Teodorescu P.P. Mechanical Systems, Classical Models Volume II: Mechanics of Discrete and Continuous Systems

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15 Dynamics of the Rigid Solid with a Fixed Point

()

()()

()

12 1 2 1 2

ss sese sese−+ − −± − −

⎡⎤

⎣⎦

()

11 22

sese sese=− −±− −

⎡⎤

⎣⎦

the relations (15.2.28) allow to write

()

31 3

11 22

sese sese

ωγα−

=−−+−−

⎡⎤

⎣⎦

−

()

32 3

11 22

sese sese

ωγα−

=−−−−−

⎡⎤

⎣⎦

−

(15.2.36)

Taking into account the relations (15.2.26'), (15.2.32) and (15.2.36), we get

()

seset

=−−

−

()

seset

=− − −

−

Introducing the polynomial of fifth degree

()

()()()

()

123

() ()s se se s se se se se seΦϕ=− − =− − − − −

(15.2.37)

and effecting a linear combination, it results

() ()

ΦΦ

() ()

11 22

2id

ss ss

ΦΦ

(15.2.37')

We can apply Jacobi’s theory of the last multiplier to this system of differential

equations. If

()

∫

()

∫

(15.2.38)

then we get

() ()

121

Fs Fs C+=,

() ()

12 2

iGs Gs t C+=−+,

(15.2.38')

where

C and

C are integration constants. Finally, it results

()

11 12

;,sstCC= ,

()

22 12

;,sstCC= .

If an integrand is a rational function of the form

()

,()Ss sΦ , where ()sΦ is a

polynomial of degree greater than four, the corresponding integral is called, in general,

hyperelliptic integral; if the degree of the polynomial is five or six, the respective

integrals are called also ultraelliptic integrals. In the Sonya Kovalevsky case, the

363

MECHANICAL SYSTEMS, CLASSICAL MODELS

problem of the rotation of a heavy rigid solid about a fixed point is reduced thus to the

inversion of a system of two ultraelliptic integrals.

If the roots

123

,,eee are distinct and if ()sϕ

′

is the derivative of the function ()sϕ ,

then one can calculate the components of the vector

ω, contained in the equatorial plane

of the ellipsoid of inertia, in the form

()

′

=−

′

∑

()

′

∑

αβγα≠≠≠, βγ< , ,, 1,2,3αβγ= ,

(15.2.39)

where we have introduced the functions

()()

Psese

ααα

=− −, 1, 2, 3α = ,

(15.2.39')

symmetric with respect to

s and

s , hence which can be expressed by means of the

theta elliptic functions. Analogously, we get

()

βγ

′

∑

()

βγ

′

=−

′

∑

αβγα≠≠≠, βγ< , ,, 1,2,3αβγ= ,

(15.2.40)

with

()

11 22

sese sese

βγ γβ

γγ

ββ

ΦΦ

⎡⎤

== −

⎢⎥

−

−− −−

⎣⎦

βγ≠

,1,2,3βγ=

(15.2.40')

The direction cosines are then given by the first two first integrals (15.2.20).

15.2.3 Other Cases of Integrability

The success of Sonya Kovalevsky is due, especially, to a new more general formulation

of the problem of motion of the rigid solid with a fixed point by means of the concepts

of the theory of analytic functions of complex variable. The general solution can be

expressed, in this case, with the aid of the elliptic functions of the time

t; the elliptic

functions are uniform analytic functions for all finite values of

t, excepting some points

in the complex plane

t, in which these functions have poles of first order. In this order

of ideas, Kovalevsky has worked to the problem of finding all the cases in which the

364

15 Dynamics of the Rigid Solid with a Fixed Point

general solution, which contains five arbitrary constants, is uniform and has no other

singularities, excepting the poles, for all the finite values of

t, t being a complex

variable; this solution has been searched in the form of the expansion into a power

series

()

ωω

∞

∑

()

αα

∞

∑

, ,

nm∈  , 1,2, 3i = ,

(15.2.41)

where the coefficients

()

ω and

()

α must satisfy some conditions. Sonya Kovalevsky

has chosen the values

n = and 2

m = ,

1,2, 3i =

, the problem of uniqueness of

this system of values remaining open.

Kovalevsky said that only the three cases of integrability considered above are

possible, as well as the case of kinetic symmetry in which the ellipsoid of inertia is a

sphere (

123

IIII===). In the latter case, starting from Euler’s equation (15.1.21'')

and by means of a scalar product by

ρ, we may write

()

[

]

d/d0

t⋅=ω

I ,

wherefrom

()

const

⋅ω

=I ; the tensor

I being spheric, we have

hωρ Ω⋅= =ω

, ,consthΩ = , (15.2.42)

hence a fourth algebraic first integral. Taking into account that any co-ordinate axis is a

principal axis of inertia, that case is a Lagrange-Poisson one. These statements have

been justified, using other methods than those of Sonya Kovalevsky, by A.M.

Lyapunov and G.G. Appelrot, the general theorems concerning the uniformity of the

solutions being considered in what follows.

We will present also the most important particular cases of integrability, as well as

other cases of loading in the dynamics of the rigid solid with a fixed point.

15.2.3.1 General Theorems Concerning the Uniformity of the Solution

We consider that the four cases of integrability mentioned above (the Euler-Poinsot

case, the Lagrange-Poisson case, the Sonya Kovalevsky case and the case of kinetic

symmetry) are the classical cases of integrability. In what concerns the uniformity of

the solution, we can – firstly – state

Theorem 15.2.3 (S. Kovalevsky) In general, excepting the classical cases of

integrability, the equations of Euler and Poisson do not allow uniform solutions which

depend on five arbitrary constants and have not other singularities excepting poles in

the whole complex plane

We notice that Sonya Kovalevsky did not consider also the case in which the

solutions can be uniform, having essential singularities besides the poles. Lyapunov

showed that such solutions cannot be uniform on the whole complex plane t (excepting

the four mentioned cases) because, by a convenient choice of the initial values, they can

become multiform functions of the time

t. We can thus state

Theorem 15.2.4 (A.M. Lyapunov) If the principal moments of inertia

I , 1, 2, 3j = ,

are real and non-zero quantities and if the co-ordinates

ρ ,

1,2, 3i =

, of the centre of

365

MECHANICAL SYSTEMS, CLASSICAL MODELS

mass are real quantities too, then the classical cases of integrability are the only cases

in which the functions

()

tω and ()

tα , 1, 2, 3i = , are uniform functions of the time t

for arbitrary initial values of these unknowns.

We consider the particular solutions

ω =

α =

1,2, 3i =

(15.2.43)

where the constants

a ,

b are not all zero; replacing in the equations of Euler and

Poisson, we find the conditions

() ( )

11 2 3 23 32 2 3

Ia I I aa Mg b bρρ+− = − ,

() ( )

22 3 1 31 13 31

Ia I I aa Mg b bρρ+− = − ,

() ( )

33 1 2 12 21 12

Ia I I aa Mg b bρρ+− = − ,

(15.2.43')

ijl l

bba+∈ = ,

1,2, 3i =

. (15.2.43'')

If the ellipsoid of inertia relative to the pole

O is not of rotation, one uses the particular

solution

()()

1213

IIII

=−

−−

()()

1223

IIII

=−

−−

()()

1323

IIII

=−

−−

(15.2.44)

123

0bbb===. (15.2.44')

In the case in which the ellipsoid of inertia is of rotation, one of the components

123

,,ρρρ vanishes and we can use the solution (for

0ρ = )

0aa==

2ia =

()

ρρ

0b =

(15.2.45)

0ρ =

, then one uses analogous solutions.

To the initial moment

tt= correspond the initial conditions

ωω=

and

αα= , 1, 2, 3i = ; in this case, the general solution of the system (15.1.21),

(14.1.54) varies continuously with these parameters. At a variation of at least one of

these parameters correspond variations of the solution, which will be of the form

ωδω+ ,

αδα+ , 1, 2, 3i = . These new solutions must verify the considered

system of equations; subtracting now the latter system thus obtained, we find the

conditions which must be verified by the variations

δω and

δα , 1,2, 3i = , in the

form

()

()( )( )

13223323223

III Mg

δω

ω δω ω δω ρ δα ρ δα+− + = −

366

15 Dynamics of the Rigid Solid with a Fixed Point

()

()( )( )

21331131331

III Mg

δω

ω δω ω δω ρ δα ρ δα+− + = −

()

()( )( )

32112212112

III Mg

δω

ω δω ω δω ρ δα ρ δα+− + = −

(15.2.46)

()

ijk k k

δα

ωδα αδω+∈ + =

, 1, 2,3i = .

(15.2.46')

Replacing the particular solutions (15.2.43), we obtain the system

()

()( )( )

13223323223

It I I a a Mgt

δω

δω δω ρ δα ρ δα+− + = −

()

()( )( )

21331131331

It I I a a Mgt

δω

δω δω ρ δα ρ δα+− + = −

()

()( )( )

32112212112

It I I a a Mgt

δω

δω δω ρ δα ρ δα+− + = −

(15.2.47)

()

ijk k k

tatb

δα

δα δω+∈ + =

, 1, 2, 3i = .

(15.2.47')

This system allows non-zero solutions of the form

ktδω =

ktδα

−

1,2, 3i =

(the system of six linear equations in

k ,

1,2, 3i =

, has non-trivial

solutions) if the exponent

k is a root of the equation of six degree

()()

() ()

()()

1323322 32

133 2 131 3 1

212 211 3 2 1

32 32

3131

21 21

I k I I a I I a Mg Mg

I I a I k I I a Mg Mg

I I a I I a I k Mg Mg

bbkaa

bbaka

bb aak

ρρ

−− −

≡=

−−−

(15.2.48)

We notice that

ω and

α are uniform functions if the variations

δω and

δα have the

same property too; in this case, it is necessary and sufficient that the roots of the

equation (15.2.48) be integers (

k ∈  ) and that a multiple root of mth order equates to

zero all the minors of an order greater than

m of the determinant Δ.

We assume firstly that

123

0III>>>. Using the solution (15.2.44), (15.2.44'),

the equation (15.2.48) takes the form (product of two minors of third order)

()

140kk k−−=

(15.2.48')

The roots of this equation are

0,1, 2±∈ ; on the other hand, all the minors of fourth

and fifth order must vanish for the triple root

1k =

. Thus, the minor of fourth order,

367

MECHANICAL SYSTEMS, CLASSICAL MODELS

obtained by eliminating the second and the third columns and the third and the fourth

lines, leads to (for

1k = )

()

132

133 3 1

IMgMg

IIaMg Mg

ρρ

−

−−

−

Developing and taking into account (15.2.44), we get

() () ()

112 3 223 1 331 2

0II I II I II Iρρρ−+ −+ −=

(15.2.49)

The middle term being imaginary (we take into account the relation of order of the

moments of inertia), it results

0ρ = ,

() ()

112 3 331 2

0II I II Iρρ−+ −=.

(15.2.49')

This case, which has not been noticed by Kovalevsky and which has been put in

evidence by Appelrot in 1892, has been found for the first time by W. Hess in 1890.

Afterwards, N.E. Jukovskiĭ, B.K. Mlodzeevski and P.A. Nekrasov have made studies in

this direction, the latter one showing that for

,0ρρ≠

one obtains multiform

solutions of the time

t if one chooses convenient values for the initial conditions.

Indeed, for

0ρ = and for the solutions (15.2.45), the equation (15.2.48) leads to

(permuting lines and columns, to can represent

Δ as a product of two determinants of

third order)

()

(

)

()

231

21 3 1

31 31

12i 0

2i 1

22i

I k Mg Mg

Ik Mg

II Ik Mg

Mg Mg

ρρ

ρρ ρρ

−

−=

−−−

−

−−

Developing, we obtain

()()()

[]

() ()

()( )

2131223

123 2 12 2Ik k k k IIkk I I I I

⎧

⎡

−+− − ++− −

⎨

⎢

⎩

⎣

()

11 3

21 3

20II I

ρρ ρ

ρρ

⎫

⎤

−− =

⎬

⎥

⎭

⎦

. (15.2.50)

Equating to zero the second right parenthesis, we get

k ∈  for

,0ρρ≠ , in the

hypothesis made concerning the relation of order of the principal moments of inertia

368

15 Dynamics of the Rigid Solid with a Fixed Point

one can have uniform solutions only for

123

0ρρρ===, hence only in the Euler-

Poinsot case.

Assuming now that the ellipsoid of inertia relative to the fixed point is of rotation (to

fix the ideas, let be

IIJ==), we can take

0ρ = without any loss of generality;

making

0ρ = in (15.2.48), we find again the equation (15.2.50). The equation in k

obtained by equating to zero the second right parenthesis has real roots if we have also

II=

(case of kinetic symmetry) or if we have

0ρ =

too (Lagrange-Poisson case)

or, finally, if

0ρ =

; in this last case, it results

()

12/ 1kk J I+= −, so that we

must have

/JI∈  . If

0ρ =

, then the system (15.2.43'), (15.2.43'') allows also the

solution

0aa==,

2ia = ,

()

Mg ρρ

0b = ,

(15.2.45')

corresponding the equation of second degree in

()

()( )()

()

22 1

12 2 3 1 3 3 1 2

12 2 2 0IIk k I I I I I I I

ρρ ρ

ρρ

+− − − + − =

(15.2.50')

IIJ==, then this equation is reduced to

()()

210

+− −+ =

(15.2.50'')

so that we must have

2/IJ∈  too. These conditions can be fulfilled simultaneously

only if

/1JI= (case of kinetic symmetry) or if

/2JI= (Sonya Kovalevsky

case).

From the study made for the three cases of integrability it results that the solutions

thus obtained are uniform, the above obtained necessary conditions of uniformity being

sufficient too. The Theorems 15.2.3 and 15.2.4 are thus completely justified. Lyapunov

showed that these results hold also for arbitrary real initial conditions, as well as in the

case of initial conditions of the form (15.1.19'), which verify the condition

αα =

A great number of scientists searched for a century (till now) various cases of

integrability, corresponding to particular initial conditions (restrictions imposed to the

energy constant

h, to the constant

′

of the moment of momentum etc.). In a

unfinished manuscript, S.A. Chaplygin tried to find all these cases of integrability by a

unique method; as a matter of fact, this was not possible till now. One can say that the

literature in this direction is one of special cases.

369

MECHANICAL SYSTEMS, CLASSICAL MODELS

W. Hess has considered in 1890 the case in which: (i) the mass centre of the rigid solid

is on the normal at the fixed point to one of the planes of circular section of the

ellipsoid of gyration, assuming that the latter one is not of rotation; (ii) at the initial

moment, the moment of momentum is situated in the respective plane of circular

section. This case has been studied again by G.G. Appelrot, P.A. Nekrasov, B.K.

Mlodzeevski, N.E. Jukovskiĭ, S.A. Chaplygin, R. Liouville and others, due to its

importance. T. Manacorda considered in 1950 the case in which the moment of the

given external forces with respect to the fixed point is normal to the straight line

OC,

concluding that the moment of momentum

′

K must have the same property (verifying

the considerations of Hess and Appelrot).

The equations of the planes of circular section are obtained by the intersection of the

ellipsoid of gyration (15.1.64) with the sphere

222

1232

/xxx IM++=

(the radius of

which is the mean semi-axis of the ellipsoid of gyration), being thus given by

12 32

11 11

0xx

II II

⎛⎞⎛⎞

−+ −=

⎜⎟⎜⎟

⎝⎠⎝⎠

Let

() ()

131 2 312 3

0xII I xII I−− −=

be the equation of one of these planes; putting the condition that the vector

ρ be normal

to this plane (corresponding to the hypothesis i)), we find again the relations (15.2.49').

Euler’s equations (15.1.21) become

()

11 3 2 23 32

III Mgωωωρα+− =



() ( )

22 1 3 31 13 31

III Mgωωωραρα+− = −



()

33 2 1 12 12

III Mgωωωρα+− =−



(15.2.51)

Eliminating

α between the first and the third equations, it results

()()

[

]

11 1 33 3 2 2 3 1 3 1 2 3 1

II II IIρω ρω ω ρω ρω+= − +−



We notice that the second relation (15.2.49') can be decomposed in the form

12 11

IICIρ−= ,

23 33

IICIρ−= , constC =

being thus led to the differential equation

()()

11 1 33 3 13 2 11 1 33 3

II CII

ρω ρω ρρω ρω ρω+= +

We obtain a fourth first integral

11 1 33 3

0IIρω ρω+=

(15.2.52)

if we have

15.2.3.2 The Hess’s Case. The Loxodromic Pendulum

370

15 Dynamics of the Rigid Solid with a Fixed Point

11 1 33 3

0IIρω ρω+=

(15.2.52')

at the initial moment

tt= . The components of the moment of momentum at the initial

moment

′

K being

000

11 22 33

,,IIIωωω, we notice that the relation (15.2.52') is

equivalent to

′

⋅=ρ

K , corresponding to the hypothesis (ii).

By particularizing the conditions (15.2.49'), we obtain the Euler-Poinsot case

(

123

0ρρρ===) or the Lagrange-Poisson case (

II= ,

0ρρ==); unlike

these cases, in which the initial conditions are arbitrary, in Hess’s case the initial

conditions must verify the relation (15.2.52'), so that one does not obtain a general

solution for a certain repartition of masses, but only a particular one.

N.E. Jukovskiĭ gave an interesting geometric interpretation to Hess’s motion. We

notice that, in the frame of reference

′

R , the velocity

′

=×ω

v of the mass centre

has the components

v ωρ

′

= ,

31 13

v ωρ ωρ

′

=−,

v ωρ

′

=− .

()

′′

=×ρKv,

111 222 333

II Iωω ω

′

=++Kiii,

()

TMv

′

= ,

()

222

11 22 33

TIIIωωω

′

=++

where

()

′

K and

()

′

are the moment of momentum with respect to the pole O and

the kinetic energy of the mass centre, at which we suppose that the whole mass of the

rigid solid is concentrated, respectively, and if we take into account the second

condition (15.2.49') and the first integral (15.2.52), then we find the relation

()

′′

=KK,

()

TTε

′′

ε =

(15.2.53)

Noting that we can write

ρρα

′

=⋅=

i in the frame of reference

′

and

introducing the polar co-ordinates

′

, ψ for the projection C

′

of C on the fixed plane

Ox x

′′

, we express the first integrals (15.1.42'), (15.1.43') in the form

3CO

Kρψ

′′

′



Mv Mg hρ

′

=− +

33OO

KKε

′′

hhε=

ggε= .

(15.2.53')

Taking into account the results in Chap. 7, Sect. 1.3.7, we see that the centre of mass

is moving as a spherical pendulum acted upon by a gravitational field of conventional

acceleration

g .

To put in evidence the rotation of the rigid solid about the straight line

OC, we

consider the velocity

v of the extremity

(0, ,0)Ai of the mean axis of the ellipsoid

371

MECHANICAL SYSTEMS, CLASSICAL MODELS

of gyration, of components

viω=− ,

v = ,

viω= and of magnitude

given by

()

222

132

viωω=+ . We obtain

()( )

()

13123

32 3

IIIII

II I

−−+

−

()

21 2 3

sin

II I I

θ =

−+

(15.2.54)

where

θ is the angle made by the velocity

v with the plane of the circular section

(given by

()

sin cos , vers vers /

AAA

vθ ==⋅

ρρ

vv ). We notice that the angle θ is

constant. Thus, the considered motion is entirely characterized.

If, in particular, at the initial moment, the moment of momentum

′

K is horizontal,

then the constant of areas

′

vanishes and the centre of mass C moves as a

mathematical pendulum; the plane of the circular section is rotating about a fixed

horizontal straight line, normal to the trajectory of the centre

C. The trajectory of the

point

A on the sphere

(0, )i is a loxodrome (with the characteristic property

constθ = ). If the mass centre C has an oscillatory motion, then the point A oscillates

on an arc of the loxodrome and if the centre

C has an asymptotic motion, then the

motion of the rigid solid tends asymptotically to a rotation about the mean axis of

inertia. The rigid solid is, in this case, a loxodromic pendulum.

15.2.3.3 The Goryachev-Chaplygin case. The Merkalov case

D.N. Goryachev considered in 1900 the case in which

12 3

4II I==

, the centre of mass

being the plane of equal moments of inertia at

O (the plane

Ox x , hence

0ρ = );

without any loss of generality, we take

0ρ = too. One assumes that the moment of

momentum

′

K is situated in the horizontal plane

Ox x

′′

, hence

′

, which

represents a particularization of the initial condition. In 1901, S.A. Chaplygin took back the

problem, giving a solution which contains an arbitrary fourth constant. Results in this

direction have given L.N. Sretenskiĭ and Yu.A. Arkhangelskiĭ too.

Euler’s equations are of the form

123

43ωωω=



2313

43 aωωωα+=



aωα=−



(15.2.55)

The first integrals (15.1.42'), (15.1.43') become

()

11 22 33

ωα ωα ωα

′

++==

()

22 2

12 3 1

ωω ω α++=−+.

(15.2.55')

372