the dynamics of the chaplygin ball with a fluid-filled cavity

ISSN 1560-3547, Regular and Chaotic Dynamics, 2013, Vol. 18, No. 5, pp. 490–496. c© Pleiades Publishing, Ltd., 2013.

The Dynamics of the Chaplygin Ball with a Fluid-filled Cavity

Alexey V. Borisov* and Ivan S. Mamaev**

Institute of Computer Science;Laboratory of Nonlinear Analysis and the Design of New Types of Vehicles,

Udmurt State University,ul. Universitetskaya 1, Izhevsk, 426034 Russia

A.A.Blagonravov Mechanical Engineering Research Institute of RAS,ul. Bardina 4, Moscow, 117334 Russia

Institute of Mathematics and Mechanics of the Ural Branch of RAS,ul. S. Kovalevskoi 16, Yekaterinburg, 620990 RussiaReceived November 25, 2011; accepted January 18, 2012

Abstract—We consider the problem of rolling of a ball with an ellipsoidal cavity filled withan ideal fluid, which executes a uniform vortex motion, on an absolutely rough plane. We pointout the case of existence of an invariant measure and show that there is a particular case ofintegrability under conditions of axial symmetry.

MSC2010 numbers: 70E18, 76B47

DOI: 10.1134/S156035471305002X

Keywords: vortex motion, nonholonomic constraint, Chaplygin ball, invariant measure, integra-bility, rigid body, ideal fluid

Dedicated to our esteemed friend and outstanding scientistSergey V.Gonchenko on the occasion of his 60th birthday.

Contents

INTRODUCTION 490

1 EQUATIONS OF MOTION 491

2 FIRST INTEGRALS AND AN INVARIANT MEASURE 493

3 AN INTEGRABLE CASE OF AXIAL SYMMETRY 494

ACKNOWLEDGMENTS 496

REFERENCES 496

INTRODUCTIONThe problem of motion of a rigid body with cavities filled with a fluid executing potential motion

in the case of nonsimply connected cavities and uniform vortex motion in the case of ellipsoidalcavities was first dealt with by N. Y. Zhukovsky [6]. This problem, formulated in different ways, waselaborated upon by A. Poincare [14], V. A. Steklov [17, 18], V. Volterra [21], and others. A reviewof these classical treatments can be found in [3] and the recently published collection [11] (for adiscussion of the dynamics of bodies with fluid-filled cavities, see also [9] and references therein).Theoretical studies on these topics were dictated by the need to substantiate the phenomenon ofprecessional motions of celestial bodies, in particular, the Earth, under the assumption that thebody consists of a hard shell (mantle) surrounding the liquid core. From recent works we pointout [16], where the model problem for research into Mercury’s librations is considered.

*E-mail: [email protected]**E-mail: [email protected]

490

THE DYNAMICS OF THE CHAPLYGIN BALL WITH A FLUID-FILLED CAVITY 491

We also note that the investigation of the dynamics of a body with nonsimply connected cavitiesfilled with an ideal fluid was one of the reasons why the notion of cyclic variables was introduced andthe Routh – Kelvin reduction procedure was developed. In addition, the problem of motion of a bodywith an ellipsoidal cavity led A. Poincare to obtain a new form of equations in quasi-velocities [15](Poincare’s equations on the Lie group).

Another class of problems associated with the above-mentioned problem concerns the dynamicsof a top and goes back to William Thomson (Lord Kelvin), who took a great interest in designingand experiments with various models of tops with dynamical effects [19, 20] (the best known andunusual effect discovered by him is that of a rising top, called Thomson’s top).1) Various experimentswith tops are described in detail in the book of J. Perry [13]. For example, it is well known that ifa boiled egg is spun fast enough on its side it will rise up along the longer axis and spin on end.But an unboiled egg can by no means be induced to rise up and spin along the longer axis. Thisillustrative example naturally brings us to the question of behavior of a body with a fluid-filledcavity on a plane. Recent research in this vein includes [7, 8, 10, 12] dealing with the problem of abody rolling on a plane and having an ellipsoidal cavity filled with an ideal fluid executing uniformvortex motion. These publications explore the stability of various steady motions of the system.In addition, the paper [12] addresses the dynamics of the Chaplygin ball with a fluid-filled cavityin the case where the friction force at the point of contact produces no torque. It is shown thatthe angular momentum relative to the point of contact and the area integral are preserved, whilethe energy does not increase. In [7] it is also shown that in the case of a spherical shell and anaxisymmetric cavity the Jellett integral is preserved for an arbitrary law of friction between theshell and the plane in the absence of friction torque.

In this note we consider the problem of rolling of a spherical shell with an ellipsoidal cavitycontaining an ideal fluid on an absolutely rough plane. We show that in the case where the centerof mass of the system coincides with the geometric center of the shell (i.e., for the Chaplygin ballwith a fluid-filled cavity), the equations of motion admit an invariant measure; furthermore, if theshell’s mass distribution and the cavity are axisymmetric about the same axis, then there is aninvariant submanifold where the equations of motion can be integrated by quadratures.

1. EQUATIONS OF MOTION

Fig. 1

Consider a generalization of the Chaplygin problemof a dynamically asymmetric ball rolling on a horizontalplane. Assume that the ball contains an ellipsoidal cavityfilled with an ideal fluid executing motion with uniformvorticity. Choose a moving coordinate system Gx1x2x3whose origin coincides with the center of mass of thesystem and whose axes are aligned with the principalaxes of inertia (see Fig. 1). In this coordinate system theequation for the cavity can be represented as

(x − xc,B−2(x − xc)) � 1, (1.1)

where B2 is a symmetric matrix whose eigenvalues co-incide with the squares of the principal semi-axes of thecavity b2

1, b22, b2

3.Following A. Poincare, we represent in the coordinate

system Gx1x2x3 the distribution of absolute velocity ofthe flow of a fluid in a cavity as [5]

v(x, t) = V + ω × x + BΞB−1(x − xc), (1.2)

where V and ω are the velocity of the center of mass and the angular velocity of the ball, and Ξ isthe skew-symmetric matrix.

1)See Chapter 18 of the book “The Life of William Thomson, Baron Kelvin of Largs” [20] and the article byW.Thomson [19].

REGULAR AND CHAOTIC DYNAMICS Vol. 18 No. 5 2013

492 BORISOV, MAMAEV

The no-slip condition at the point of contact Q is written as

V + ω × r = 0, r = a − Rγ, (1.3)

where r is the vector from the center of mass to the point of contact, a is the vector from the centerof mass to the center of mass of the ball, and γ is the unit vector of the vertical.

We define the vector ξ corresponding to the matrix Ξ whose components in the coordinatesystem Gx1x2x3 are given by the relation

ξk = −εkijΞij.

The equations of motion for this system are represented as [5](∂T

∂V

)·+ ω × ∂T

∂V= −mgγ + N ,

(∂T

∂ω

)·+ ω × ∂T

∂ω= r × N ,

(∂T

∂ξ

)·− ξ × ∂T

∂ξ= 0,

(1.4)

where N is the reaction of the constraint, T is the kinetic energy of the system (with no constraintimposed), m is the mass of the ball with the fluid, and g is the acceleration of the gravitationalfield.

Remark. Given the dependence ξ(t), we can obtain the trajectories of fluid particles in the cavityrelative to its center using (1.2). Indeed, let us perform the linear change of variables

x′ = B−1(x − xc),

which transforms the ellipsoidal cavity into a sphere. In this case, in the coordinate system attachedto the shell the corresponding velocities are determined by the relation

v′ = B−1(v − ω × x) = ξ × x′,

that is, the vortex flow of the fluid in the cavity corresponds to the rotation of some (imaginary)sphere (x′,x′) � 1 with angular velocity ξ(t).

Calculating the kinetic energy of the system, we find

T =12mV 2 +

12(ω, Iω) +

12

Tr(BΞB−1J(BΞB−1)T

)− Tr

(BΞB−1JΩ

), (1.5)

the components of the matrices Ω and J are defined by

Ωij = −εijkωk, Jij =∫

Cavity

(x − xc)i(x − xc)jρ dV =mc

5

∑k

BikBjk,

where the integration extends over the entire cavity with a fluid of density ρ, and Bij are theelements of the matrix B, and mc is the mass of the fluid in the cavity. Due to the symmetry of B,we have J = mc

5 B2, whence

T =12(mV 2 + (ω, Iω)

)− mc

10Tr(Ξ2B2) − mc

5Tr(BΞB2Ω)

=12(mV 2 + (ω, Iω)

)+ (ξ,B′ω) +

12(ξ,B′′ξ),

B′ =2mc detB

5B−1, B′′ =

mc

5((TrB2)E − B2

).

(1.6)

In the case where the principal axes of the cavity and the principal axes of inertia of the systemcoincide

B′ = diag(

25mcb2b3,

25mcb1b3,

25mcb1b2

), B′′ = diag

(mc

5(b2

2 + b23),

mc

5(b2

1 + b23),

mc

5(b2

1 + b22)

).



Eliminating the reaction of the constraint N with the help of the first of Eqs. (1.4) and theconstraint (1.3), we obtain the following system of equations:(

∂T

∂ξ

)·= ξ × ∂T

∂ξ,

(∂T

∂ω+ mr × (ω × r)

)·

=(

∂T

∂ω+ mr × (ω × r)

)× ω + mr × (ω × r) + mgr × γ.

(1.7)

In order to obtain a closed system of equations, we have to add the relation

γ = γ × ω. (1.8)

Remark. We show how the first of Eqs. (1.7) is related to the Helmholtz equation for the vorticityωc in the fluid. Using the relations (1.2) and the representation (1.6), we can show that the vorticityin the moving axes is given by

ωc =12

rot v = ω +1

2detB((Tr B2)E − B2

)Bξ = (B′)−1 ∂T

∂ξ.

Hence, using the definition of B′, we find

B−1ωc = k∂T

∂ξ, k =

52mc detB

= const. (1.9)

By the Helmholtz equation, in the moving axes Gx1x2x3 (see [9]) rotating with angular velocity ω,we have

ωc + ω × ωc = (ωc,∇)v(x, t) = Ωωc + BΞB−1ωc.

Using the relations Ωa = ω × a, Ξa = ξ × a, where a is an arbitrary vector, we obtain theequation of motion for ωc in the form(

B−1ωc

)· = ξ × (B−1ωc).

According to (1.9), this equation is identical, up to the factor, with the first equation in (1.7).

2. FIRST INTEGRALS AND AN INVARIANT MEASUREIn the general case the system (1.7)–(1.8) admits obvious three first integrals

geometric F0 = (γ,γ) = 1,

magnitudeof vorticity F1 =

(∂T

∂ξ,∂T

∂ξ

),

energy E = T (ξ,ω) − mg(r,γ),

where T (ξ,ω) = T (V , ξ,ω)∣∣V =r×ω

.If we also set a = 0, the system of Eqs. (1.7)–(1.8) admits an invariant measure ρ dξ dω dγ,

where the density is given by

ρ(γ) =

(det

∥∥∥∥ ∂T

∂zi ∂zj

∥∥∥∥)1

2

,

where a = (ξ,ω) is a six-dimensional vector.In addition, as follows from (1.7), for a = 0 the angular momentum vector of the system relative

to the point of contact

M =∂T

∂ω=

∂T

∂ω+ mr × (ω × r)

remains constant in the fixed coordinate system (since M = M × ω). Hence, a pair of additionalfirst integrals appears in the system (1.7)–(1.8) when a = 0:


494 BORISOV, MAMAEV

square of the angular momentum F2 = (M ,M),area integral F3 = (M ,γ).

Analogous integrals for the case where there is friction without torque between the shell and theplane are found in [12].

The equations of motion (1.7)–(1.8) are similar in form to the equations of other well-knownintegrable nonholonomic systems [4], for which the integrability is established by the generalizedEuler – Jacobi theorem [1], and the corresponding invariant manifolds are two-dimensional tori .Nevertheless, the mechanism of integrability must apparently be different in this case. Indeed, seta = 0 and consider the quantity m in Eqs. (1.7) as an independent parameter, setting it to be equalto zero: m = 0; we obtain

K = ξ × K, M = M × ω,

K = B′′ξ + B′ω, M = Iω + B′ξ.(2.1)

These equations can be represented in Hamiltonian form with the degenerate Lie –Poisson bracketcorresponding to the algebra so(4) = so(3) ⊕ so(3); the well-known integrable cases of this systemare presented in [5]. In integrable cases the invariant manifolds of the system (2.1) are two-dimensional tori; hence, for a complete system which also contains the equation for γ

γ = γ × ω,

the integral manifolds are three-dimensional tori .Thus, one would expect that in the initial system (1.7)–(1.8) the nondegenerate integral

manifolds should also be three-dimensional tori for m �= 0 in the case of preservation of the invariantmeasure. In particular, for the system to be integrable when a = 0, it is necessary to find anadditional first integral. Such a situation arises in nonholonomic systems discussed in [2].

3. AN INTEGRABLE CASE OF AXIAL SYMMETRY

We shall assume that the following conditions are satisfied:

1. the center of mass of the system coincides with the geometric center of the shell, that is,

a = 0;

2. the shell and the cavity are axisymmetric about the same axis, and hence in the system ofprincipal axes

I = diag(I1, I1, I3), B = diag(b1, b1, b3);

3. the square of the angular momentum of the system relative to the point of contact is zero

M21 + M2

2 + M23 = 0.

Obviously, the last relation implies that each single component of the angular momentum vectoris equal to zero:

M1 = M2 = M3 = 0. (3.1)

We restrict the equations of motion to the invariant manifold defined by (3.1). To do this, weexpress the vorticity ξ and the moment of vorticity K as

ξ = −(B′)−1(Iω + Dγ × (ω × γ)

), D = mR2,

K =∂T

∂ξ= B′ω − B′′(B′)−1

(Iω + Dγ × (ω × γ)

)

and substitute into the equations

K = ξ × K, γ = γ × ω. (3.2)



The system of equations thus obtained admits the vector symmetry field

vs = ω2∂

∂ω1− ω1

∂

∂ω2+ γ2

∂

∂γ1− γ1

∂

∂γ2. (3.3)

Choose a system of invariants (first integrals) of this vector field in the form γ3, γ2, K3, Kn, K2,where

Kn = Jn(ω,γ),

J2n = Ic

1(I1 − Ic3) − Ic

3(Ic1 − Ic

3) −D

(Ic1(I1 − I3) − Ic

3(Ic1 − Ic

3))

I3 + D − Ic3

γ23 ;

here Ic1 = mc

5 (b21 + b2

3), Ic3 = 2

5mcb21 are the principal moments of inertia of the cavity relative to its

center.

Remark 1. The function Jn(γ3) identically vanishes only under the condition

I3 = Ic3,

which is possible only in the case of a weightless shell.

In terms of these variables, the equations of motion become(γ2

)· =(K2

)· = 0, γ3 = γ1ω2 − ω2γ1,

K3 = (γ1ω2 − ω2γ1)D

JnKn, Kn = (γ1ω2 − ω2γ1)

D

JnK3,

D =Ic3J1J3 + 2Ic

1Ic3J3 − Ic

1J1J3 − (Ic3)

2(J3 + J1)Ic3(J3 − Ic

3),

where the notation Ji = Ii + D has been introduced for brevity.Dividing the last two equations by γ3, we obtain the system

dK3

dγ3=

D

Jn(γ3)Kn,

dKn

dγ3=

D

Jn(γ3)K3. (3.4)

The constants of integration of these equations define a pair of first integrals linear in the angularvelocity in the initial system (3.2).

Remark. Equations (3.4) admit a simple quadratic first integral of the form

F = DK2n − DK2

3 .

Expressing the energy integral in terms of the functions γ3, γ3, K3, Kn, we obtain

E =425

mcb21b

23(E − mgR) =

12(1 − γ2

3)−1

[J1

(Ic1(J1 − Ic

3) − Ic3(I

c1 − Ic

3))γ23

+

(J3(2Ic

1 − Ic3)

J3 − Ic3

(1 − γ23) +

J1

(Ic1(J1 − Ic

3) − Ic3(I

c1 − Ic

3))

(J3 − Ic3)2

γ23

)K2

3

+(

J1 − D(1 − γ23) − DJ1

J3 − Ic3

γ23

)K2

n + 2J1γ3Jn(γ3)

J3 − Ic3

KnK3

].

Substituting the general solution of (3.4) into this equation and expressing γ23 from the resulting

equation, we obtain the gyroscopic function of the system under consideration.

Remark. In the case of axial symmetry the equations of motion of the general system (1.7)–(1.8)also admit a symmetry field analogous to (3.3), however, the system reduced to a symmetry field isnonintegrable and exhibits chaotic behavior. The problem of integrability of the system (1.7)–(1.8)with zero angular momentum without axial symmetry remains an open problem.


496 BORISOV, MAMAEV

ACKNOWLEDGMENTS

This work was carried out at the Udmurt State University and was supported by Grant ofthe President of the Russian Federation for Support of Leading Scientific Schools NSh-2519.2012.1“Dynamical Systems of Classical Mechanics and Control Problems”, Analytic Departmental TargetProgram “Development of Scientific Potential of Higher Schools” (1.1248.2011), Analytic Depart-mental Target Program “Development of Scientific Potential of Higher Schools” (1.7734.2013),Federal Target Program “Scientific and Scientific-Pedagogical Personnel of Innovative Russia”(Agreement No14.A37.21.1935).

REFERENCES1. Bolsinov, A. V., Borisov, A. V., and Mamaev, I. S., Hamiltonisation of Non-Holonomic Systems in the

Neighborhood of Invariant Manifolds, Nelin. Dinam., 2010, vol. 6, no. 4, pp. 829–854 (Russian).2. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., Rolling of a Homogeneous Ball over a Dynamically

Asymmetric Sphere, Nelin. Dinam., 2010, vol. 6, no. 4, pp. 869–889 (Russian).3. Borisov, A. V., Gazizullina, L. A., and Mamaev, I. S., On V. A. Steklov’s Legacy in Classical Mechanics,

Nelin. Dinam., 2011, vol. 7, no. 2, pp. 389–403 (Russian).4. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., Generalized Chaplygins Transformation and Explicit

Integration of a System with a Spherical Support, Nelin. Dinam., 2011, vol. 7, no. 2, pp. 313–338(Russian).

5. Borisov, A. V. and Mamaev, I. S., Rigid Body Dynamics, Moscow–Izhevsk: R&C Dynamics, Institute ofComputer Science, 2001 (Russian).

6. Zhukovskii, N. E., Motion of a Rigid Body Containing a Cavity Filled with a Homogeneous ContinuousLiquid, in Collected Works: Vol. 2, Moscow: Gostekhteorizdat, 1949, pp. 31–152 (Russian).

7. Karapetyan, A. V. and Prokomina, O. V., The Stability of Permanent Rotations of a Top with a CavityFilled with Liquid on a Plane with Friction, Prikl. Mat. Mekh., 2000, vol. 64, no. 1, pp. 85–91 [J. Appl.Math. Mech., 2000, vol. 64, no. 1, pp. 81–86].

8. Markeev, A. P., The Stability of the Rotation of a Top with a Cavity Filled with Liquid, Izv. Akad. NaukSSSR. Mekh. Tverd. Tela, 1985, vol. 20, no. 3, pp. 19–26 (Russian).

9. Moiseev, N. N. and Rumiantsev, V. V., Dynamic Stability of Bodies Containing Fluid, New York:Springer, 1968.

10. Rudenko, T. V., The Stability of the Steady Motion of a Gyrostat with a Liquid in a Cavity, Prikl. Mat.Mekh., 2002, vol. 66, no. 2, pp. 183–191 [J. Appl. Math. Mech., 2002, vol. 66, no. 2, pp. 171–178].

11. Steklov, V. A., Works on Mechanics 1902-1909: Translations from French, Moscow–Izhevsk: R&CDynamics, Institute of Computer Science, 2011 (Russian).

12. Liu, Y. Z., The Stability of a Fluid-Filled Top Rotating on a Horizontal Plane, Arch. Appl. Mech., 1992,vol. 62, pp. 487–494.

13. Perry, J., Spinning Top and Gyroscopic Motions, New York: Dover, 1957.14. Poincare, H., Sur la precession des corps deformables, Bull. Astron., 1910, vol. 27, pp. 321–356.15. Poincare, H., Sur le forme nouvelle des equations de la mecanique, C. R. Acad. Sci. Paris, 1901, vol. 132,

pp. 369–371.16. Rambaux, N., Van Hoolst, T., Dehant, V., and Bois, E., Inertial Core-Mantle Coupling and Libration

of Mercury, Astron. Astrophys., 2007, vol. 468, no. 2, pp. 711–719.17. Stekloff, V. A., Sur la theorie des tourbillons, Ann. Fac. Sci. Toulouse Math. (2), 1908, vol. 10, pp. 271–

334.18. Stekloff, V. A., Sur le movement d’un corps solide ayant une cavite de forme ellipsoidale remplie par un

liquide incompressible et sur les variations des latitudes, Ann. Fac. Sci. Toulouse Math. (3), 1909, vol. 1,pp. 145–256.

19. Thompson, S. Ph., The Life of William Thomson, Baron Kelvin of Largs: Vol. 2, London: McMillan,1910, pp. 736–752.

20. Thomson, W., On the Precessional Motion of a Liquid, Nature, 1877, vol. 15, pp. 297–298.21. Volterra, V., Sur la theorie des variations des latitedes, Acta. Math., 1899, vol. 22, pp. 201–358.


the dynamics of the chaplygin ball with a fluid-filled cavity

Documents