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A. V. BORISOV
Department of Theoretical Mechanics Moscow State University, Vorob'ievy Gory 119899, Moscow, Russia E-mail: b orisov@rcd.ru

I. S. MAMAEV
Lab oratory of Dynamical Chaos and Nonlinearity Udmurt State University, Universitetskaya, 1 426034, Izhevsk, Russia E-mail: mamaev@rcd.ru

EULER ­ POISSON EQUATIONS AND INTEGRABLE CASES
Received April 18, 2001

DOI: 10.1070/RD2001v006n03ABEH000176

In this paper we propose a new approach to the study of integrable cases based on intensive computer methods' application. We make a new investigation of Kovalevskaya and Goryachev ­ Chaplygin cases of Euler ­ Poisson equations and obtain many new results in rigid body dynamics in absolute space. Also we present the visualization of some special particular solutions.

1. Euler ­ Poisson equations and integrable cases
In this pap er we prop ose a new approach to the study of the classical problem of integrable cases in rigid b o dy dynamics. It is based on computer metho ds' application to b oth analytical and numerical investigation of the systems in question. This approach allows to obtain many new results, some of which are presented in the pap er (for detailed presentation, see [9]). Even in the analysis of integrable cases that, basically, allow complete classification of all the solutions, the computer research metho ds have, in some sense, started a new age. Earlier investigations of integrable systems involved mostly the analytical metho ds p ermitting to obtain the explicit quadratures and geometrical interpretations which in many cases were very artificial (consider, for example, Zhukovsky's interpretation of Kovalevskaya top's motion [12]). By contrast the combination of the ideas of top ological analysis (bifurcation diagrams), stability theory, metho d of phase sections, and immediate computer visualization of "particularly sp ecial" solutions is quite able to demonstrate the sp ecific character of an integrable situation and to single out the most typical prop erties of motion. With such investigation, it is p ossible to obtain a variety of new results, even in the field that seems to b e thoroughly studied (for example, for the tops of Kovalevskaya and Goryachev ­ Chaplygin, and for Bobylev ­ Stekloff solution). The matter is that it is very hard to see these results in those cumb ersome analytical expressions. The pro ofs of these facts can probably b e obtained analytically as well, but only after their revealing by computer metho ds. Here we should esp ecially note the analysis of motion in the absolute space. Some sp ecific motions of such integrable tops can probably initiate some concrete ideas, concerning their practical investigation. Let's remind, for example, that the Kovalevskaya top discovered more than a century ago still can not find any application just b ecause its motion remains practically unknown despite of its complete solutions in elliptic functions.
Mathematics Sub ject Classification 37J35, 70E17



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We also give certain unstable p erio dic solutions that generate a family of doubly-asymptotic motions, whose b ehavior is most complicated and lo oks irregular even in cases with an additional integral. Under p erturbation, such solutions are first to collapse and the whole areas, filled with now "real" chaotic tra jectories, app ear near them in the phase space. The computer investigations force "the revision" of many asp ects of analytical investigations and help to understand their real meaning. While some analytical results, such as separation of variables, are very useful for the study of bifurcations and classical solutions, their further "development" to explicit quadratures (through -functions) is practically useless. These results are collected, for example, in [21, 17], but they are more useful as exercise in differential equations than as metho ds of dynamics analysis.

1.1. A rigid b o dy with a fixed p oint
Euler ­ Poisson equations describing the motion of a rigid b o dy around a fixed p oint in homogeneous gravity field have the form I + â I = µr â , (1.1) = â ,

point in gravity field.

where = (1 , 2 , 3 ), r = (r1 , r2 , r3 ) = (1 , 2 , 3 ) are, resp ectively, the comp onents of the angular velo city vector, the comp onents of the radius vector of the center of mass, and the comp onents of the vertical unit vector in the frame of the principal axes, rigidly b ound with the rigid b o dy and passing through the p oint of fixation, I = diag(I1 , I2 , I3 ) is the tensor of inertia in relation to the p oint of fixation in the same axes, µ = mg is the weight of the b o dy (Fig. 1). Using the pro jections of the momentum vector M = I in the same axes, equations (1.1) can b e presented in the Hamiltonian form Fig. 1. A rigid body with a fixed (1.2) Mi = {Mi , H }, i = {i , H }, i = 1, 2, 3, with a Poisson bracket corresp onding to algebra e(3)
ij k

{Mi , Mj } = -

Mk ,

{Mi , j } = -

ij k k

,

{i , j } = 0,

(1.3)

and with the Hamiltonian equal to the full energy of b o dy H = 1 (AM , M ) - µ(r , ). 2 where A = I
-1

(1.4)

.

Remark 1. Euler (1758) already knew the equations of motion in form (1.1), he also had found the elementary case of integrability, when the rigid body moves under inertia (r = 0). The integrability of an axially symmetric top with the center of gravity on the symmetry axis was established by Lagrange and a little bit later by Poisson, the latter's name being included in the term for the general equations (1.1).

Lie ­ Poisson bracket (1.3) is degenerated, it has two Casimir functions, commuting with any function of variables M , in the structure of brackets (1.3) F1 = (M , ), F2 = 2 . (1.5)



In the vector form, equation (1.2) can b e written as M = M â H + â H , M = â H . M
254

(1.6)

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The form of equations (1.2), (1.6) is induced by the Poincar´ ­ Chetayev equations, written on e group S O (3) (see [9]). Functions F1 and F2 are integrals of equations (1.6) for any Hamiltonian function H . For Euler ­ Poisson equations they have a natural physical and geometrical origin. Integral F 1 represents a pro jection of the momentum vector on the fixed vertical axis and it is referred to as area integral in rigid b o dy dynamics, it represents the symmetry in relation to the rotations around the fixed vertical axis. The origin of integral F2 = const is purely geometrical, it is equal to the squared absolute value of the vertical unit vector. For real motions the value of the constant of this integral is equal to one: F2 = 2 = 1. When bracket (1.3) is restricted on the common level of integrals F 1 and F2 , it b ecomes nondegenerate, and Darb oux theorem ([9]) implies that the bracket can b e represented in the usual canonical form in certain symplectic co ordinates. For the various purp oses it is p ossible to use b oth canonical Euler variables ( , , , p , p , p ) and Andoyer ­ Deprit variables (L, G, H, l, g , h). In b oth cases on the symplectic leaf defined by p = const (resp ectively, H = const) we obtain the canonical system with two degrees of freedom. For Liouville integrability ([9]) of system (1.1), and system (1.6) as well, the presence of one more additional integral is necessary b esides Hamiltonian (1.4), which is also a first integral of the system.

2. Kovalevskaya case
It is known that this integral exists in the cases of Euler, Lagrange and Kovalevskaya, and with additional restriction (M , ) = 0 in Goryachev-Chaplygin case. While in the first two cases the motion has b een studied thoroughly enough, Kovalevskaya and Goryachev-Chaplygin cases are still p o orly investigated. The additional integrals in Euler and Lagrange cases have natural physical origin. In the first case the integral is the squared absolute value of the momentum vector, in the second case the integral is the pro jection of this quantity on the axis of dynamical symmetry. In the case of integrability found by S. V. Kovalevskaya (1888) the additional integral has no explicit symmetrical origin. It was found almost a century after two previous integrals, and it is incomparably more complicated b oth from the p oint of view of explicit integration and for the qualitative analysis of motion. The rigid b o dy in this case is dynamically symmetric: a 1 = a2 , and the center of mass is situated on the equatorial plane of the inertia ellipsoid r 3 = 0. In this case relation a3 = 1 = 2 is also valid. 1 I3 The Hamiltonian and the additional integral found by Kovalevskaya are given by:
2 2 H = 1 M1 + M2 + 2M 2 2 3

a

I

- x1 , (2.1)

F3 =

2 2 M1 - M 2 + x 2

2 1

+ (M1 M2 + x 2 )2 = k 2 ,

where the comp onents of the radius vector of the center of mass are r = (x, 0, 0) and the weight is equal to µ = 1 without loss of generality.

2.1. Explicit integration. Kovalevskaya variables
Together with the additional integral S. V. Kovalevskaya has found the remarkable variables that transform the equations of motion (1.1) to Ab el ­ Jacobi form (see [29]). With this form the further integration in -functions (of two variables) can b e p erformed according to a certain general pattern (see [28]). Here we shall present only the corresp onding change of variables.


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Kovalevskaya variables s1 , s2 are defined by the following formulas s1 = R - R 1 R2 , 2(z1 - z2 )2 s2 = R + R 1 R2 , 2(z1 - z2 )2 (2.2)

z1 = M1 + iM2 ,

2 22 2 2 R = R(z1 , z2 ) = 1 z1 z2 - h (z1 + z2 ) + c(z1 + z2 ) + k - 1, 4 2 4 R1 = R(z1 , z1 ), R2 = R(z2 , z2 ),

z2 = M1 - iM2 ,

where F1 = (M , ) = c,H = h. To simplify the evaluations further we will assume x = 1. The equations of motion b ecome ds
1

P (s1 ) where P (s) = 2s + h 2
2

=

dt , s1 - s 2

ds

2

P (s2 )

=

dt , s2 - s 1

(2.3)

2 -k 16

2 2 2 4s3 + 2hs2 + h - k + 1 s + c . 4 16 4 16

Because P (s) is the p olynomial of the fifth degree the quadrature for (2.3) is referred to as ultrael liptic (hyperel liptic). In pap er [BorMam Rhd] we present the generalized Kovalevskaya variables for the similar integrable case on the bundle of brackets containing algebras e(3), so(4), so(3, 1).

Fig. 2. Bifurcation diagrams of Kovalevskaya case for various c. Roman numbers denote Appelrot classes. The continuous curves correspond to the stable periodic solutions, dashed -- to the unstable ones and to the separatrices.

2.2. Bifurcation diagram and App elrot classes
The values of integrals h, c, k for which p olynomial P (s) has multiple ro ots determine on the common space of these integrals the bifurcation diagram the collection of two-dimensional surfaces, on which the typ e of motion changes (see Fig.2). At the same time the ultraelliptic quadratures in (2.3) transform


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to the elliptic ones, and the corresp onding (particularly sp ecial) motions are called Appelrot classes [5]. Different branches of the bifurcation diagram corresp ond to various App elrot classes. It is easy to show (and it is a well-known fact) that the App elrot classes defined by the multiplicity of the ro ots of p olynomial P (s) = 0 coincide with the set of special Liouvil le tori on which integrals H, F1 , F2 , F3 are dep endent, i. e. the Jacobi matrix rank
(H, F2 , F3 , F4 ) (M , )

drops [23]. It is obvious

that these sp ecial tori in the phase space of the reduced system (i. e. for Euler ­ Poisson equations) determine the stable and unstable p erio dic motions and their asymptotic tra jectories. The stability of branches is indicated on the bifurcation diagram presented in Fig.2. Combined with the ab ove describ ed Poincar´ phase sections, the diagram is very useful for the study of dynamics e b ecause it allows the visual understanding of qualitative b ehavior of all tra jectories of an integrable system in the phase space. The explicit solutions for App elrot classes can b e obtained directly without equations (2.3). Their construction, involving nonobvious transformations, was started by App elrot himself [4], and it was obtained in the most complete form by mechanician A. I. Dokshevich [12] from Donetsk. Let's present a part of his results, related mainly to the p erio dic and asymptotic motions (the most imp ortant for dynamics), and try to clarify their mechanical sense. There are four App elrot classes. I. Delone solution [11]: Here k 2 = 0, h > c2 and two additional invariant relations app ear
2 2 M1 - M 2 + x1 = 0, 2

M1 M2 + x2 = 0,

(2.4)

which define the p erio dic solution of Euler ­ Poisson equations. It turns out that the motion in this case at the zero value of area integral c = 0 is p erio dic not only for the reduced system (on the Poisson sphere), but in the absolute space [17] as well (see Figs. 7-10). To derive the explicit quadrature, we express all the variables on the common level of integrals and invariant relations (2.4)as the functions of M 1
2 2 M2 = 2z - M1 , 2 M3 = h - M 1/2 2 1 1/2

2 x1 = -M1 + z ,

x1 = -M1 (2z - M1 )
1/2

,

x3 = (x2 - z 2 )

(2.5)

2 M 2 + M2 2 2 z= 1 = (1 + 2 ) 2

=x

- cM1 ±

2 (h - c2 )(h - M1 ) . h

Then we obtain the quadrature for M

1 1/2

2 2 M1 = M2 M3 = (h - M1 )(2z - M1 )

,

(2.6)

which is elliptic at h = c2 . For c = 0 it is also p ossible to obtain a simpler explicit solution, if we use variable M3 instead of M1 . It follows from Fig. 2 that under magnification of c up to c =
3 4
3/4

branch IV of App elrot class

"runs" into Delone solution and under further magnification up to c < 2, the branch divides it into three parts. For c2 = 2, the branches of all four App elrot classes merge in p oint h = 2, k 2 = 0. To the p oint of their intersection corresp ond the unstable fixed p oint on the Poisson sphere (the Staude rotation) ([9]) and to the one-dimensional motion asymptotic to this p oint, which is easily calculated from (2.6) in elementary functions M1 = 3 + ch2 u ± 4 ch u 2x , 9 - ch2 u u = 2 xt. (2.7)

2

For c2 > 2, one branch of class IV also "runs" into Delone solution, while its other branch intersects the part of a parab ola, corresp onding to the class I I.


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Fig. 3. Phase portraits (sections by plane g = /2) for Kovalevskaya case at the zero value of area integral c = 0 (three qualitatively different types are shown). One can see reorganizations of the portraits and the bifurcations of the periodic solutions, which happen during the intersection of critical levels of energy h = 0 and h = 1. (The gray color fills the nonphysical range of values of l, L/G for the given values of integrals h, c.)

I I. The solutions of the second class are on the lower branch of parab ola (h - c 2 )2 = k 2 , note that 1 c2 - 1 h c2 . For c = 0, the stable p erio dic tra jectories b elong to this class, and the rigid 2 b o dy p erforms flat oscillations in the meridional plane passing through the center of mass, and the conditions M1 = M3 = 0, 2 = 0 hold. For c = 0, there are additional invariant relations (2.8) and the explicit integration is presented in [12]. Starting from c > 2, the branches of classes I I and IV b egin to intersect. I I I. The branch of the parab ola ab ove the tangency p oint with the axis k 2 = 0 corresp onds to this class. It ob eys the conditions (h - c2 )2 = k 2 , c
2

M3 = c3 ,

M 2 2 M1 + M2 + c 1 = k ,

h

c 2 + 12 . 2c

(2.9)

For c = 0, these requirements determine the whole upp er branch of the parab ola, and for c = 0 this branch is b ounded from ab ove by one of the branches of class IV.


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Fig. 4. Phase portraits (sections by plane g = ) for Kovalevskaya case for c = 1.15 and for the fixed values of energy h, which correspond to the phase portraits of qualitatively various types. The variables l and L/G correspond to the cylindrical involute of the sphere and the phase portrait is symmetrical in relation to the meridian l = /2, 3 . (The bifurcation diagrams on the right figures are drawn roughly and not to scale.) 4


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Physically, class I I I corresp onds to the unstable p erio dic solutions and to the solutions asymptotic to them. For c = 0, the p erio dic motion for the part of the branch denoted as I I I a) is oscillations of a physical p endulum in the meridional plane passing through the center of masses, and for the part I I I b) it is rotations in the same plane. These solutions meet in the p oint h = 1, which is the upp er unstable equilibrium. Its instability can b e strictly proved by various approaches [36]. Later this pro of will b e obtained by explicit construction of the asymptotic solution. Let's use the following parametrization of the common level of the integrals of motion corresp onding to the third App elrot class for the zero value of area integral c = 0 [12] M1 =
2 2 M1 + M3 sin ,

M3 =

2 2 M1 + M3 cos

(2.10)

k1 = k cos 2 , where k1 = 1 +
2 2 M1 - M 2 ,k 2

k2 = k sin 2 ,

2

= 2 + M1 M2 (at x = 1), Kovalevskaya integral having the form

2 2 k1 + k 2 = k 2 . Differentiating with resp ect to time, we obtain

= M2 -

M1 k2 . 2 2 M1 + M 3

(2.11)

After one more differentiation (2.11) and elimination of M 2 with the help of (2.11), taking into account h = k > 0, we have the equality 2 cos + sin = 2h cos 2 sin . ¨ Multiplying (2.12) by
and integrating with resp ect to time, we obtain cos2

(2.12)

2 cos + 2h cos = c1 = const. The integration constant is obtained from the condition = 0, which imply that M 1 = 0, = M2 , 2 = 4x2 . Thus, and therefore c1 2 = 2(x - k cos ) cos , k > 0. (2.13)

Remark. For c = 0, we obtain equation [12] for a similar (but a little different) angular variable 2 = 2(x - (k + c2 ) cos ) cos .

For angle , we have the equation = -M3 = -
2 2 M1 + M3 cos ,

2 2 which after taking into account integral of energy M 1 + M3 - k1 = h and condition h = k resulting 2 2 in equality M1 + M3 = ± 2k cos , is reduced to the following form: = 2k cos cos .

After substitution cos = (ch u)

-1

we can write it as u = 2k cos .
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Thus, the complete system of equations defining the asymptotic tra jectories of App elrot class I I I under condition c = 0, h = k > 0 is reduced to the form 2 = (1 - 2 )(x - k + (x + k ) 2 ), = tg 2 -1 u = 2k cos , ch u = (cos ) . Its solutions are: 1. k < x, = cn( xt, k0 ), 2. k > x, = dn 3. k = x, = (ch xt)
x+k t, k 2
-1 0 2 k0 = x + k

(2.14)

,

2x 2 k0 = 2x , x+k

,

where k0 is the absolute value of the corresp onding Jacobi elliptic functions. Using 1­3, it is p ossible to show, that u is a function of fixed sign, i. e. these solutions in cases 1­2 describ e the motions asymptotic to the p erio dic solution, and in case 3 describ e the motions asymptotic to the fixed p oint. (The analytical quadratures in the case c = 0 are more cumb ersome [12].) IV. This class consists of two branches (see Fig. 2), one of which corresp onds to stable p erio dic motions, and another -- to unstable motions and to separatrices. For c = 0, these branches meet in p oint k 2 = x2 = 1, h = 0. For c = 0, the parametric equations of the branches are
4 k 2 = 1 + tc + t , 4 2 h = t - c, t 2

(2.15)

t (-, 0) (c, +), for c > 0, t (-, +) \ {0}, for c < 0, For c = 0 1. k 2 = x2 , h < 0, h2 = k 2 + x 2. k 2 = x2 , h > 0
2

(branch IVa); (branch IVb).

The stable and unstable p erio dic solutions for App elrot class IV in the Kovalevskaya case (as well as in a more general case, when the tensor of inertia has the form I = diag(1, a, 2), a = const, and the solution do es not dep end on a) were found by D. K. Bobylev [7] and V. A. Stekloff [38] (see also [9]). Bobylev ­ Stekloff solution. For this solution the following relations always hold M2 = 0, which allow to express as a function of M c 2 1 = m - M 3 ,
3 2 3 2 1/2

M1 = m = const,

c 2 = k 2 - 1 m2 - m + M 2
3

,

3 = mM

3

and to obtain an elliptic quadrature for M

c M3 = - k 2 - 1 m2 - m + M 2 Here h and k 2 are defined by parametric equations c h = 1 m2 - m , 2

2 3

2 1/2

.

(2.16)

k 2 = 1 + 1 m4 + cm, 2
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i. e. they coincide with (2.15). For c = 0, the motions o ccur in the fourth class that corresp ond to oscillations and rotations ob eying the law of physical p endulum in the equatorial plane of the inertia ellipsoid. For these solutions, M1 = m = 0, 3 = 0,
2 M3 = -(1 - (h - M3 )2 ) 1/2

.

The asymptotic solutions for arbitrary values of c = 0 can b e found in [12], but they are very cumb ersome. Let's sp ecify these solutions under the additional conditions k 2 = x2 , h > 0, c = 0. (2.17)

For this purp ose we use an interesting involute transformation, (M , ) (L, s) (its square is equal to identity), found by A. I. Dokshevich: L1 = - L2 = - M1 , 2 + M2 M1 2 M2 , 2 M + M2
2 1 3

s1 = -1 + 2x s2 = -2 + 4x
2 1

2 3

2 (M1 + M

2 M1 - M

2 2 22 2)

, (2.18)

2 3

M1 M2 , 2 2 (M1 + M2 )2

L3 = M3 + 2x

M1 , 2 M + M2

s3 =

3 . 2 M + M2
2 1

In new variables (L, s) the equations of motion have the form L1 = L 2 L3 , L2 = -L1 L3 - xs3 , L3 = -2xcL2 + xs2 , s1 = 2L3 s2 - 4(k 2 - x2 )s3 L2 , s 3 = s 1 L2 - s 2 L1 . s2 = -2L3 s1 + 4(k 2 - x2 )s1 L3 ,

(2.19)

Under condition (2.17) in system (2.19) the equations for L 3 , s1 , s2 are separated and are reduced to quadratures s2 = (1 - s2 )1/2 , L3 = (h + xs1 )1/2 , 1 (2.20) s1 = 2 (h + xs1 )(1 - s2 ). 1 To obtain the solution of complete system (2.19) it suffices to find the solution of a linear second-order equation with co efficients explicitly time-dep endent L1 = s
-1 1

(-L3 s

3

s

hs2 - 1 s1 ), L2 = 3 4x s3 = -x(1 + 2s1 )s3 . ¨
2

hs2 - 1 s1 , 3 4x

(2.21)

Equations (2.20), (2.21) describ e the solutions asymptotic to p erio dic motions under conditions (2.17) (see Fig. 15). For h = x, which corresp onds to the energy of the upp er unstable equilibrium, we shall obtain one more (in addition to class I I I) solution asymptotic to the equilibrium expressed in elementary functions th u , L = - 2x , u = 2xt. s1 = 1 - 2 th u, s2 = 2 3 ch u ch u App elrot classes define the most simple motions b oth in reduced, and in absolute phase space. The other motions of Kovalevskaya top have quasip erio dic character and dep end on the corresp onding domain of the bifurcation diagram. Under p erturbation of the Kovalevskaya case, a sto chastic zone app ear near the unstable solutions and their separatrixes. Unfortunately, the (asymptotic) solutions presented in this paragraph do not yet allow (b ecause of various reasons ) to advance in the analytical investigation of nonintegrability of the p erturb ed Kovalevskaya top (the pro of of nonintegrability for c = 0 is obtained by variational metho ds in [8]).


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2.3. Phase p ortrait and visualization of particulary sp ecial solutions
For each fixed value of area integral (M , ) = c, defining various typ es of the bifurcation diagrams on the plane (k 2 , h), there is its own collection of phase p ortraits. Fixing the level of energy h we obtain several various typ es of phase p ortraits, which are defined by intersections of straight line h = const with the bifurcation diagram. Here we present two sets of the phase p ortraits corresp onding to the
3/4 , Fig. 4) bifurcation most simple (c = 0, Fig. 3) and to the most complicated (1 < c < 4 3 diagrams. Also, the form of some "particulary sp ecial" solutions on the Poisson sphere and in the absolute space is presented in the following paragraphs.

Remark. The investigation of invariant tori topology with the help of Poincar´ sections is also presented e in [14], in different variables and without explanation of mechanical meaning of various motions (in particular, without there the analysis of stability).

Phase p ortrait for c = 0. In this case the bifurcation diagram consists of two parts of parab olas and two straight lines (see Fig. 2 a). The physical sense of branches corresp onding to the parab ola h2 = k 2 and to the straight line k 2 = 1 is esp ecially clear and is describ ed ab ove. On the parab ola there are solutions describing flat oscillations and rigid b o dy rotations in the meridional plane (around axis O y , p erp endicular to axis O x, on which the center of mass is situated), and on the straight line -- the one describing flat oscillations and rotations in the equatorial plane (around axis O z ). On the remaining branches k 2 = 0 and h2 = k 2 - 1, Delone and Bobylev ­ Stekloff solutions are situated, accordingly. Ab ove we have presented the phase p ortraits and indicated where they are situated on the bifurcation diagram. It follows from Fig. 2 a) that there are three intervals for the constant of energy h: (-1, 0), (0, 1), (1, ), to each of which the qualitatively various typ es of phase p ortraits corresp ond (see Fig. 3). Phase p ortrait for c = 1.15 1

4 3

34

. With the bifurcation diagram (Fig. 2 c) it

is p ossible to establish that there are five energy intervals with corresp onding typ es of phase p ortrait (see Fig. 4). In this case the p erio dic solutions corresp onding to the branches of the bifurcation diagram do not have the forms as simple, as for c = 0, though they tend to it at h c.

Fig. 5. Delone solution. Motion of the unit vector for the zero value of area integral (c = 0) and for various values of energy. Remark. For the construction of the phase portraits we use Poincar´ sections and Andoyer ­ Deprit e variables. For c = 0 we choose g = as an intersecting plane, and for c = 1.15 we choose g = . We change 2


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Fig. 6. Delone Solution. Motion of the unit vector for the nonzero value of area integral (c = 1.15) and for various values of energy h. the intersecting plane because in this case not all periodic solutions intersect plane g = . Let's also mention 2 the different types of symmetry of phase portraits on sphere (l, L/G): for g = the portrait is symmetrical 2 with respect to the equator (axis L/G = 0), and for g = it is symmetrical with respect to the meridional plane (l = , 3 ). 22

Let's pro ceed to visualization of some most interesting motions of a rigid b o dy in the reduced and absolute spaces. Delone solution (k 2 = 0). In this case the tra jectory of the vertical unit vector on the Poisson sphere is represented by curves with the figure-of-eight typ e (see Fig. 5, 6), and for c = 0 (Fig. 5) the p oints of self-intersection of these "figure-of-eight typ e curves" coincide and have co ordinates = (1, 0, 0). This p oint determines the lower p osition of the center of mass of a rigid b o dy. When c increases the irregular "figure-of-eight typ e curves" also app ear on the Poisson sphere, all of them have the same two p oints of intersection on the equator of the Poisson sphere (see Fig. 6).

Fig. 7. Delone solution. Motion of the apexes in the fixed frame of reference for the zero value of area integral (c = 0).

Fig. 8. Delone solution. Motion of the apex of the center of mass for c = 0 and various h.

It is known that for c = 0 Delone solution determines the p erio dic motions in b oth the reduced system and the absolute space [18]. For c = 0 this assertion is not valid and the motion of a rigid b o dy in the absolute space is quasi-p erio dic. In figures 7­10 the tra jectories of three ap exes of a rigid


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Fig. 9. Delone solution. Motion of the apex situated in the equatorial plane perpendicular to the radius-vector of the center of mass for c = 0 and various h.

Fig. 10. Delone solution. Motion of the apex of the axis of dynamic symmetry for c = 0 and various h.

b o dy are shown for c = 0 and for various values of energy. In all figures the fixed axes O X Y Z are arbitrarily rotated to show the b est view of obtained tra jectories. Bobylev ­ Stekloff solution. Bobylev ­ Stekloff solution on the bifurcation diagram (see Fig. 2) is situated on the lower right branch and corresp onds to the stable p erio dic solution on the Poisson sphere (see Fig. 11, 12). It is clearly visible in Fig. 11, that for c = 0 all tra jectories on the Poisson sphere pass through the p oints of its equator (0, 1, 0) and (0, -1, 0), not intersecting the meridional plane 1 = 0. The remarkable motion of the center of mass in absolute space corresp onds to this case: the center of mass describe the curves with cusps, which for al l energies are situated on the equator (see Fig. 13).

Fig. 11. Bobylev ­ Stekloff solution. Motion of the vertical unit vector on the Poisson sphere for c = 0 and various values of energy.

The tra jectories on the Poisson sphere for c = 0 are presented in Fig. 12, in this case the ap ex of the center of mass traces in the absolute space the curves with cusps on the same latitude, which dep ends on a constant value of energy h (see Fig. 14). Physically, Bobylev ­ Stekloff solution can b e implemented as follows: a b o dy is twisted around the axis passing through the center of mass and arbitrarily p ositioned in the absolute space, then it is released without an initial impulse.


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Fig. 12. Bobylev ­ Stekloff solution. Motion of the vertical unit vector on the Poisson sphere for c = 0 (c = 1.15) and various values of energy.

Fig. 13. Bobylev ­ Stekloff solution. Motion of the apex passing through the center of mass in the absolute space for c = 0 and various h.

Fig. 14. Bobylev ­ Stekloff solution. Motion of the apex passing through the center of mass in the asolute space for c = 0 (c = 1.15) and various h.

Remark 2. Motion of the remaining apexes in the absolute space is very complicated, therefore we do not present it.

Unstable p erio dic solutions and the separatrices for Kovalevskaya case have very complicated form b oth on the Poisson sphere, and in the absolute space. In Fig. 15, the tra jectories of motion corresp onding to the separatrices for c = 0 (c = 1.15) and for the same value of energy h = 2 are presented. It is clearly visible that most of the time the tra jectory is staying near the p erio dic solution, in the figure this is shown by darker shading. These tra jectories in some sense represent the complexity of Kovalevskaya integrable case, some motions in this case have visually chaotic character (in the absolute space the motion lo oks even more irregular).
Remark 3. Let's give one more representation of Kovalevskaya integral, this time as a sum of squares. For this purpose we use the pro jections of the moment on semimoving axes S1 = M 1 1 + M 2 2 , 266 S3 = M 1 2 - M 2 1 . 3, 2001


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Fig. 15. Tra jectories on the Poisson sphere for the solutions asymptotical to the unstable periodic solutions. It is possible to show that we can write Kovalevskaya integral in the form F=
2 M1 + M 2 2 2 2 2 2 + x(M1 S1 + M2 S2 ) + x2 (1 + 2 ).

Letting S = (S1 , S2 ) and M = (M1 , M2 ) be two-dimensional vectors, we denote the angle 2 2 between them as (see Fig. 16). Taking into account, that 1 + 2 = sin2 , where is the angle between the vertical axis and the symmetry axis of the inertia ellipsoid, we can write Kovalevskaya integral in the form
2 F = 1 G4 sin2 + G cos + x sin 4 2 2

= k2,

G2 = M 2 . Fig. 16

Remark 4. Let's also present the interesting nonlinear transformation preserving the structure of algebra so(3): 2 2 M1 - M 2 M1 M2 K1 = , K2 = , K3 = 1 M 3 . 2 + M2 2 + M2 2 2 M1 M1 2 2 It is possible to show that, for the system Adoyer ­ Deprit canonical variables, the transformation corresponds to the canonical transformation of the type (L, l) L , 2l . 2

2.4. The historical comments
Kovalevskaya metho d. When S. V. Kovalevskaya discovered the general case of integrability she was not guided by physical reasons, instead she develop ed the ideas of K. Weierstrass, P. Painlev´ e and H. Poincar´ concerning the investigation of the analytic continuation of the solutions of a system e of ordinary differential equations into the complex plane of time. S. V. Kovalevskaya assumed that in


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integrable cases the general solution on the complex plane has no other singularities, except for p oles. This assumption allowed to obtain the conditions when the additional integral exists. In addition to the determination of the first integral, S. V. Kovalevskaya found the quite nontrivial system of variables, in which the equations have the Ab el ­ Jacobi form, and b esides she obtained the explicit solution in - functions. The reduction to quadratures in Kovalevskaya case is still considered to b e very complicated and do es not allow any essential simplification. A. M. Lyapunov in pap er [31] improved Kovalevskaya's analysis, having required for the sake of integrability the single-valuedness (the meromorphic prop erty) of the general solution as a complex function of time, and studying the solutions of variational equation. Lyapunov metho d is slightly different from Kovalevskaya's approach, which was further develop ed in pap ers by M. Adler, P. van Mo erb eke, who asso ciated the presence of the full parametric set of single-valued Laurent (p olar) expansions with the algebraic integrability of the system (in some narrow sense [1, 2]). The most complete analysis of the full parametric expansions in Euler ­ Poisson equations is contained in pap er [30]. The classical presentation of Kovalevskaya and Lyapunov results is included in several textb o oks [3, 16]. The investigations of Kovalevskaya laid the foundations of a new metho d of integrability analysis of a system, and at the same time they were the first example of the search of obstructions to integrability that evolved recently into a separate science [26]. Let's also note that in spite of the fact that there exist certain strict results concerning the relation of the branching of the general solution with the nonexistence of first integrals [26], Kovalevskaya metho d nevertheless remains as a test of integrability. It is ambiguous in many asp ects, and its application to various problems requires particular skills and additional arguments. In the physical literature this metho d is usually referred to as Painlev´ ­ e Kovalevskaya test. Kovalevskaya case, its analysis and generalizations. A geometrical interpretation of Kovalevskaya case, which is not, however, natural enough and an original metho d of reduction of Kovalevskaya case to quadratures were suggested by N. E. Zhukovsky [22]. He also used Kovalevskaya variables to construct some curvilinear co ordinates on a plane (plane M 1 , M2 ), which corresp ond to the separable variables of Kovalevskaya top. His arguments were simplified by W. Tannenb erg and K. Suslov [39, 40]. F. K¨ otter also simplified slightly the metho d of explicit integration in Kovalevskaya case [25] and suggested investigation of the motion in a frame of reference, uniformly rotating around the vertical axis. From the mo dern p oint of view, the intro duction of Kovalevskaya variables and the reduction to Ab el equations is presented in [27]. The qualitative analysis of the motion of the axis of dynamic symmetry is presented in [27], the top ological and bifurcational analysis is presented in [23]. The action-angle variables for Kovalevskaya top are constructed in [41] (see also [13]). We discuss them in our b o ok [9]. N. I. Mertsalov carried out the natural exp eriments, but he did not reveal, however, any particular prop erties of the top's motion [27, 32]. The structure of complex tori is explored in [6] with the help of algebraic geometry metho ds. The bifurcation diagrams for Kovalevskaya case in connection with Kolosoff analogy are considered in [15]. The quantization of Kovalevskaya top is a problem which is discussed from the very moment of creation of quantum mechanics (Lapp orte, 1933), but still it is not completely solved [24, 34]. In pap er [13], the Picard ­ Fuchs equation, originating from the integration of Kovalevskaya case, was written out. The first Lax representation for the n-dimensional Kovalevskaya case without a sp ectral parameter was constructed by A. M. Perelomov [33]. The representation with a sp ectral parameter in the general formulation (for motion in two homogeneous fields) was suggested by A. G. Rejman and M. A. Semenov ­ Tyan-Shansky [35]. This generalization of Kovalevskaya case is still p o orly investigated (in particular, it is not integrated in quadratures, and lacks top ological and qualitative analysis).
Remark 5. In paper [20], K. P. Hadeler and E. N. Selivanova show the family of systems on sphere S 2 , which allow an integral of the fourth degree with respect to the momentum components, which can not be


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reduced to Kovalevskaya case (or to its generalization, described by Goryachev). In paper [37], the similar construction is suggested for the systems with an integral of the third degree. Note only that in these papers no explicit form of an additional integrals was given, and the corresponding family is determined in the result of the solution of a certain differential equation, for which the existence theorems are proved.

3. Goryachev ­ Chaplygin case
Let's consider the particular integrable case of Goryachev ­ Chaplygin case, with the momentum vector situated on the horizontal plane, i. e. (M , ) = 0. It has almost the same restrictions on the dynamical a parameters, as Kovalevskaya case, but the ratio of the moments of inertia is now equal to four ( a3 = 4), 1 instead of two. The Hamiltonian and the additional integral are written as
2 2 2 H = 1 (M1 + M2 + 4M3 ) - x , 2 2 2 F = M3 (M1 + M2 ) + xM1 3 .

3.1. Explicit integration
The variables of Kovalevskaya typ e that reduce the system to Ab el ­ Jacobi equations were found by S. A. Chaplygin [10]. They are determined by formulas
2 2 M1 + M2 = 4uv ,

M3 = u - v dv = 0, P2 (v )

(3.1)

and satisfy the system of differential equations du - P1 (u)

2u du + 2v dv = dt, P1 (u) P2 (v ) P1 (u) = - u3 - 1 (h - x)u - 1 f 2 4 P2 (v ) = - v 3 - 1 (h - x)v + 1 f 2 4 u3 - 1 (h + x)u - 1 f , 2 4 v 3 - 1 (h + x)v + 1 f , 2 4

(3.2)

where h, f are the constant values of energy integal and Chaplygin integral (H = h, F = f ).
Remark. Essentially, by introducing variables u, v Chaplygin constructed the system of Andoyer ­ variables, or more precisely, the variables connected with them by the relation L = u - v , G = u + v . the generalization of Goryachev ­ Chaplygin case is constructed with the help of analysis of Andoyer ­ variables for a bundle of Poisson brackets, which include algebras so(4), e(3), so(3, 1), and the corresp separable variables are found. Deprit In [9] Deprit onding

3.2. Bifurcation diagram and phase p ortrait
Using functions P1 (u), P2 (v ) and the condition of multiplicity of these p olynomials' ro ots it is easy to construct the bifurcation diagram [23]. On the plane (f , h) it consists of three branches (Fig. 17): I. I I. I I I. f = 0, h > -1, f = t3 , t (-, +), f = t3 , t (-, +).

2 h = 3 t2 - 1, 2

h = 3 t2 + 1,



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Fig. 17. Bifurcation diagram of Goryachev ­ Chaplygin case. The nonphysical area of the integrals' values is marked by grey color. Also we indicate two levels of energy, for which the phase portraits are constructed (see Fig. 18, 19). Letters Ai , Bi , Ci , . . . denote the periodic solutions and separatrices, which are similarly denoted on phase portraits.

Fig. 18. Phase portrait of Goryachev ­ Chaplygin case for h = 0.3 (section by plane g = /2). Letters A1 , B1 , C1 denote the periodic solutions situated on the branches of the bifurcation diagram (Fig. 17). Point B1 on the bifurcation diagram, for which f = 0, corresponds, at first, to the two pendulum periodic solutions (they are situated on the phase portrait in the poles of sphere L/G = ±1 and in point l = 0, L/G = = 0) and, secondly, to the whole straight line L/G = 0, l = 0 that is also filled by periodic solutions (Goryachev solution) of pendulum type.

Fig. 19. Phase portrait of Goryachev ­ Chaplygin case for h = 1.3 (section by plane g = /2). Letters A2 , B2 , C2 , D2 , F2 denote the periodic solutions situated on the branches of the bifurcation diagram (Fig. 17). By contrast with the previous portrait the unstable solutions (and the separatrices to them) D2 and F2 are added. Also, same as above, point B2 on the bifurcation diagram corresponds to four rotational periodic solutions (rotations in equatorial and meridional plane with taking into account the direction of rotation). They are represented by points L/G = ±1 and l = 0, , L/G = 0, and by the whole straight line L/G = 0, which is filled by periodic solutions (Goryachev solutions) of reduced system.

Three p erio dic solutions b elong to the first class (I): 1) Rotations and oscillations in the equatorial plane of the inertia ellipsoid (M 1 = M3 = 0, 2 = 0); 2) Rotations and oscillations in the meridional plane of the inertia ellipsoid (M 1 = M2 = 0, 3 = 0); 3) The particular Goryachev solutions, corresp onding to f = 0. Unfortunately, the solutions situated on the branches I I, I I I are practically not investigated at all. The phase p ortraits corresp onding to various values of energy are presented in Fig. 18, 19.


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Fig. 20. Perturbation of Goryachev ­ Chaplygin case for the fixed value of energy (h = 1.5) and for the increase of the constant value of area integral (section by plane g = /2). The figures show that there is a stochastic layer near separatrices, which at first increases, and then decreases together with the area of possible motion. It is interesting that under further magnification of c, the area of possible motion decreases together with the stochastic layer up to the complete disappearence. Remark 6. The absence of explicit analytical expressions for asymptotic solutions is also an obstruction to investigation of a perturbed system. Note that N. I. Mertsalov in paper [32] has made the statement concerning the integrability of Goryachev ­ Chaplygin top equations for c = (M , ) = 0. The computer experiments presented in Fig. 20, show that this statement is wrong, and there is a stochastic layer near unstable manifolds for c = 0, which implies nonintegrability.

3.3. Visualization of particulary sp ecial solutions
Among the p erio dic solutions in Goryachev ­ Chaplygin problem Goryachev solution is very sp ecial. On the bifurcation diagram it is situated on straight line f = 0, b esides this line contains the p erio dic solutions of Euler ­ Poisson equations that corresp ond to the oscillations (for h < 1) and rotations (h > 1) of the rigid b o dy in planes O xy and O xz , ob eying the comp ound p endulum law. Let's discuss in detail Goryachev solution and the solutions situated on branches I I and I I I (see Fig. 17). Goryachev solution. For this solution there are two additional invariant relations [12]
2/3 1

2 2 M1 + M2 = bM

,

2 2 f = M3 (M1 + M2 ) + M1 3 = 0,

(b > 0).

(3.3)



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Fig. 21. "Goryachev solution" represents the whole torus, filled by periodic solutions of the reduced system (M , ) (the so called resonance 1 : 1); for h < 1 they are pendulum type solutions, and for h > 1 they are rotary type solutions. On this and the following figures the tra jectories on the Poisson sphere corresponding to various solutions on this torus are presented.

Fig. 22. This figure illustrates the behavior of the principal for Goryachev solutions at a fixed value of energy h < 1 (h are periodic in the absolute space, which with the increase Oxy to oscillations in plane Oxz . (Letters x, y , z denote the

axes of a rigid body in the fixed frame of reference = -0.7). It can be clearly seen that this solutions of parameter b changes from oscillations in plane axes bound with the body.)



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Fig. 23. This figure illustrate periodic motion in the absolute motion of the principal axis Oy for Goryachev solution for h > 1

the quasi- Fig. 24. Motion of the vertical unit vector on the Poisson sphere space (the for the stable periodic motion in Goryachev ­ Chaplygin case for is shown) various values of energy. (h = 1.7).

Fig. 25. Motion of the vertical unit vector on the Poisson sphere for the unstable periodic solution in Goryachev ­ Chaplygin case for various values of energy.

An arbitrary constant b in these relations sp ecifies a parameterization of the whole set of p erio dic solutions: in the phase space it is a degenerated torus filled by p erio dic solutions. Relations (3.3) were obtained by D. N. Goryachev, which made S. A. Chaplygin understand that the condition f = 0 is to o strong and obtained solution (3.2) in the conventional form. For h < 1 and with increase of b from 0 up to bmax , the solution changes from oscillation in the equatorial plane to oscillation in the meridional plane (Fig. 17). On the phase p ortrait (see Fig. 18) it corresp onds to straight line L/G = 0 and to the meridian connecting it with p oles. For h > 1 and with the increase of b from 0 up to b max the solution changes from a rotation in the equatorial plane to another one (in the opp osite direction, Fig. 19). The motion of the ap ex on the Poisson sphere is presented in Fig. 17. The remarkable phenomenon, which was unnoticed earlier, is the fact that for Goryachev solutions in the absolute space for h < 1, the motion is p erio dic one of oscillatory typ e (see Fig. 19). And for h > 1 the corresp onding motion is quasip erio dic and bifrequency (Fig. 20). All indicated facts practically cannot b e seen immediately from the analytical solution, which for the first time was obtained by Goryachev in a very cumb ersome form [19]. Despite of some


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Fig. 26. Motion of case for the stable different points of the same values of

the apexes of the principal axes of a rigid body in the absolute space in Goryachev ­ Chaplygin periodic solution on branch III in Fig. 17, for two various values of energy h 1 , h2 from the view. Letters xi , yi , zi , i = 1, 2 denote the tra jectories of the relevant axes corresponding to energy.

simplifications available, for example, in [12], the explicit formulas only partially allow to understand the character of the motions, obtained by the computer metho ds.

Fig. 27. Motion of the apexes of the principal axes of a rigid body in the absolute space in Goryachev ­ Chaplygin case for the unstable periodic solution on branch II in Fig. 17, for a single value of an energy. Letters x, y , z denote the tra jectories of the corresponding axes. (Motion for other values of energy do not differ qualitatively, therefore we do not present them.)

The stable and unstable p erio dic solutions of Euler ­ Poisson equations for Goryachev ­ Chaplygin case are situated in the bifurcation diagram on branches I I I and I I, accordingly (see Fig. 17, 24­27). The numerical investigations show that the motions of the initial system in the absolute space, corresp onding to these solutions are also periodic at any value of energy (see Fig. 26, 27). This fact, apparently, can not b e found in the present literature and reflects the sp ecific character of the rigid b o dy dynamics for the zero value of area integral (M , ) = 0 (compare with Delone solutions for Kovalevskaya case, [9]). Instead of the formal pro of we present a series of figures visually confirming this statement. On them the tra jectories of system are presented b oth on the Poisson sphere and as the tra jectories of the ap exes in the absolute space, the ma jority of them are complicated enough.


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The general conclusion about Goryachev ­ Chaplygin case is the observation that in its analysis we deal with interesting oscil latory (rotary ) motions in the absolute space, i. e. it is possible to speak about a certain complicated pendulum. However, the area of application of such oscil lations, is not very clear yet. Note also the comparative simplicity of Goryachev ­ Chaplygin top's motions in comparison with those of Kovalevskaya top. Few analytical results obtained by study of Goryachev ­ Chaplygin case cannot give the visual representation of the motion. On the contrary, the computer investigation of the motion discovers its remarkable properties that are also typical for the related integrable systems.

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1

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