nonintegrability (the method of transversally intersecting separatrices). MSC2000 numbers: 70Hxx DOI: 10.1134/S1560354709020014 Key words: liquid and gas self-gravitating ellipsoids, integrability, chaotic behavior

Contents I. DYNAMICS OF A SELF-GRAVITATING FLUID ELLIPSOID
1 INTRODUCTION 2 TH 2.1 2.2 2.3 2.4 E DIRICHLET AND RIE The Dirichlet Equations . . The Riemann Equations . . Gravitational potential . . The Roche Problem . . . . ST INTEGR Vorticity . . Momentum . Energy . . . MAN .... .... .... .... N . . . . EQUATIONS .......... .......... .......... .......... 182 182 184 184 185 186 187

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3 FIR 3.1 3.2 3.3

ALS 188 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 189 . 189 . 191 . 191

4 LAGRANGIAN AND HAMILTONIAN FORMALISM 4.1 Hamiltonian Principle and Lagrangian Formalism . . . . . . . . . . . . . . . . . . . 4.2 Symmetry Group and the Dedekind Reciprocity Law . . . . . . . . . . . . . . . . . 4.3 Hamiltonian Formalism and Symmetry-based Reduction . . . . . . . . . . . . . . . .
* ** ***

E-mail: borisov@ics.org.ru E-mail: aka@ics.org.ru E-mail: mamaev@ics.org.ru

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BORISOV et al. o . . . in . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 193 194 195 196 197

5 PARTICULAR CASES OF MOTION 5.1 Shape-preserving Motions of the Ellips 5.2 Axisymmetric Case (Dirichlet [2]) . . 5.3 Riemannian Case [3] . . . . . . . . . . 5.4 Elliptic Cylinder (Lipschitz [13]) . . .

6 CHAOTIC OSCILLATIONS OF A THREE-AXIAL ELLIPSOID

I I. DYNAMICS OF A GAS CLOUD WITH ELLIPSOIDAL STRATIFICATION 198
1 INTRODUCTION 2 EQUATIONS OF MOTION OF A GAS CLOUD FIELD 2.1 The Ovsyannikov Model [74] . . . . . . . . . . . 2.2 The Dyson Model [77] . . . . . . . . . . . . . . . 2.3 Model of a Cooling Gas Cloud (Fujimoto [78]) . 2.4 Model of a Dust Cloud (Gravitational Collapse) WITH A LINEAR VELOCITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 201 203 204 205 2 05 206 207 207 211 212 214 214 198

3 LAGRANGIAN FORMALISM, SYMMETRIES, AND FIRST INTEGRALS 4 SYMMETRY-BASED REDUCTION AND HAMILTONIAN FORMALISM 5 PARTICULAR CASES OF MOTION 5 5.1 Case of = 3 (Monoatomic Gas) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Case of Axial Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Generalization of the Riemannian Case . . . . . . . . . . . . . . . . . . . . . . . . . . 6 ACKNOWLEDGMENTS REFERENCES

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This review article is dedicated to the dynamics of self-gravitating liquid and gas ellipsoids. This area of mechanics is represented by extensive studies, which were in many cases carried out indep endently by scolars from different countries and published in various journals, not always readily available. Certain results in this area are still controversial. We make here an attempt of a systematic exp osition of fundamental, b oth classical and fairly recent, results concerning the dynamics of ellipsoidal figures. We also present our fresh results demonstrating a chaotic b ehavior of oscillating fluid ellipsoids. Some unresolved problems are formulated. In the first part of the article, we describ e, in a closed, systematic form, basic results in the dynamics of ellipsoidal figures of ideal, incompressible fluid, starting from the foundational works of Dirichlet and Riemann. As is known, these studies were preceded by a p eriod of investigations of the static equilibrium of a rotating fluid mass, which were dictated by the scientific interest in the figure of the Earth and traced back to Newton's Mathematical Principles of Natural Philosophy and works by Clairaut and Maclaurin. A brief historical survey of this p eriod (also associated with the names of Jacobi, Mayer, and Liouville) can b e found in a monograph by Chandrasekhar [1]. Fundamental discoveries by Dirichlet [2] and Riemann [3] (1857-1861) were a turning p oint in developing the theory of figures of equilibrium; Dirichlet and Riemann were the first to investigate the dynamics of fluid ellipsoids. They noted the existence of a finite-dimensional solution of the Euler equations of the ideal-fluid dynamics according to which the ellipsoid preserves its shap e but deforms. This is an exact hydrodynamic solution, so that the question of investigating the dynamics of such fluid ellipsoids can b e correctly p osed. We describ e here the basic results of classical studies in a brief, closed form. We present the equations of motion in various representations and the integrals of motion; we also discuss the Lagrangian and Hamiltonian forms of the equations, which makes it p ossible to naturally carry out a symmetry-group-based reduction of the system. Next, we analyze partial solutions starting from the standard configurations found for the first time by Maclaurin, Jacobi, and Riemann. We note a new class of chaotic motions of ellipsoids in the form of irregular pulsations, with the axes of the ellipsoid remaining motionless in the absolute space. In this case, we numerically find p eriodic solutions, construct separatrices, and therefore present a computer proof of nonintegrability of the system in the general case. To analytically and numerically investigate such motions, we use the regularization of the equations of motion and formulate the hyp othesis of the nonexistence of analytical integrals. A particular case of this hyp othesis is the problem (tracing back to Riemann) of the nonintegrability of a geodesic flow on a very simple two-dimensional cubical surface in R3 (a cubic, in contrast to quadrics, quadratic surfaces). In the second part of the article, we consider problems of the dynamics of a gas cloud with an ellipsoidal stratification. The account of initial results on equilibrium and stability of compressible liquid and gas masses was given by Jeans in his fundamental essay [4], to which the Adams Prize for 1917 was awarded. This b ook also considers p ossible applications of these studies to the problems of cosmogony. Note (see p. 147 of [4]) that the most elementary model which is different from the classical incompressible model of figures of equilibrium was given by Roche [5], who largely utilized this model in construction of his cosmogonic theory generalizing the Laplace hyp othesis. The Roche model (as we call it following [4]) contains a gravitating Newtonian center surrounded by weightless atmosphere inside which self-gravitation is neglected. In this part, in a unified form, we present results obtained by Ovsyannikov, Dyson, Linden-Bell, Zel'dovich, Fujimoto, etc. Various forms of the equations of motion of the gas cloud are derived under certain thermodynamic assumptions. Some questions of the Lagrangian-Hamiltonian formalism are also discussed. A generalized Dedekind law of reciprocity is formulated. An analog of the Riemann equation for the case of a compressible fluid is obtained. A particular case of the expansion of an ellipsoidal cloud of ideal monoatomic gas in the absence of gravitation is analyzed in detail. This case was recently considered by Gaffet, who noted new integrals of this system. We formulate Gaffet's results more accurately, which enables us to establish the Liouvillian integrability in an extended space (Gaffet found integrals of the system presented, which can b e obtained by a reduction with use of a nonautonomous Jacobi-typ e integral). In particular, the involutivity of all the Gaffet integrals is revealed. An analogy b etween the general system considered by Gaffet and the generalized Euler- Calogero system is noted. The question of the Lax representation in this case is discussed. In conclusion, we present new partial solutions that are axisymmetric with or without the presence of gravitation. For the case without gravitation, the system is reduced to quadratures at arbitrary initial conditions. For the presence of gravitation, chaotic motions are noted, which is
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due to the nonintegrability of this system. Chaotic motions are also revealed in more general cases of oscillating gaseous ellipsoids.

I. DYNAMICS OF A SELF-GRAVITATING FLUID ELLIPSOID
1. INTRODUCTION The studies by L. Dirichlet in the dynamics of a self-gravitating fluid ellipsoid are dated back to 1856-1857. Ditichlet rep orted these studies in his lectures in 1857 and simultaneously in the Nachrichten von der Gesel lschaft der Wissenschaften zu Gottingen as a brief note [6]. Unfortunately, ? he had no time to describ e and publish his results in full (due to his illness and untimely death in 1859). These studies were prepared for publication and p osthumously published by Dedekind in 1861 [2]. Three basic results can b e isolated in Dirichlet's study: 1. A new partial solution of the hydrodynamic equations is presented, which describ es the motion of a homogeneous, self-gravitating ellipsoid, and the equations of motion (of fluid particles) in motionless axes are derived. 2. Seven first integrals of the obtained equations are found; six of them, linear in velocities, corresp ond to the conservation laws of vorticity and total momentum, and the seventh integral is the total energy of the moving fluid. 3. The motion of an axisymmetric ellipsoid is integrated in quadratures with the inclusion of Newton's and Maclaurin's spheroids as partial solutions (in this case, Dirichlet also analyzes the p ossibility of existence of the solutions found in the case of no external pressure, i. e., in vacuum). It is interesting to note that Dirichlet noted the integrals corresp onding to the conservation of the vorticity vector prior to the publication of the well-known study of 1858 by Helmholtz. [7]. As can b e judged by the form of the obtained integrals, Dirichlet was aware (b efore Helmholtz) of the conservation of vorticity not only for a particular solution but also for the general hydrodynamic equations (Dirichlet's note [6] is also evidence for his awareness). This fact was also noted by Klein in his well-known lectures [8]. Dedekind, while preparing Dirichlet's results for publication, discovered the reciprocity law according to which each solution of the Ditrichlet equations is corresp onded with a reciprocal solution in which the variables that describ e the rotation of the ellipsoid and the fluid motion inside it are p ermutated; in particular, he presented a solution (the Dedekind ellipsoid) reciprocal to the Jacobi ellipsoid, with the coordinate axes remaining motionless in space and with the fluid moving inside this invariable region [9]. An enormous contribution to the investigation of the dynamics of the fluid ellipsoid was made by an outstanding work by Riemann [3], which app eared in 1861, virtually immediately after the publication of Dirichlet's studies. The basic results of this work can b e briefly formulated as follows: 1. The equations of motion in moving axes (the principal axes of so that the order of the system was lowered and a linear-integr Furthermore, Riemann represented the equations of motion Hamiltonian form with a linear Lee-Poisson bracket (Riemann reduction to a b etter observable form). the ellipsoid) were obtained, al-based reduction was done. of the reduced system in a himself called this procedure

2. All partial solutions corresp onding to the motion of the ellipsoid without changes in its form were presented and conditions of their existence were analyzed (i. e., the p ossible lengths of the ma jor semiaxes). All these solutions imply that the ellipsoid rotates ab out an axis immovable in space. They included all solutions known by that time -- those obtained by Newton, Maclaurin, Jacobi, and Dedekind (for which the rotational axis coincides with one of the principal axes) and also new solutions (Riemann ellipsoids) for which the rotational axis lies in one of the principal planes of the ellipsoid.
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3. Riemann used the energy integral of the system as the Lyapunov function (in modern terminology) to investigate the stability of shap e-preserving motions (in the class of motions preserving the ellipsoidal shap e); in this way, he found the Lyapunov-stability limits for the Maclaurin spheroids and Jacobi ellipsoids. 4. A particular cas of the principal system with two a material p oint forces (it is this e was noted in which a three-axial ellipsoid (unsteadily) rotates ab out one axes, and its semiaxes vary with time. This gives rise to a (Hamiltonian) degrees of freedom for which Riemann noted an analogy with the motion of on a two-dimensional surface of the form xy z = const in a p otential field of case that we will consider b elow in detail).

The study by Riemann was unique in terms of the imp ortance of its results and p ossibilities of further generalizations; it was well in advance of its time. There is also a study by Brioschi of 1861 [10], which was dedicated to lowering the order in the Dirichlet equations with the use of a decomp osition into a p otential and a vortical comp onent. However, no substantial advance in the problem was associated with this work. In his lectures in mechanics of 1876, Kirchhoff [11] also considered the motion of self-gravitating fluid ellipsoids. He noted that the d'Alemb ert principle is applicable to the Dirichlet motion (although he did not use it to derive the equation of motion). Kirchhoff presents a quadrature for the axisymmetric case and separately analyzes the case where the ellipsoid preserves the directions of its axes in space (a particular case of the motion considered by Riemann); Kirchhoff (following Riemann) conjectures that this problem also cannot b e integrated in quadratures. The p ossibility of applying the variational principle to the derivation of the equations of motion of a fluid ellipsoid was indep endently shown by Padova in 1871 [12] and Lipschitz in 1874 [13]. In the latter study [13], the problem of the motion of an elliptic cylinder was also formulated and integrated in quadratures. Betti [14] also used the variational principle to derive the equations of motion of a fluid ellipsoid and represented these equations in a Lagrangian and a Hamiltonian form. However, as Tedone noted in his extensive survey [15], Betti made a mistake in his study when applying the variational principle to the derivation of the equation of motion of a homogeneous ellipsoid with an ellipsoidal fluid-density stratification. In this case, the hydrodynamic equations for the stratified, self-gravitating ellipsoid do not admit a solution with a linear dep endence on the initial coordinates, which Betti considered (in view of the complex dep endence of the gravitational p otential inside the stratified ellipsoid). Nevertheless, all Betti's results remain valid for a constant density. Betti also represented the equations of motion in a Hamiltonian form (explicitly using the Poisson brackets on the so(3) algebra) with a linear Poisson bracket and carried out a linear-integral-based reduction. The ab ove-listed results are the principal achievements of the classical p eriod of the investigation of the dynamics of the Dirichlet ellipsoids. General problems of the dynamics and statics of fluid ellipsoids, including the issues of stability, were investigated in classical treatises by Basset [16], Lamb [17], Thomson and Tait [18], Routh [19], in b ooks by App ell [20], Lyttleton [21], in certain studies by Basset [22-24], Duhem [25], Hagen [26], Hicks [27], Hill [28], Love [29, 30], etc. Note also the following related sub jects that constitute particular lines of research in this area. Ç The investigation of figures of equilibrium bifurcating from the ellipsoid, e. g., p ear-shap ed figures, and the analysis of their stability (Lyapunov [31, 32], PoincarÄ [33, 34], Darwin [35], Jeans [4], e and Sretenski [36]). As is known, a comprehensive analysis of this problem led Lyapunov to i the development of a general theory of stability of motion, which goes under his name. Results concerning the figures of equilibrium were also obtained in classical studies by Giesen [37], Bryan [38], and Liouville [39]. Here, we do not touch up on the theory of stability, where many problems still remain op en. Ç Figures of equilibrium of a homogeneous fluid that is, however, stratified in a sp ecial manner. The theoretical analysis reached here its summit with a p osthumous study by Lyapunov [32] (published by Stekloff ), which still remains p oorly comprehended. These investigations are closely related to the theory of the p otential of stratified fluids (see Dyson [40], Ferrers [41], Volterra [42]). In [42], as in Lyapunov's studies on this sub ject (Lyapunov's results are presented from a more modern,
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functional-analysis standp oint un a b ook by Lichtenstein [43]), integral equations arise; they were later studied in the framework of functional analysis. In the context of this problem, let us mention a valuable but virtually forgotten study by Veronnet [44], a b ook by Pizzetti [45], and certain modern works [46-48] and [49] (the last publications do not contain any references to classical results and are highly controversial). A new p eriod associated with astrophysical investigations was marked with investigations by Chandrasekhar and his school (see, e. g., [1, 50, 51]). Ç In many problems that trace back to the classical Plateau exp eriments and the Bohr- Wheeler model of the atom [52], the surface-tension forces are considered instead of Newtonian attraction. For the comprehension of this question, we can recommend b ooks by App ell [20] and Chandrasekhar [1]; it should b e emphasized, however, that the theory presented there is much less advanced (see also [53, 55, 56], where a Pade approximation for the p otential of the tension surface is given). At least, we are not aware of any results concerning the dynamics of fluid (masses) drops sub jected to the action of surface-tension forces. Effects of viscosity on the dynamics in the Dirichlet-Riemann problem are discussed, e. g., in [57]. Early computer investigations of selfgravitating nonellipsoidal figures can b e found, e. g., in [58]. Ç Note an interesting study by Narlikar and Larmor (1933) [59], where -- likely, for the first time -- the classical results (by Maclaurin, Jacobi, Dirichlet, PoincarÄ, etc.) are revised in the context e of stellar-dynamical problems rather than planetary evolution. These investigators assumed that energy dissipation occurs in the process of stellar evolution. Later, this line of research was further develop ed by Chandrasekhar. 2. THE DIRICHLET AND RIEMANN EQUATIONS 2.1. The Dirichlet Equations We recall here the principal steps represent them in a modern matrix f The equations of the dynamics of a Lagrangian form are in the case of in the derivation of the Dirichlet and Riemann equations and orm. a homogeneous, incompressible, ideal fluid of unit density in p otential forces applied to the fluid as follows: x a
T

? x=-

(U + p) , a

(1)

where a = (a1 , a2 , a3 ) are the initial p ositions of the material p oints of the medium (the so-called Lagrangian coordinates), x(a, t) are the coordinates of the p oints of the medium at the time t (i. e., x(a, 0) = a), U (a, t) is the density of the p otential energy of the external forces, p(a, t) is
xi the pressure, and x = aj is the matrix of the partial derivatives. These equations must b e a supplemented with the incompressibility condition, which can b e written is the case at hand as

det

x a

= 1.

(2)

Thus, we obtain a system of partial differential equations in which four quantities, viz., x1 , x2 , x3 , and p, are unknown as the functions of the variables a and t. To determine them, except initial conditions (x(a, 0) = a, x(a, 0) = v0 (a)), also b oundary conditions must b e sp ecified; in our case, the latter reduce to the statement that the pressure has the same value indep endent of a everywhere on the free surface. Dirichlet noted that, if the p otential of the external forces U (a, t) is a homogeneous quadratic function of the Lagrangian coordinates, i. e. U (a, t) = U0 (t) + (a, V(t)a), (3) where U0 (t) is indep endent of a and V(t) is a symmetric matrix, then the equations of motion (1), (2) admit a partial solution x(a, t) = F(t)a, Here, F(t) is a 3 ç 3 matrix.
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det F(t) = 1.

(4)


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In this case, the b oundary conditions will b e satisfied provided that the fluid has initially an ellipsoidal shap e, (a, A-2 a) 0 1, (5)

where A0 = diag(A0 , A0 , A0 ) is the matrix of the initial semiaxes and the pressure has the form 1 2 3 p(a, t) = p0 (t) + (t)(1 - (a, A-2 a)). 0 (6)

We substitute (3), (4), and (6) into (1) and (2) to obtain equations for the matrix F(t) and the function (t) in the form ? FT F = -2V - 2 A-2 , 0 det F = 1. (the Dirichlet equations) (7)

As Dirichlet showed, the system of ten equations (7) for ten unknown functions Fij (t), (t), i, j = 1, 2, 3, is compatible. Obviously, the transformation (4) changes the original ellipsoid (5) into the ellipsoid sp ecified by the quadratic form (x, (FA2 FT )-1 x) 0 1. (8)

2.2. The Riemann Equations Before writing an explicit expression for the p otential (3) and, accordingly, the right-hand side of equations (7), let us show how the equations of motion can b e written in a Riemannian form. To this end, we pass to the moving system of the principal axes of the ellipsoid. It is known that such a transformation is given by the orthogonal matrix = Qx, QT = Q
-1

.

(9)

In the new coordinates , the ellipsoid is sp ecified by the relationship ( , A-2 ) 1, (10)

where A = diag(A1 , A2 , A3 ) is the matrix of the principal semiaxes at the given time. We also note that, since the transform (4) is linear, the fluid particles constantly move over ellipsoids for which ( , A-2 ) = (a, A-2 a) = n2 = const, 0 0 n2 < 1. (11)

(In particular, the fluid particles that were initially at the b oundary remain at the b oundary at any time). Therefore, the modulus of the vector A-1 does not vary, so that the vectors A-1 and A-1 a are also related by the orthogonal transformation 0 A-1 = A
-1 0

a,

T = -1 .

(12)

Thus, we obtain the following decomp osition of the matrix F: F = QT AA
-1 0

.

(13)

Remark 1. Multiplying by a constant matrix A0 yields a decomp osition of the form FA0 = QT A known in linear algebra as a singular decomp osition [60].
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We introduce the angular velocities corresp onding to the orthogonal transformations, w = QQT , = T ,

(15)

which are known to b e antisymmetric matrices [63]. The substitution of (15) into equations (7) yields the Riemann equations, which can b e written in the following matrix form: ^ v - wv + v = -2VA + 2 A-1 , (the Riemann equations) (16) v = A - wA + A, A1 A2 A3 = 1, ^ where V = A-1 A0 VA0 T A-1 . We complement this system with the equations of evolution of Q = wQ, = , to obtain the complete system of equations of motion describing th Remark 2. For an arbitrary matrix G GL(3) with differing G = QA with Q, S O(3), A = diag(a1 , a2 , a3 ), a1 > a2 > matrices Q and admit discrete transforms of the form [60] R0 = E,

the matrices Q and , (17) e dynamics of the fluid ellipsoid. eigenvalues, the decomp osition a3 is not unique. Indeed, the (18)

Therefore, the space R2 S O(3) S O(3) (which is diffeomorphous to the configuration space of the Riemann system) is a four-sheet covering of the real configuration space S L(3). A similar procedure is used for a quaternion representation of the equations of a rigid b ody [63]. 2.3. Gravitational p otential Now, we determine the right-hand sides of equations (7) and (17). We use the known representation of the gravitational p otential for the interior of the ellipsoid in the system of the principal axes 3 U ( ) = - mG 4
0

Q = QRi , = Ri , i = 0, 1, 2, 3, R1 = diag(1, - 1, -1), R2 = diag(-1, 1, -1), R3 = diag(-1, -1, 1).

d ()

1-

i

i2 A2 + i

,

2 () =
i

(A2 + ), i

(19)

where G is the gravitational constant and m = 4 A1 A2 A3 is the mass of the ellipsoid. 3 It is now necessary to represent (19) in terms of the elements of the transformation matrix F and in the Lagrangian coordinates a. We use (13) to find A = QFA0 T and obtain A2 = AAT = QFA2 FT QT , 0 2 () = det(A2 + E) = det(FA2 FT + E), 0
i

i2 2 -1 T 2T -1 2 + = ( , (A + E) ) = (a, F (FA0 F + E) Fa). Ai equations: 3 mG; 4

(20)

Thus, we find the following representation for the matrix V in the Dirichlet d V= FT (FA2 FT + E)-1 F, = 0 2T 0 det(FA0 F + E)

(21)

it can b e shown by direct calculations (see [2]) that V dep ends on the elements of the matrix F only through symmetric combinations of the form ij = k Fik Fj k , which are the dot products of columns of the matrix F. The relationship a = A0 T A-1 can b e used to easily show that, in the Riemann equations, ^ = diag(V1 , V2 , V3 ), where ^^^ V ^ Vi =
0

1 d 1 =- Ai Ai + A2 () i

0

d . ()
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2.4. The Roche Problem By the Roche problem, according to Jeans's terminology [4] (see also [1, 35]) we mean the problem of the interaction of a deformable b ody (satellite) and a spherical rigid b ody which move along circular Keplerian orbits. Actually, in [64] Roche considered the motion of the liquid mass under the action of a gravitating center (the notion of Roche zones traces back to this work). More general problem, where the second b ody does not have a spherical symmetry (i.e. the motion of two arbitrary b odies with mass centers moving along circular orbits), is called the Darwin problem [35]. Let a self-gravitating fluid mass move in the field of a spherically symmetric rigid b ody and b oth of these b odies rotate ab out their common center of mass in circular orbits. We choose a (moving) coordinate system Ox1 x2 x3 with its origin at the center of mass of the ellipsoid and direct the Ox1 axis toward the common center and the Ox3 axis normally to the plane of rotation (see Fig. 1).

R Z C X Y O x1

x

3

x

2

Fig. 1.

The equations of motion of incompressible fluid can b e written in this case in the following Lagrangian form x a
T

? (x + 2 e3 ç x) = -

a

1 ms p + U + Us - 2 (x2 + x2 ) + 2 Rx 1 2 2 me + ms x det = 1, a

1

,

(23) (24)

where, as b efore, a are the Lagrangian coordinates of fluid elements, x(a, t) are their p ositions at the given time, p(a, t) is the pressure, R is the distance b etween the centers of mass of the b odies, me and ms are the masses of the ellipsoid and the sphere, resp ectively, is the angular velocity of rotation of the system ab out their common center of mass, and U is the gravitational p otential (19). The gravitational p otential of a spherical b ody Us has the form Us = - ms G (x1 - R )2 + x2 2 + x2 3 =- ms G R 1+ x1 1 1 + 2x2 - x2 - x2 + . . . , 1 2 3 R 2 R2

where G is the gravitational constant.
x We omit higher-order terms in |R| and use the well-known relationship for a circular Keplerian orbit R3 2 = G(me + ms ) to obtain finally (after collecting like terms) the equation

x a where B = diag
me ms

T

? (x + 2 e3 ç x) = -
3ms +me me me +ms , me +ms

a

1 p + U - 2 (x, Bx) , det 2

x a

= 1,

(25)

,-

ms me +ms

. In the limiting case of a motionless Newtonian center

0 , we have B = diag(3, 0, -1).
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By substituting (4) into (6), we obtain the equations of motion in Roche's problem in the form ? FT (F + 2F) = -2V + 2 A-2 + 2 FT BF, 0 det F = 1, where = - ij
k

(26)

is the matrix of the rotational velocity.

Remark 3. Equations (25) are given in the b ook by Chandrasekhar, who uses them only to find hydrostatically equilibrated configurations of fluid masses and analyze their stability. Chandrasekhar does not present the dynamical equations (26). Obviously, equations (26) can also b e written in the Riemann form as ? ^ ? v - wv - vw + 2v = -2VA + 2 A-1 + 2 ABA, v = A - wA + Aw, A1 A2 A3 = 1,

(27)

? ? where = QQT andB = QBQT are the matrices reduced to the principal axes of the ellipsoid. As b efore, equations (17) should b e added. Note that equations (27) in this case do not form a closed system (in contrast to the Riemann equations), and the system closes only up on adding equations (17) for the evolution of the matrix Q.

3. FIRST INTEGRALS Let us return to the Dirichlet-Riemann problem on dynamics of the self-gravitating ellipsoid. The first integrals of the equations, linear in the velocities, can b e obtained from the conservation laws for vorticity and angular momentum (the law of areas).

3.1. Vorticity We write the law of conservation of vorticity for the hydrodynamic equations in the Lagrangian form (1), thus obtaining xi xi xi xi - ak al al ak = kl = const,
kl

(28) and find (29)

i

with the condition kl = -lk satisfied. We denote this antisymmetric matrix as = for the Dirichlet equations (7) that = FT F - FT F = const. In the Riemannian variables, we obtain = A0 A0 = T (A2 + A2 - 2AwA) = const.

(30)

A straightforward proof of the conservation of vorticity based on the Dirichlet equations (7) is obvious (since the right-hand side is a symmetric matrix). As already mentioned, the conservation of vorticity in this problem was noted by Dirichlet even b efore the app earance of a classical study by Helmholtz in which this law was extended to ideal hydrodynamics on the whole.
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3.2. Momentum The angular momentum relative to the center of the ellipsoid can b e represented as Mij = (xi xj - xj xi )d3 x = m 5 (Fik Fj k - Fj k Fik )(A0 )2 . k (31)

k

In a matrix form, with the unimp ortant multiplier omitted, we have where M =
5 m

M = FA2 FT - FA2 FT = const, 0 0 Mij . Similarly, in the Riemannian variables, we have M = QT (A2 w + wA2 - 2AA)Q = const.

(32)

(33)

The use of the Riemann equations (16) is convenient in proving the invariability of the momentum M (the relevant calculations are also straightforward in this case). 3.3. Energy In addition to the linear integrals, the equations of motion also admit another, quadratic integral, viz., the total energy of the system. The integration of the kinetic and the p otential energy of the fluid particles over the volume of the ellipsoid yields m E = (Te + Ue ), 5 1 1 2T Te = Tr(FA0 F ) = Tr(A2 - w2 A2 - 2 A2 + 2AwA), (34) 2 2 d Ue = -2 . 2 )( + A2 )( + A2 ) ( + A1 0 2 3 4. LAGRANGIAN AND HAMILTONIAN FORMALISM 4.1. Hamiltonian Principle and Lagrangian Formalism It is known (see, e. g., [11]) that the motion of ideal fluid satisfies the Hamilton principle; therefore, Dirichlet's solution also satisfies this principle. This makes it p ossible to represent the equations of motion in a Lagrangian and, next, in a Hamiltonian form. The Hamiltonian principle for the considered problem was used for the first time by Lipschitz [13] and Padova [12]. As the Lagrangian function, it is necessary to choose the difference b etween the kinetic and p otential energies of the fluid in the ellipsoid; within the unimp ortant multiplier, we have where Te and Ue were defined ab ove in (34). The elements of the matrix F app ear as generalized coordinates. We write the Lagrange-Euler equations taking into account the constraint det F = 1 to obtain L F
Ç

L = Te - Ue ,

(35)

-

L = , F F
f F

(36) =
f Fij

where = det F, and use the following matrix notation for any function:

,

f F

=

b eing the undefined Lagrangian multiplier. The differentiation in view of the formula F
-1

f Fij T F

, =

yields ? FA2 0 = 2 F
0

d det(FA2 FT + E) 0
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+ (F

-1 T

) det F.

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We can easily make sure that these equations coincide with the Dirichlet equations (7) if we set = 2 . The matrix of the initial semiaxes of motion of the system as a set of p to the initial conditions; indeed, up on Lagrangian function and the equation A0 app ears in the Lagrangian function and the equations arameters. Obviously, these parameters can b e transferred the substitution G = FA0 (suggested by Dedekind [9]), the of constraint can b e written as
0

1 L = Tr(GGT ) + 2 2

d det(GG + E)
T

,

(38)

= det G = det A0 = const. The initial conditions have obviously the form G|t=0 = A0 , and the equation of motion preserves its form,
L G Ç L G G

-

=H

.

It can also b e shown that the substitution G (det A0 )1/3 G, t (det A0 )1/3 t 2

reduces the system (38) to the case of = 1/2, = 1. Thus, the dynamics of the self-gravitating fluid ellipsoid is describ ed by a natural Lagrangian system without parameters on the S L(3) group. The first integrals -- vorticity (30), momentum (32), and energy (34) -- can b e represented in the form = GT G - GT G, 1 E = Tr(GGT ) - 2 2
0

M = GGT - GGT , d det(GGT + E) . (39)

Riemann used the decomp osition (13) to represent the equations of motion on the configuration space R2 SO(3) SO(3) (the direct product of the Ab el group of translations and two copies of the group of rotations of three-dimensional space), with the elements of the matrices w and corresp onding to the velocity comp onents with resp ect to the basis of left-invariant vector fields. The equations of motion assume the form of the PoincarÄ equations on the Lie group [63]; in e view of the fact that the Lagrangian function (38) is indep endent of the elements of the matrices Q and and with due account for the constraint = A1 A2 A3 = const, we obtain the following representation of the Riemann equations: L Ç L = + ~ , i Ai Ai A L L Ç ij k wk , = wj i

L wi

Ç

=
j,k

ij
j,k

k

L k . j

(40)

where is the Lagrangian undetermined multiplier (which coincides with within a multiplier) ~ and ij k is the Levi-Civita antisymmetric tensor. From here on, the comp onents wi and i are related to the elements of the antisymmetric matrices (15) according to the regular rule wij = ij k wk , ij = ij k k .
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4.2. Symmetry Group and the Dedekind Reciprocity Law The Lagrangian representation of the Dirichlet equations (36) offers a very simple way to finding the symmetry group of the system. Indeed, it can b e shown that the Lagrangian with the constraint [see (38)] and, therefore, the equations of motion are invariant with resp ect to transformations of the form G = S1 GS2 , Thus, the system is invariant with resp ect to the group = S O(3) S O(3). Clearly, the Noether integrals corresp onding to the transformations (42) are the integrals of vorticity and total momentum (39). Accordingly, as will b e shown b elow, the Riemann equations describ e a system reduced based on the given symmetry group. Furthermore, it can easily b e shown using (38) that the equations of motion are invariant with resp ect to the discrete transformation of transp osition of matrices: G = GT . Therefore, we have Theorem 1 (The Dedekind reciprocity law). Any solution, G(t), of the Dirichlet equations can be placed in correspondence with the solution G (t) = GT (t) for which the rotation of the el lipsoid and the rotation of the fluid inside the el lipsoid (i. e., and Q; see (13)) are interchanged. The most widely known example is the Dedekind ellipsoid reciprocal to the Jacobi ellipsoid. In this case, the axes of the three-axial ellipsoid are spatially invariable and the fluid inside it moves around the minor axis in closed ellipses [3, 9]. 4.3. Hamiltonian Formalism and Symmetry-based Reduction We represent the Riemann equations in a Hamiltonian form. To this end, we first use the constraint equation = const to find a representation of one semiaxis, v0 A3 = , (43) A1 A2 where v0 is the volume of the ellipsoid (within a multiplier). We carry out the Legendre transformation L L L pi = , mk = , Åk = , i = 1, 2, k = 1, 2, 3, i wk k A (44) H= pi Ai + (mk wk + Åk k ) - L | A,,wp,m,Å .
i k

S1 , S2 S O(3).

(42)

It can b e shown using the expressions for the integrals, (30) and (33), that the vectors m = (m1 , m2 , m3 ) and Å = (Å1 , Å2 , Å3 ) are related to the momentum and vorticity of the ellipsoid via the formulas m = QT M , Å = T , (45) where the vectors M and are constituted by the comp onents of the antisymmetric matrices M and according to the normal rule (41). In the new variables, the equations of motion assume the form H H Ai = , pi = , i = 1, 2, pi Ai (46) H H m=mç , Å =Åç . m Å Here, the Hamiltonian is H = HA + H
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HA = H
mÅ

1 A-2 (p2 + p2 ) + (p1 A-1 - p2 A-1 )2 1 2 3 2 1 , 2 A-2 i mi + Åi Aj - Ak
2

(47)

1 = 4

+

cycle

mi - Åi Aj + Ak

2

,

where Ue is sp ecified by formula (34) and it is assumed that A3 is defined according to (43). In addition, equations (46) must necessarily b e supplemented with equations describing the evolution of the matrices Q and ; they have the form Qij =
k ,l

ikl Q

kj

H , ml

ij =
k ,l

ikl

kj

H . Ål

(48)

Equations (46) and (48) form a Hamiltonian system with eight degrees of freedom and uncanonical Poisson brackets, {Ai , pj } = ij , {mi , mj } = ij k mk , {Åi , Åj } = ij k Åk , (49) (50)

{mr , Qj k } = ikl Qj l , where zero brackets are omitted.

{Åi , j k } = ikl j l ,

Remark 4. The elimination of one semiaxis (43) results in the loss of symmetry of the Hamiltonian (47); therefore, the equations for the semiaxes Ai are normally left in the Lagrangian form with an undetermined multiplier [1, 3]. It can b e seen from the ab ove relationships that the system of equations (46), which describ es the evolution of the variables Ai , pi , m, and Å, separates; in addition, the Poisson bracket of these variables, (49), also proves to b e closed. It is not difficult to show that that equations (46) describ e a system reduced over the symmetry group (42). Limitation: the brackets (49) obviously have two Casimir functions,
m

= (m, m),

Å = (Å, Å),

(51)

and have a rank of eight (provided that m = 0, Å = 0). Therefore, the reduced system has general ly four degrees of freedom. In particular cases where one of the integrals (51) is zero, the reduced system has three degrees of freedom. These are so-called irrotational (Å = 0) and momentum-free (m = 0) ellipsoids. If both of the integrals (51) vanish, the reduced system has two degrees of freedom and describ es oscillations of the ellipsoid without changes in the directions of the axes and without inner flows (this case will b e considered b elow in detail). Remark 5. The canonical variables in the Riemann equations were introduced for the first time by Betti [14], who used the commutation representations of the so(4) algebra long b efore the advent of the modern theory of Hamiltonian systems on the Lie algebras. With the use of commutation, he introduced, in a quite modern way, canonical variables to reduce the integration of the Riemann equations to the integration of the Hamilton-Jacobi equations. The Hamiltonian nature of the Riemann equations is also considered in modern studies [53-55], which are related to the representation of the equations of motion on an extended Lie algebra for which the actual motions are in sp ecial orbits; the value of such a calculation for dynamics is not yet clear to us. A more formal procedure of reduction and Hamiltonization of the Riemann equations nearly relevant to our study is describ ed in [60]. An akin analysis is done in [61] in the context of the Dirichlet motions in ideal magnetohydrodynamics. An alternative approach to the Hamiltonian nature, which also should b e discussed, is presented in [62].
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Remark 6. The linear transformation L = m + Å, reduces the angular part H
mÅ

of the Hamiltonian (47) to a diagonal form H
mÅ

=m-Å

=

1 (A2 - 1 = diag (A2 + = diag {Li , Lj } = ij k Lk ,

1 1 (L, L) + ( , 4 4 1 , A3 )2 (A3 - A1 )2 1 , 2 (A + A )2 A3 ) 3 1

), , 1 (A1 - A2 )2 1 , (A1 + A2 )2 , ; (52)

in this case, the Poisson brackets reduce to the form and corresp ond, as is known, to an so(4) algebra. The corresp onding equations of motion can b e represented in the matrix form X = [X, ], where 0 X= -L L
2 3 H L3 H L2 H L1 H 1 H 2 H 3

{Li , j } = ij k k ,

{i , j } = ij k L

k

(53)

L

3

- L2 L
1

1 2 3

0 , = -
H L3 H L2 H - 1

-

0 -L
1



0 - -
H L1 H 2

,

(54)

0
3

0 -
H 3

- 1 - 2 -

0

0

and the equations for Ai and pi preserve their previous form (46). Equations (53) coincide in their form with the equations of motion of a free four-dimensional rigid b ody. In the dynamics of the rigid b ody, the momentum and angular velocity are in this case linked by linear relationships of the form 1 X = (J + J), (55) 2 where the constant symmetrical matrix J is the moment of inertia of the b ody with resp ect to axes fixed to the b ody. It can easily b e shown that the matrices (54) for the Dirichlet-Riemann problem do not satisfy the relationship (55) at any matrix J, i. e., the analogy with the dynamics of the rigid b ody is purely formal in this case. Recall that the matrices and for a four-dimensional rigid b ody have the form [63] = diag 1 1 1 , , 2 + 3 1 + 3 1 + 2 , = diag 1 1 1 , , . 0 + 1 0 + 2 0 + 3

This form of equations (53) was noted by Dyson [77] for the case of the dynamics of a compressible ellipsoid (see b elow). 5. PARTICULAR CASES OF MOTION 5.1. Shap e-preserving Motions of the Ellipsoin The simplest motions of the fluid ellipsoids are represented by a family of solutions for which all the three axes of the ellipsoid are time-indep endent, Ai = const, i = 1, 2, 3. (56) Clearly, the Maclaurin and Jacobi ellipsoids are examples of such motions. In these cases, the ellipsoid rotates as a rigid b ody ab out the principal axis (the symmetry axis for the Maclaurin ellipsoid and the shortest axis for the Jacobi ellipsoid).
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The Dedekind ellipsoid offers another example of such motions, the axes b eing invariable in b oth their lengths and directions. As noted ab ove, the Dedekind ellipsoid is reciprocal to the Jacobi ellipsoid in terms of Theorem 1 (while the Maclaurin ellipsoid is self-reciprocal). For all the ab ove-mentioned solutions (the Maclaurin, Jacobi, and Dedekind ellipsoids), two pairs of comp onents of the vectors m and Å vanish, the remaining comp onents b eing constant (for example, it can b e assumed without loss of generality that m1 = Å1 = m2 = Å2 = 0, m3 = const, Å3 = const). Riemann [3] has proved the following, more general result: Theorem 2. Let (56) be satisfied and let al l the Ai be different. Then m and Å are timeindependent and at least one pair of components of these vectors vanishes (i. e. mi = Åi = 0 for some i). As a consequence, we find that any motion of a shap e-preserving fluid ellipsoid whose axes do not coincide, is a fixed p oint of the reduced system (46) or, which is the same, of the Riemann equations (27). Another proof of this statement is given in [12]. Riemann also noted new solutions -- the Riemann ellipsoids -- for the case where only one pair of comp onents of m and Å vanishes (i. e. m1 = Å1 = 0, Å2 , m2 , Å3 , m3 = 0). V. A. Stekloff [65, 66] analyzed in detail the case of equality of a pair of axes (Ai = Aj = Ak ) and showed that no shap e-preserving motions other than Maclaurin ellipsoids (spheroids) exist in this case. In this sense, he generalized the Riemann result to the axisymmetric case (Riemann himself gave no detailed proof for this case). An attempt of revising Riemann's results was made in [67]. 5.2. Axisymmetric Case (Dirichlet [2]) It can easily b e shown that the equations of motion determined by the Lagrangian function (38) admit a (two-dimensional) invariant manifold that consists of matrices of the form u v0 G = -v u 0 , 0 0w where det G = (u2 + v 2 )w = v0 = const is the volume of the ellipsoid. This manifold corresp onds to an axisymmetric motion of the fluid ellipsoid (see [2]). In this case, the matrix of the principal semiaxes is A = (GGT )1/2 = diag( u2 + v 2 , u2 + v 2 , w).

In view of the condition det G = v0 , we make the substitution of variables u=
1/3 v0

r cos ,

v=

1/3 v0

r sin ,

v w= 0 r

1/3 2

and find that the Lagrangian function (38) is L = v0 where 2 Us = - v0
0 2/3

1+

2 r6

r 2 + r 2 2 + Us , 2arctg r 6 - 1 , r6 - 1
6 6

r > 1,

2 = - r2 ç 2 ) + 1/r 4 ln 1+1-r v0 ( + r 1 - 1-r 1 - r6

d

,

r < 1.
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The variable is cyclic; therefore, we have a first integral of the form p =
2/3 v0

1 L = 2r 2 ,

which coincides within a multiplier with the single nonzero comp onent of the momentum M12 (32). With the use of the energy integral (34), we obtain a quadrature that sp ecifies the evolution of r :

1+ where h =
E 2/ mv0
3

2 r6

r 2 = h - U ,

U = Us +

c , r2

and c =

p 4

are fixed values of the energy and momentum integrals. The minimum

of the reduced p otential U corresp onds to the Maclaurin spheroid. 5.3. Riemannian Case [3] There is an invariant manifold more general than the ab ove-describ ed one. It is sp ecified by the block-diagonal matrix of the general form u1 v1 0 G = u2 v2 0 0 0 w3 We compute the integrals (30) and (32) obtaining
M12 = u1 u2 - u2 u1 + v1 v2 - v2 v1 , M23 = M13 = 0,

.

(57)



12

= u1 v1 - v1 u1 + u2 v2 - v2 u2 ,



23

=

13

= 0,

It is also obvious that iQ and have in this case a block-diagonal form similar to (57); therefore, this case corresp onds to that noted by Riemann, for which, in equations (46), we should set m1 = m2 = 0, Å1 = Å2 = 0, m3 = const, Å3 = const.

Thus, we obtain a Hamiltonian system with two degrees of freedom, which describ es the evolution of the principal semiaxes A1 and A2 ; its Hamiltonian is H= 1 A-2 (p2 + p2 ) + (p1 A-1 - p2 A-1 )2 1 2 3 2 1 + U (A1 , A2 ), 2 A-2 i c2 c2 1 2 + , (A1 - A2 )2 (A1 + A2 )2 (58)

where the reduced p otential is U = Ue +

and c2 = 1 (m3 + Å3 )2 , c2 = 1 (m3 - Å3 )2 are fixed constants of the integrals. 1 2 4 4 The particular version of the system (58) for c1 = c2 = 0 (i. e., for invariable directions of the principal axes of the ellipsoid) was also noted by Kirchhoff [11], who suggested that the problem does not reduce to quadratures. At U = 0, the Hamiltonian (58) describ es a geodesic flow on the cubic A1 A2 A3 = const. This remarkable analogy b etween two different dynamical systems was also noted by Riemann.
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5.4. Elliptic Cylinder (Lipschitz [13]) This case can b e obtained through a limiting process in the Riemannian case, with one axis of the ellipsoid going to infinity (A3 ). It is, however, more convenient to start with considering the case of a two-dimensional motion of fluid assuming that the matrix F has the form F= ? F0 01 ? where F is a 2 ç 2 matrix with unit determinant. , ? det F = 1, (59)

Obviously, the considerations on which the derivation of the Dirichlet equations [31] was based can b e applied to this case without modifications; only the right-hand side of the equations should b e prop erly changed. To this end, it is necessary to use the well-known representation of the p otential of the interior p oints of the elliptic cylinder with a large length l in the system of principal axes U ( ) = U0 (l) - ?
2 2 1 2 - A1 (A1 + A2 ) A2 (A1 + A2 )

+ O(1/l),

omitted.

where = Gm, G is the gravitational constant and m = A1 A2 is the mass p er unit length of ? ? ? the cylinder. The constant U0 (l) - does not app ear in the equations of motion and can b e
l

By analogy with the ab ove considerations, we pass to the Lagrangian representation and make ? ? ?? the substitution G = FA0 , where A0 = diag(A0 , A0 ), to obtain the Lagrangian of the system in 1 2 the form 1 ?? ? L = Tr GGT - Ue , 2 ?? ? ? Ue = -2 ln(A1 + A2 )2 = -2 ln(Tr(GGT ) + 2 det G). ? ? ? ? ?? Based on the singular decomp osition of the matrix G = QT A with cos - sin ? Q= , sin cos cos - sin ? = , sin cos A= A1 0 0 A2 ,

explicitly substituted, we obtain a Lagrangian function in the form L= 1 2 2 ? A1 + A2 + (A1 - A2 )2 + (A2 - A1 )2 - Ue (A1 , A2 ). 2 L = p ,
1/2

We can see that the variables and are cyclic, and there are two linear integrals L = p .
1/2

(60)

We parametrize the relationship A1 A2 = v0 using hyp erb olic functions, ? A1 = v0 (ch u + sh u), ? A2 = v0 (ch u - sh u). ?

We use the energy integral and the integrals (60) to obtain a quadrature for the variable u: c2 ?1 c2 ?2 ? U = 2 ln(ch u) + 2 + 2 , ? ch u sh u where c2 = ?1
1 16

? v0 (ch 2u)u2 = h - U , ?

(p - p )2 , c2 = ?2

1 16

(p + p )2 , and h are fixed constants of the first integrals.
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6. CHAOTIC OSCILLATIONS OF A THREE-AXIAL ELLIPSOID Let us consider in more detail the oscillations (pulsations) of a fluid ellipsoid in the Riemannian case (57). We now represent the equations of motion of the system (58) in a Hamiltonian form most convenient for a numerical investigation of the system. We parametrize the surface A1 A2 A3 = v0 using cylindrical coordinates 2v0 , r sin2 2 . p p sin , p2 = pr sin - cos p1 = pr cos - r r The Hamiltonian (58) can b e represented in the form A1 = r cos , A2 = r sin , A3 =
2

(61)

1 H= 2

c2 1 + 6 04 r sin 2

-1

p+

2 r

p r

2 2

+

p c2 0 sin 2 pr cos 2 - 6 sin4 2 r r

2

+ U (r, ),

(62)

where c0 = 4v0 . Since the original system is defined in the quadrant A1 > 0, A2 > 0, A3 > 0, for this case we have 0 < < /2. In this system, the transformation of variables = r2, enables obtaining the Hamiltonian in the form H= 2(2 (c2 cos2 + 3 sin4 )p2 + sin2 (c2 + 3 sin2 )p2 - 2c2 cos sin p p ) 0 0 0 (c2 + 3 sin4 ) 0 + U (, ). (64) (65) = 2, (63)

Up on passing to new Cartesian coordinates according to the formulas x = cos , we obtain H = 2 p2 + x y 4 p2 y y 4 + c2 ?0 + U (x, y ), (66) y = sin ,

where = x2 + y 2 ; obviously, the system (66) is defined in the upp er semiplane (y > 0). In this case, as we can see, the kinetic energy of the system has the simplest form. Remark 7. The transformation (63) is the Levi-Civita transformation (known also as the Bolin transformation) known in celestial mechanics, which is usually written in the complex form x + iy = ei = (A1 + iA2 )2 . As already noted ab ove, at U = 0, the Hamiltonian (66) describ es a geodesic flow on the cubical surface A1 A2 A3 = const, emb edded in Euclidian space R3 . Almost all tra jectories (geodesics) of this system are not compact; therefore, computer simulations for a numerical proof on nonintegrability at U = 0 cannot b e done. As shown recently by S. L. Ziglin [68], this system (at U = 0, i. e., a geodesic flow) does not admit a meromorphic additional integral. Remark 8. Various algebraic surfaces in three-dimensional space and singular lines on them (asymptotic lines, lines of curvature, and geodesics) were actively investigated by mathematicians in the 19th century. They were highly enthused by the integrability of the problem of geodesics on the ellipsoid and, generally, on quadrics, discovered by Jacobi. This integrability also refers to multidimensional cases. The problem at hand is a classical example of the separation of variables. Extensive literature has b een dedicated to studying this problem from b oth analytical (integration using theta functions) and qualitative standp oints. However, the mathematicians of the 19th century succeeded little, and nearly nothing was added in the 20th century to finding geodesic flows in higher-order surfaces. Likely, in the context of this problem, Darbu develop ed a theory of orthogonal families of surfaces investigating, in this avenue, interesting surfaces of
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the third (Darbu cyclides) and fourth degree (which were also discovered by Rob erts [69] and Wangerin [70]). A distinctive prop erty of these families of surfaces is in the fact that they are LamÄ families, form a triorthogonal network in three-dimensional space, and a solution for e asymptotic and curvature lines can b e written for them in the form of elliptic quadratures. However, geodesics for these surfaces have not b een found. As our numerical simulations show, the reason for this fact is the nonintegrability of the geodesic flow. In the context of this problem, we also note studies by Schl?fli [71] and Cayley [72] dedicated to the classification of various cubic a surfaces in space, which, following the fundamental study by Jacobi on geodesics in quadrics, laid the foundation of modern algebraic geometry. Recently, V. V. Kozlov [73] found top ological obstacles for the integrability of geodesic flows on noncompact algebraic surfaces (in particular, of the third and fourth degrees). Unfortunately, his results do not apply to Riemannian surfaces A1 A2 A3 = const. Shown in Fig. 2 are phase p ortraits of the system (66). As the plane of the PoincarÄ map, the e plane x = 1 is chosen. It can b e seen from the diagrams that the phase p ortrait is virtually regular at energies close to the minimum energy (see Figs 1a and 1c). As the energy is increased, the phase p ortrait b ecomes chaotized, which can b e clearly seen from Figs 1b and 1d. In Figs 1e and 1f, the intersection of unstable invariant manifolds (separatrices) is also depicted, which can serve as a numerical proof of the nonintegrability of this system. Remark 9. In the numerical integration of the equations of motion, it is convenient to use the representation of the p otential Ue (34) and its derivatives in terms of elliptic functions,


I (A1 , A2 , A3 ) =
0

d 2 = F (, k), () ö = arcsin A2 - A2 1 3 A2 1 ,

ö=


A2 - A2 , 1 3 d

k=

A2 - A2 1 2 , A2 - A2 1 3

is the first-kind elliptic integral. 1 - k2 sin2 Due to the invariance with resp ect to p ermutations of the quantities A1 , A2 , and A3 , they can b e ordered so as to make all values of ö, , and k real. To find the derivatives (which are obviously not invariant with resp ect to p ermutations of Ai ), a necessary ordering of the quantities A1 , A2 , and A3 should b e made first, after which the derivative of the integral I with resp ect to the corresp onding argument should b e calculated. For the derivatives F (, k), the following relationships are valid: F -1/2 , = 1 - k2 sin2
0

where F (, k) =

F 1 = k 1-k


2

E (, k) - (1 - k2 )F (, k) k sin cos - k 1 - k2 sin2

,

where E (, k) =
0

1 - k2 sin2 d is the second-kind elliptic integral.

II. DYNAMICS OF A GAS CLOUD WITH ELLIPSOIDAL STRATIFICATION
1. INTRODUCTION The investigation of the dynamics of gas ellipsoids traces back to a study by L. V. Ovsyannikov [74] (1956), who analyzed the most general equations describing the motion of an ideal p olytropic gas, without taking into account gravitation, with a velocity field linear in the coordinates of the gas particles (from here on, by the gas ellipsoid, we mean the Dirichlet solution generalized to various models of compressible fluid). Note that the pap er [74] is very brief and purely mathematical:
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p
y

19 9

p

y

y

y

a) h = -1.15, c1 = c2 = 0

b) h = -1.1, c1 = c2 = 0

p

y

py

y

y

c) h = -1.09, c1 = 0.1, c2 = 0

d) h = -1.07, c1 = 0.1, c2 = 0

p

y

py

y

y

e) h = -0.9, c1 = c2 = 0

f ) h = -0.9, c1 = c2 = 0

Fig. 2. The PoincarÄ map of system (66). For all panels, c0 = 1, = 0, 6; for the map, the planes x = 1 (a-d) e and x = 0.1 (e ,f ) are chosen.

in fact, the equations of motion are obtained there, several p ossible cases of the existence of the considered solution are noted, and an incomplete set of first integrals is given for two cases. It is interesting that Ovsyannikov's pap er contains no references, so that the relationship b etween the obtained solution and the Dirichlet solution is not revealed. Later, D. Lynden-Bell [75] (1962) demonstrated, also without any references, the existence of the solution in the form of a spheroid for a self-gravitating dust cloud (i. e., for a medium not resisting to deformations, p 0). Ya. B. Zel'dovich [76] (1965) obtained the equations of motion of a self-gravitating dust ellipsoid in the general case and studied (on a physical level of rigor) the p ossibility of collapse and expansion in this problem. Likely, Ya. B. Zel'dovich also overlooked the relationship of this problem to the Dirichlet-Riemann problem, since the model of a dust cloud can b e obtained simply by setting p = 0 in the Dirichlet equations. Indep endently of Ovsyannikov (at least without a reference), F. Dyson [77] (1968) obtained the equations of motion of an ideal-gas cloud in the case of an isothermal flow (although without the assumption of a p olytropic b ehavior of the gas); a Gaussian density distribution with an ellipsoidal stratification was found. Dyson noted a relationship b etween the obtained solution and the Dirichlet problem and wrote the equations of motion of the gas ellipsoid in a Riemannian form.
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Also indep endently of Ovsyannikov, M. Fujimoto [78] (1968) describ es a model of a cooling ellipsoidal gas cloud; in essence, he obtains a generalization of a case considered by Ovsyannikov (if we assume the cooling parameter to b e ö = 0, we will obtain Ovsyannikov's equations). In addition, in Fujimoto's model, the density is constant, which enabled taking into account the gravitational interaction b etween the particles of the cloud. Fujimoto also noted a relationship of this problem to the Dirichlet problem and, in studying it, used the techniques develop ed by Chandrasekhar [1] and Rossner [79]. Let us also mention a study by Anisimov [80] (1970), who follows [74] and [77] considering two cases of the integrable dynamics of a gas ellipsoid without allowances for gravitation but with the additional condition of the monoatomic structure of the gas (a p olytropic index of = 5 ). The first 3 case is the motion of an axisymmetric ellipsoid; the second, of an elliptic cylinder. A nonautonomic Jacobi integral was found (which is due to the uniformity of the p otential with a uniformity degree of -2). This integral is essentially necessary for integration in the cases under study; as we will show b elow, these systems are not integrable in the general case. Bogoyavlenski [81] (1976) analyzes the dynamics of a gas ellipsoid on a physical level of rigor i taking into account gravitation (i. e., he considers the Fujimoto model without cooling). Explicit Lagrangian and Hamiltonian representations of the system are used. Gaffet [83-85] shows that the system that describ es irrotational gas ellipsoids without considering gravitation, for a monoatomic gas = 5 , satisfies the PainlevÄ prop erty; in these studies, first e 3 integrals are found and integration in quadratures is carried out for certain particular cases. There are also studies analyzing a spherically symmetric motion of a gas cloud; one of the most general solutions is describ ed by Lidov [86], who considers time-dep endent, one-dimensional, spherically symmetric, adiabatic motions of a self-gravitating mass of a p erfect gas. Nemchinov [87] uses a solution that describ es the ellipsoidal expansion of a gas cloud to find characteristic features of nonspherical explosions (in particular, he notes an increase in the impact of the stream in the direction of one of the principal axes compared to a similar spherical explosion); the effect of the heating of the cloud on the expansion sp eed is also investigated. Finally, let us mention a series of studies (see [88] and references therein) generalizing the problem of the expansion of an ellipsoidal cloud to vacuum (or the collapse of an ellipsoidal cavity) with the presence of a rarefaction (compression) wave. 2. EQUATIONS OF MOTION OF A GAS CLOUD WITH A LINEAR VELOCITY FIELD Now consider in a similar way the case where the motion of a compressible fluid (gas ) is also defined by a linear transformation of the Lagrangian coordinates x(a, t) = F(t)a; (67) for the compressible medium, the condition det F = 1 is obviously not valid. Clearly, the velocity field is linear in the coordinates of the fluid particles: v (x, t) = x = F(t)F
-1

(t)x.

In this case, the equations that describ e the flow in the given case (for p otential forces) have the following Lagrangian representation: x a
T

? x=-

1 p U - , a a
-1

(68)

and the continuity equation in the Lagrangian form is + Tr here, Tr
x -1 x a a

x a

x a

= 0;

(69)

= div v (x, t).
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T

20 1

For the flow of the structure under study (67), the continuity equations can easily b e integrated. Indeed, if we introduce the notation (t) = det F(t) using the relationship that Tr
x -1 x a a F

= F

-1

, we find

=



; therefore, (a, t) = f (a) , (t) (70)

where the function f (a) is time-indep endent. Except the four functions x(a, t), p(a, t), the medium at hand is describ ed by three additional scalar quantities -- the density (a, t), the sp ecific internal energy Uin (a, t), and the temp erature T (a, t). Therefore, it is necessary to complement the system (68), (69) with three other equations. As is known [89], these additional equations are of a thermodynamic rather than mechanical nature and dep end essentially on our assumptions concerning the prop erties of the medium and on the character of the flow. Dep ending on these assumptions, various gasdynamic models can b e obtained. Consider three of them that are most widely known, emphasizing the explicit assumptions. Unless the opp osite is stipulated, we assume in what follows that the p otential of the external forces applied to the system, U , is zero. 2.1. The Ovsyannikov Model [74] . The gas is ideal and can be described by the equation of state p = RT , where R is the universal gas constant. 2 . The gas is polytropic, and its internal energy depends linearly on the temperature, Uin = cV T , where cV = const is the specific heat at constant volume. 3 . The gas flow is adiabatic (i. e., there is no heat exchange between different parts of the gas volume); therefore, the energy variations are described by the equation Uin = -p 1
Ç

1

(71)

(72)

.

(73)

Remark 10. Equation (73) is a consequence of the first principle of thermodynamics, Q = dUin + p dV , where, in view of assumption 3 , it is necessary to set Q = 0, (V = 1 ). Remark 11. Recall that, due to the well-known thermodynamic identity dUi dV
n T

=

T

p T

V

-p

the internal energy of the ideal gas (71) dep ends only on the temp erature, Uin = Uin (T ). We find using equations (71)-(73) and taking into account the relationship (70) that p + = 0, p where the dimensionless constant = 1 + quantities, we have p(a, t) = g(a) , (t)
R cV

is the adiabatic index. Thus, for the thermodynamic 1 1 g(a) RT (a, t) = 1- (t) , -1 -1 f (a)
No. 2 2009

Uin (a, t) =

where g(a) is an arbitrary, time-indep endent quantity.
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Thus, for the existence of a solution of the form (67) in this case, it should necessarily b e required that 1 a g(a) = Va, (74) f (a) where V is a certain constant matrix, a = are ? FT F + (det F)1- V = 0
a1 , a2 , a3

. Then the equations of motion for F(t) (75)

(the Ovsyannikov equations).

As Ovsyannikov has shown [74], it is sufficient to consider the following solutions of equation (74). Theorem 3. Any solution of equation (74) reduces via a linear transformation of the Lagrangian coordinates to one of the fol lowing four types (depending on the rank of the matrix V): (I) V = diag(1 , 2 , 3 ), i = Á1 (i = 1, 2, 3), g(a) = g(s), f (a) = 2g (s), s = (a, Va); 1 0 (I I) V = 0 2 0 , 0 00 if = 0 then g(a) = g(s), if = 0 then g(a) = g(s), (I I I) V = diag(, 0, 0), = Á1, (IV) V = 0, g(a) = const, f (a) = f (a) =
a1 g (s) (a,Va) , 2g (s),

i = Á1

(i = 1, 2),

s = a1 s ?
a1 g

a2 a1

,

ln s() = ?

2 d 1 ++2

;

s = (a, Va); (a1 );

g(a) = g(a1 ),

f (a) =

f (a) is an arbitrary function.

Proof. We briefly note the principal steps in the proof. Rewriting the solvability condition (74) in the form a ç a g(a) = a ç f (a)Va = 0 yields a f (a) ç (Va) = f (a)V , (76) where V = (V23 - V32 , V31 - V13 , V12 - V21 ). We take the dot product by Va and represent the solvability condition as the vector equation VT V = 0. (77)

In addition, it is clear that equations (75) are invariant with resp ect to nonsingular transformations of the Lagrangian variables, for which a = Sa , V = ST VS, F = (det S)
1- 1+

FS,

det S = 0.

(78)

If rank V < 3, the most general matrix of the corresp onding rank reduces via the transformations (78) to the form given in cases I I-IV. If rank V = 3, then V = 0; therefore, V is symmetric and reduces to the form of case I. The relationships (74) and (76) make it p ossible to easily find the corresp onding functions f (a) and g(a) from the known matrix V, From the physical standp oint, case I with a sign-definite matrix V is most interesting. In particular, if we set V = diag(-1, -1, -1) (i. e., s = -(a, a)) and choose a linear function g(s), we will obtain 1 g(a) = 0 (d2 - (a, a)), f (a) = 0 , (79) 0 2
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20 3

where we must assume that 0 > 0 (since the density of the gas is p ositive). In this case, the gas is distributed with a constant density = 0 / det(F) inside the finite ellipsoidal volume (a, a) = (x, (FFT )-1 x) d2 . 0 (80) gravitational forces (75)). The problem [81]. However, the nowledge of regular Therefore, the solution of the form (67) in this case remains valid up on adding (the matrix (21) with A0 = E b eing added to the right-hand side of equation of the motion of a compressible self-gravitating gas cloud was formulated in analysis presented there app ears to b e somewhat naive in view of the modern k and chaotic motions in dynamical systems. 2.2. The Dyson Model [77] Assumptions and coincide with those polytropic behavior, we assume that 2 . The gas is isothermal at the initial time, i. We substitute the pressure from the equation obtain Uin = RT 1 3 in the preceding case, whereas, instead of the e., T (a, t = 0) does not depend on a. of state (71) into (73) and make use of (70) to -. (81)

At the same time, as mentioned ab ove (see Note 9), the internal energy dep ends only on the temp erature, and the right-hand side of (81) does not dep end on a; therefore, the gas remains isothermal at all later times and (81) can b e represented as dUin + RT = 0. d dUin dT (82)

By integrating this equation in view of (72), we obtain a relationship b etween T and : = 0 exp - (RT )-
1

dT

.

(83)

Thus, according to (70), (71), and (82), the pressure can ultimately b e written as p(a, t) = We substitute (84) into (i. e., U = 0). Thus, we b e a uniform quadratic are defined to within a RT ((t)) f (a). (t) (84)

(68) and restrict ourselves to the case where no external forces are present find that the existence of a solution of the form (67) requires that ln f (a) function of the Lagrangian coordinates. Since the Lagrangian coordinates nonsingular linear substitution (78), we can represent f (a) in the form f (a) = m 1 exp(- (a, a)), 3/2 2 (2 ) (85)

where m = (x)d3 x = f (a)d3 a is the mass of the gas. Finally, for the elements of the matrix F, we obtain the equation of motion ? FT F = RT ()E (the Dyson equations). (86) Remark 12. According to (85), we find that the gas, during its motion, has an ellipsoidal density stratification of the form 1 (87) (x, t) = f (n2 ), n2 = (x, (FFT )-1 x). (t) Thus, everywhere on the ellipsoid n2 = const, the density has the same value. The gravitational p otential of such b odies, as is known, is not a uniform quadratic function of the coordinates; therefore, no solution of the form (67) exists with the presence of gravitation.
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2.3. Model of a Cooling Gas Cloud (Fujimoto [78]) In this model, the assumptions 1 and 2 coincide with those in Ovsyannikov's case, i. e., the gas is assumed to be ideal and polytropic, while the third assumption in this case has the form 3 . The motion of the gas is not adiabatic, the variations in the internal energy satisfying the equation Uin + p Tr x a
-1

x a

= -ön T m .

(88)

Remark 13. Equations (88) differ from the equations of an adiabatic process (73) by the terms -ön T m . We use equations (70)-(72) to eliminate Uin from equation (88) and find p ö( - 1) n + =- p R
-1

T

m-1

.

(89)

To obtain a solution in the form (67), we additionally require that m = 1, (a, t) = 0 , (t)

where 0 = const is indep endent of a, i. e., the density is constant inside the cloud. The solution of equation (89) in this case has the form p(a, t) = (t)g(a), where (t) satisfies the equation + = -ö1-n , ? ö= ? ö( - 1) n 0 R
-1

.

(90)

The function g(a) should obviously satisfy equation (74), and it can easily b e shown that, according to Theorem 3, we may choose g(a) = 1 - (a, a), f (a) = const.

The condition of the finiteness of the total gas mass implies that V = diag(-1, -1, -1), the gas occupying initially the region (a, a) 1 (in the original physical variables, this inequality sp ecifies an ellipsoid of the form (x, (F, F)-1 |t=0 x) 1). Finally, we obtain the system of equations describing the dynamics of the cooling cloud in the form 2 d ? , FT F = E +2 FT (FFT + E)-1 F 0 T det(FF + E) 0 (ln( ))Ç = -ö1-n . ? The parenthesized term describ es the gravitational interaction b etween the particles of the cloud. Allowances for the gravitational interaction in the solution of the form (68) are p ossible in this case due to the uniformity of gas in the cloud (0 = const). The numerical results of [78] demonstrate a p ossibility of gravitational collapse in this system (at ö > 0).
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20 5

2.4. Model of a Dust Cloud (Gravitational Collapse) 1 . The medium (dust) does not counteract deformations, p 0. 2 . At the initial time, the particles are distributed uniformly (inside the el lipsoid), (t, a)
t=0

= 0 = const.

For a solution of the form (68), the density obviously does not dep end on the coordinates at all subsequent times, b eing determined by the relationship 0 . (t) = det F(t) Therefore, allowances for the gravitational attraction of particles in the clouds are p ossible in this model in the framework of the linear solution (68), and the equations of motion can b e written as ? F F = 2
T 0

FT (FFT + E)-1 F

d det(FFT + E)

.

(91)

This model is used in astrophysics to describ e the gravitational collapse [76]. In particular, it is applied in [75] to the description of the collapse of an elliptic has cloud at zero temp erature. 3. LAGRANGIAN FORMALISM, SYMMETRIES, AND FIRST INTEGRALS We will now show that the Dyson equations (86), the Ovsyannikov equations (75) under the condition (79), and the equations of a dust cloud (91) admit a natural Lagrangian description. It can b e shown by means of direct calculations that the equations of motion can b e written in the form L F
Ç

-

L = 0, F

1 L = Tr(FFT ) - Ug (F), 2 where

(92)

Uin () for the Dyson model, 1 d 1- - 2 for the Ovsyannikov model with gravitation, - 1 det(FFT + E) Ug (F) = 0 d - 2 for the dust-cloud model, T det(FF + E) 0 (93) where, as ab ove, = det F, F GL(3).

Remark 14. In the Dyson model, the Lagrangian representation (92) can b e directly obtained from the Hamiltonian principle for barotropic flows (see [11])
t2 t2


t1

(T - U ) dt =
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W dt,
t1

(94)

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BORISOV et al.

where T and U are the kinetic and the p otential energy of the fluid and W is the barotropic p otential that satisfies the equation W = p 3 d x. (95)

Based on the ab ove assumption, we obtain for our case within a constant: W= RT ln d3 x = Uin . (96)

These considerations can also b e generalized to the Ovsyannikov model. By analogy with the fluid ellipsoid (see Section 2, ? 4), we conclude that the system (92) is invariant with resp ect to linear transformations of the form F = S1 FS2 , which form a symmetry group = S O(3) S O(3). The Dedekind reciprocity law (Teorem 1 in Part 1), which corresponds to a discrete transformation F = FT , is also valid in the dynamics of gas clouds. According to the Noether second theorem, integrals of motion linear in velocity -- the vorticity and total angular momentum of the system - corresp ond to the transformations (97) and can b e represented in the matrix form = FT F - FT F, E= M = FFT - F FT . (98) S1 , S2 S O(3), (97)

In addition, there is also a quadratic integral, the total energy of the system 1 Tr(FFT ) + Ug (F). 2 (99)

4. SYMMETRY-BASED REDUCTION AND HAMILTONIAN FORMALISM It is not difficult to carry out a reduction based on the linear integrals (98) using the results of the preceding section. To this end, we make use of the Riemannian decomp osition F = QT A, Q, S O(3), Q = wQ, we obtain the expression 1 1 i A2 + (Aj + Ak )2 (wi - i )2 + (Aj - Ak )2 (wi + i )2 - Ug (A). 2 4 We denote the three-dimensional vector of semiaxes as q = (A1 , A2 , A3 ) and represent the equations of motion in the form L Ç L - = 0, q q L= L w
Ç

A = diag(A1 , A2 , A3 ).

For the Lagrangian function of the gas cloud (92), in view of the equations = ,

L = ç w, w

L

Ç

=

L ç .

This is an analog of the Riemann equations (16), (40) for the case of a gas cloud (the difference is in the absence of the term containing pressure). These equations can easily b e written in a matrix form similar to (16) [77]. The Lagrangian transformation p= L , q m= L , w Å= L
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20 7

yields a Hamiltonian system q= H , p
2 i

p=- 1 p+ 4

H , q

m=mç
2

H , m

Å =Åç
2

H , Å

1 H= 2

mi + Åi qj - qk

+

mi - Åi qj + qk

(100)

+ Ug (q).

The Poissonian structure of the system (100) has the form {qi , pj } = ij , {mi , mj } = ij k mk ,
m

{Åi , Åj } = ij k Åk ,

(101)

where the zero brackets are omitted. As ab ove, the bracket (101) has two Casimir functions = (m, m), Å = (Å, Å), and the vorticity of the system. Hamiltonian system with five degrees of freedom. have a system with four degrees of freedom. degrees of freedom similar to the problem of the which corresp ond to the squared total momentum In the general case (m = 0, Å = 0), we have a In the particular case of m = 0 or Å = 0, we If m = Å = 0, we obtain a system with three motion of a unit-mass p oint in R3 = {q}.
5 3

5. PARTICULAR CASES OF MOTION 5.1. Case of = (Monoatomic Gas) Consider, in greater detail, the case of the expansion of an ellipsoidal cloud of ideal monoatomic gas in the absence of gravitation; we will show that the system has additional symmetries in this case, where, as is known, cV = 3 R and, therefore, = 5 . 2 3 We use (92) to represent the Lagrangian of the system as L= 1 Tr(FFT ) - Ug (F), 2 Ug (F) = 3 1 k , 2 (det F)2/3 (102)

where k = const is a p ositive constant (introduced for convenience). The integrals -- vorticity , momentum M, and energy E -- were mentioned ab ove (98), (99). We denote the eigenvalues of the matrices FFT as A2 , A2 , A2 and call Ai the principal semiaxes 1 2 3 of th gas ellipsoid (Ai coincides with the semiaxes of the gas ellipsoid in Ovsyannikov's model at the pressure and density distribution (79); for Dyson's model with a normal density distribution (85), this term is only conventional). We define an analog of the central moment of inertia of the system by the formula I = Tr FFT = A2 + A2 + A2 . 1 2 3 (103) As we can see, according to (102), the dynamics of the could can b e describ ed in this case by a natural Lagrangian system with a uniform p otential of uniformity degree = -2 (for an arbitrary , the uniformity degree is = 3(1 - )). We use the Lagrange-Jacobi formula for uniform systems [82] to obtain ? I = 4E = const, ? where E is the energy of the system (for an arbitrary , we find I = 4E - 2(3(1 - ) + 2)Ug ). The integration of this relationship yields I = 2E t2 + at + b,

(104)

where the integration constants a and b can b e expressed in terms of the phase variables and time according to the formulas In fact, a and b are nonautonomic (explicitly time-dep endent) integrals of the system considered.
REGULAR AND CHAOTIC DYNAMICS Vol. 14 No. 2 2009

a = 2 Tr(FT F) - 4E t,

b = 2E t2 - 2 Tr(FT F)t + I .

(105)


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For the first time, the integral (104) for uniform systems of a degree of -2 was noted by Jacobi in the problem of the motion of particles in a straight line. For the problem of the motion of a gas cloud, the Jacobi integral was found in [80]. The integrals (105) for system (102) were indicated in [83], while corresp onding symmetries in the particular case of = 0 were mentioned in [84]. Prop osition 1. At t Á, at least one of the semiaxes, Ai , of the gas cloud goes to infinity. Except the nonautonomic integrals (105), the systems in this case admits an autonomic quadratic integral indep endent of the energy integral, J = 2I E - [Tr(FT F)]2 . (106)

For uniform systems of degree -2, this integral was found in a more general case in [90]. For the system (102) in the particular case of = 0, it is also given in [84]. Preliminary results on symmetries for this integral were given in [91-93]. For uniform natural systems of degree -2, a sp ecial reduction can b e made to lower the numb er of degrees of freedom by unity. We describ e it in application to the considered system (102). We carry out a substitution of time and a (pro jective) substitution of variables dt = I d , G=I
-1/2

F.

(107)

It can easily b e shown by direct calculation that the evolution of the matrix G(t) can b e describ ed by a Lagrangian system with a constraint in the following form: L= 1 Tr 2 dG dGT d d ? - Ug (G),
T

3 1 ? Ug (G) = k , 2 (det G)2/3

(108)

= Tr(GG ) = 1. A relationship b etween the "old" time t and the "new" time can b e found using (104). Note that the system (108) differs from the Dirichlet system, since the constraint is different in this case (in the Dirichlet problem, det G = 1). It is interesting that the energy integral for the system (108) coincides with the integral (106): 1 ?1 E = J = Tr 4 2 dG dGT d d ? - Ug (G).

The linear integrals in the system (108) remains the same, = GT dG dGT - G, d d M=G dGT dG T - G; d d

furthermore, the system (108) is invariant with resp ect to the same transformations (97), which form a group = S O(3) S O(3). Therefore, a symmetry-based reduction similar to the ab ovedescrib ed one is p ossible (see Part I I, Section 3), with the only difference that, in this case, the following relationship b etween the semiaxes is valid: ? ? ? A2 + A2 + A2 = 1. 1 2 3 (109)

??? ? ? We use the Riemann decomp osition of the matrix G = QT A, Q, S O(3), A = diag(A1 , A2 , A3 ), to obtain, in this case, a system similar to (100) but with an additional constraint (109). To take this constraint into account and represent the equations in the most symmetric form, we define variables q and K according to the formulas qi = Ai , K =qç dq . d
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(110)
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Then we finally obtain a reduced system in the form ? ? ? dK H H dq H =Kç +qç , =qç , d K q d K ? ? dm H dÅ H =mç . =Åç , d m d Å 1 1 ? H = (K , K ) + 2 4 mi + Åi qj - qk
2

(111)

+

mi - Åi qj + qk

2

+ Ug (q).

The (nonzero) Poisson brackets corresp onding to the system (110) are as follows: {Ki , Kj } = ij k Kk , {Ki , qj } = ij k qk , {mi , mj } = ij k mk , {Åi , Åj } = ij k Åk .

Thuis Poisson structure, as is known, corresp onds to the algebra e(3) so(3) so(3) and has four Casimir functions, K = (K , q), m = (m, m), Thus, we ultimately conclude that 1. if m , Å = 0, equations (100) correspond to a Hamiltonian system with four degrees of freedom; 2. if 3. if
m m

q = (q, q), Å = (Å, Å); q = 1.

in view of the definition, (110), of the variables K and q, we have in this case
K

= 0,

= 0 (or Å = 0), we obtain a system with three degrees of freedom; = Å = 0, we obtain a system with two degrees of freedom.

As already mentioned ab ove, Gaffet [85] noted, for the case of Å = 0, two additional first integrals (of the sixth degree in the velocities) indep endent of the energy integral and put forward the hyp othesis of the integrability of the system in this case. Moreover, it was stated in [85] that the system (102) at = 0 (or M = 0) is integrable in Liouville's sense; missing integrals are presented, although their commutativity is not shown. The missing integrals are p olynomials of the sixth degree in momenta and have the form I6 = 36k
2

1 Y0 Y2 - Y12 + 3X2 + T (X0 + Y02 ) 4

+6k(4T 2 Y0 + 3P Y1 + 6T Y2 ) + 27P 2 + 4T 3 , 3k 2 L6 = A2 m, V0 A2 m ç (V0 A2 m + m) , (q1 q2 q3 )2/3 where A = diag(q1 , q2 , q3 ) and the quantities Xi , Yi , P, and T can b e expressed in terms of the symmetric matrix 3 Ki K1 m3 m3 1 2 2 2 3 q 1 -q 2 q 3 -q 1 3 i=1 qi - q1 3 m3 Ki K2 m1 1 V0 = - q2 2 -q 2 2 -q 2 3 qi q1 2 q2 3 i=1 3 Ki m2 m1 K3 1 2 -q 2 2 -q 2 3 qi - q3 q q
3 1 2 3

i=1

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as follows:
k k Xk =(q1 q2 q3 )2(k-1)/3 Tr(V0 A2 ), Yk =(q1 q2 q3 )2(k+1)/3 Tr(V0 A-2 ), 1 2 T = - (q1 q2 q3 )4/3 Tr(V0 ), P =(q1 q2 q3 )2 det V0 . 2 In the case of Å = 0, the system (100) is an Euler-Calogero-Moser system of typ e D3 [94, 95] with the p otential Ug . The Lax representation of the given system without a p otential can b e found, e. g., in [96]. In the case where Ug = 3 (q q k )2/3 , equations (100) can b e written as 2 1 2 q2 k L = [L, A] + D-1 , (q1 q2 q2 )2/3 (112) D = [D, A] + L, l = [l, A],

where the matrices

L=



p

1

m3 p2 q 2 -q 1 m1 -m2 q 3 -q 1 q 3 -q 2 -m2 m1 q 3 +q 1 q 3 +q 2 m3 0 q 2 +q 1 m 0 q-+q31 q3m2q + 2

m3 q 2 -q

1

-m2 -m2 m3 q 3 -q 1 q 3 +q 1 q 2 +q 1 m1 m1 0 q 3 -q 2 q 3 +q 2 m1 p3 0 q-+q2 3 m1 0 -p3 q--q2 3 -m1 -m1 q 3 +q 2 q 3 -q 2 - p 2
1

0
-m3 q 2 +q 1 m2 q 3 +q 1 m2 q 3 -q 1 -m3 q 2 -q 1



m2 q 3 -q

1

-m3 q 2 -q 1 3

-p

1

form an L - A pair for the system without a p otential, and the matrix D has the form D = diag(q1 , q2 , q3 , -q3 , -q2 , -q1 ). We failed to write the general equations (112) in the form of a normal L - A pair. For the ab ove-presented system with a third-degree integral, the L - A pair was obtained in [97] using a completely different technique. Here, the question of generalizing the construction of this L - A pair to the Gaffet system should b e raised.
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A=



l= 0
-m3 (q2 -q1 )2



0 - m3 0 m1 - m1 0 m3 m2 - m1 0 0 m1 - m2 , - m2 m1 0 0 - m1 m2 m3 0 - m1 m1 0 - m3 0 - m3 m2 - m2 m3 0
3 2 2 m3 -m2 m2 -m3 (q2 -q1 )2 (q3 -q1 )2 (q3 +q1 )2 (q2 +q1 )2

0

m

-m

m

-m



,

0
m3 (q2 +q1 )2

0

m2 -m1 (q3 -q1 )2 (q3 -q2 )2 -m2 m1 (q3 +q1 )2 (q3 +q2 )2 m3 (q2 +q1 )2

m1 -m1 (q3 -q2 )2 (q3 +q2 )2

0



0 0

0 0

m1 -m2 (q3 +q2 )2 (q3 +q1 )2 -m1 m2 (q3 -q2 )2 (q3 -q1 )2

0

0

-m3 m2 -m2 m3 (q2 +q1 )2 (q3 +q1 )2 (q3 -q1 )2 (q2 -q1 )2

-m1 m1 (q3 +q2 )2 (q3 -q2 )2

0

-m3 (q2 -q1 )2

0


THE HAMILTONIAN DYNAMICS OF SELF-GRAVITATING ELLIPSOIDS

21 1

5.2. The Case of Axial Symmetry As in the case of a fluid ellipsoid, it can easily b e shown that the system (92) admits a threedimensional invariant manifold formed by matrices of the form u v 0 F = -v u 0 . (113) 0 0w The liner integrals (98) simplify in this case b ecoming
12

= -M12 = 2(uv - v u), 1 u = r cos , 2



13

=

23

= M13 = M23 = 0.

(114)

Consider the Ovsyannikov model with gravitation (93) and make the substitution of variables 1 v = r sin , 2 w = z.

Then the Lagrangian function of the system assumes the form 1 L = (r 2 + r 2 2 + z 2 ) - Ug (r, z ), 2 k 1 Ug = + Ue (r, z ), 2 z ) -1 - 1 (r

(115)

1 where = 2 cyclic integral

where the energy of the gravitational field Ue can b e expressed in terms of elementary functions: 2 arctg 2 - 1 , > 1, 2 - 1 d 2 2 Ue = -2 =- ç r2 ln 1+1- z ( + 2 ) + z 2 1- 1-2 0 , < 1, 1 - 2
r z

is the semiaxis ratio. Since the Lagrangian (115) is indep endent of , there is the L = r 2 = c = const,

which coincides with the integrals (114) within a multiplier. For a fixed value of this integral, we make the Legendre transformation pr = L = r, pz = r and obtain a Hamiltonian system with two degrees of freedom in the canonical form H= 12 (p + p2 ) + U (r, z ), z 2r U = c2 + Ug (r, ); 2r 2
L z

=z

(116)

here, U is the reduced p otential. Consider the simplest (integrable) case, the motion of a monoatomic gas ( = 5 ) without 3 allowances for gravitation (i. e., Ue = 0; see also the preceding section). It was shown ab ove that the system in this case admits a reduction by one more degree of freedom and, therefore, reduces to a quadrature. Indeed, we make a substitution of variables and time of the form r = R cos ,
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where, in view of the conditions r > 0 and z > 0, the variable (0, /2). We obtain the following equations for R and : d2 (R2 ) = 4H = const, dt2 1 c2 3 k + = h1 = const. + 2 2 sin )2/3 2 cos 2 (cos

1 2

d dt

2

The quadrature for at c = 0, with certain limitations on the initial conditions, was obtained in [80]. As we can see, the evolution of (t) can b e determined by the reduced p otential 1 c2 3 k ? U ( ) = + . 2 2 sin )2/3 2 cos 2 (cos At all values of the parameters c and k, this function has one critical value 0 in the interval ? (0, /2), in which U reaches its minimum. This value corresp onds to the self-similar expansion of a spheroidal gas cloud. In other cases, the expansion of the cloud is accompanied by oscillations in the semiaxis lengths, with varying in the interval (1 , 2 ), where i are the roots of the equation ? U ( ) = h1 . ra In the general case, Ue = 0, the t jectories of the system (116) are not finite. However, it can easily b e shown that, at k > 91 22/3 ( 665 - 21)c2 0, 43c2 , the reduced p otential has a minimum 6 at the p oint 1 32 0 = arctg , R0 = (c + 3 Ç 21/3 k). 8 2 Therefore, near the minimum of the energy U (0 , R0 ), the tra jectories of the system are finite and a PoincarÄ map can b e constructed. Such a map in the plane = as the plane of section is shown e 4 in Fig. 3. A chaotic layer that originates from the splitting of resonant tori can b e clearly seen in this figure, which testifies to the nonintegrability of the system (116).
pR

R

Fig. 3. The PoincarÄ map of the system (116) at k = c = = 1 in the section plane = e

4

.

5.3. Generalization of the Riemannian Case An invariant manifold of the form (57) also exists for gas ellipsoids, i. e., u1 v1 0 F = u2 v2 0 0 0 w3 As in the Riemannian case, it can b e shown for a fluid ellipsoid that, in the case of gas, the following relationships are also valid: m1 = m2 = Å1 = Å2 = 0, m3 = const, Å3 = const.
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.

(117)


THE HAMILTONIAN DYNAMICS OF SELF-GRAVITATING ELLIPSOIDS

21 3

Thus, we conclude that, according to (100), the evolution of the semiaxes Ai = qi , i = 1, 2, 3 can b e describ ed by the third-degree Hamiltonian system 1 H = p2 + U (q ), 2 U = c2 c2 1 2 + + Ug (q), (q1 - q2 )2 (q1 + q2 )2 (118)

where q, p are canonically conjugate variables and c1 = 1 (m3 + Å3 ), c2 = 1 (m3 - Å3 ) are fixed 2 2 constants. It was shown ab ove that, for a monatomic gas ( = 5 ), without taking into account gravitation 3 (Ue = 0), the system admits a reduction by one more degree of freedom. As a result, we obtain in this case a system of the form ? ? ? dK H H dq H =Kç +qç , =qç , d K q d K 3 c2 c2 1 k 1 2 ? ? ? H = K 2 + U (q), U (q) = + + . 2 2 (q1 q2 q3 )2/3 (q1 - q2 )2 (q1 + q2 )2 This system is equivalent to the problem of the motion of a spherical top in an axisymmetric p otential [63]. As shown in [98, Sec. 4], this system is integrable provided c2 = c2 . At c1 = c2 = 0, 1 2 the additional integral of the third degree in the velocities has the form F3 = K1 K2 K3 - 3k K1 q2 q3 + K2 q3 q1 + K3 q1 q2 . (q1 q2 q3 )2/3
2 2 2 2 f (q1 + 3kq3 )(q2 + 3kq3 ) , 4 q3

If c1 = c2 = c = 0, we have an additional sixth-degree integral F6 = (F3 + Fa )2 + 4 where Fa =
2 4c2 q1 q2 q3 2 2 K3 , (q1 - q2 )2

f=

4c2 (q1 q2 q3 )2/3 2 q, 2 2 (q1 - q2 )2 3

=

(q1 q2 q3 )2/3 K1 K2 - 3k + f . q1 q2

In the more general case of c2 = c2 , the system (117) b ecomes nonintegrable. 1 2 Figure 4 shows the corresp onding PoincarÄ map in the Anduaye variables, which are traditionally e used for reductions in the problems of rigid-b ody motion with a fixed p oint [63]. The break down of the resonant tori and the birth of isolated p eriodic solution can b e clearly seen from the figure, which is evidence for the nonintegrability of the problem.

? Fig. 4. PoincarÄ map of the system (117) at an energy level of H = 30 at k = 1/3, c1 = 1, c2 = 0.3 in the e section plane g = .

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6. ACKNOWLEDGMENTS Authors thank B. Gaffet, V. V. Kozlov and F. FassÄ for useful discussions. This work was o supp orted by the Russian Foundation for Basic Researcvh (pro ject codes 08-01-00651 and 07-0192210). The work of I. S. Mamaev and A. A. Kilin was done in the framework of the program of the President of the Russian Federation for the supp ort of young scientists, Doctors of Science (pro ject code MD-5239.2008.1) and Candidates of Science (pro ject code MK-6376.2008.1), resp ectively. REFERENCES
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2009