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

Hence,

()

ln /MM is a first integral of the differential system (15.1.24); but a

function of a first integral is a first integral too. We can thus state

Theorem 15.1.4 If

and

are two multipliers of the differential system (15.1.24),

then their ratio

/MM is also a first integral of this system.

Starting from the conditions (15.1.25') and (15.1.29), we may write

()

MfX

∂

∑

obtaining thus another form of the Theorem 15.1.4. We state

Theorem 15.1.4' If M is a multiplier of the differential system (15.1.24) and f is a first

integral of this system, then

Mf is a multiplier of the respective system too.

∂

∑

(15.1.30)

hence if the divergence of the

n-dimensional vector of components

, 1,2,...,in= ,

vanishes, then the equation of Jacobi’s multiplier is verified for

1M = (any constant is

a Jacobi multiplier). Taking into account the above result, we can state

Theorem 15.1.4'' Any non-constant multiplier is a first integral of the differential

system (15.1.24) which verifies the condition (15.1.30).

15.1.1.5 Properties of Invariance. Theory of the Last Multiplier

Let be the change of variables

()

,,...

xxyyy= , 1,2,...,in= ,

(15.1.31)

where

x are functions of class

C , the functional determinant being non-zero

()

, ,...,

det 0

, ,...,

yy y

xx x

∂

⎡⎤

≡≠

⎢⎥

⎣⎦

(15.1.31')

Writing again the differential system (15.1.24) in the form (15.1.22), we have

(

xt= , 1

X = )

ii i i

yy y y

tt xt x

∂∂ ∂

=+ = =

∑∑

D , 1,2,...,in= .

Denoting

yY=D , where

Y , 1,2,...,in= , are functions of

y , 1,2,...,jn= , by

means of the relations (15.1.31), the differential system (15.1.24) takes the form

...

YY Y

===

(15.1.32)

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MECHANICAL SYSTEMS, CLASSICAL MODELS

We notice that

111 1

nnn n

jj i

ij i

jji i

fX X Y

xyxy

∂∂∂ ∂

∂∂∂∂

=== =

== =

∑∑∑ ∑

D .

Hence, the differential operator

D is invariant to a change of variable (15.1.31). Let

M be a multiplier of the differential system (15.1.24), which satisfies the relation

(15.1.27'), written in the form

Mf D=D ,

(15.1.33)

to put in evidence the differentiation with respect to the variables

x , 1,2,...,in= , in

the functional determinant. We observe that the matric relation

()

12 1 12 1 1 2

12 12 12

, , ,..., , , ,..., , ,...,

, ,..., , ,..., , ,...,

nnn

ff f fff f yy y

xx x yy y xx x

∂∂∂

−−

⎡⎤⎡⎤⎡⎤

⎢⎥⎢⎥⎢⎥

⎣⎦⎣⎦⎣⎦

takes place, whence we have

DDJ= ; taking into account (15.1.33), we can write

MJ f D

−

=D ,

−

= ,

(15.1.33')

because the operator

D remains constructively invariant. If f is a first integral of the

differential system (15.1.24), then

MMf= is a multiplier corresponding to this

system; hence, we get

MJ M J f

−−

= . But

−

verifies the relation (15.1.33'),

which is of the form (15.1.27'), being thus a multiplier of the differential system

(15.1.32); on the other hand,

f is a first integral for this differential system too. Hence,

we can state

Theorem 15.1.5 If M is a multiplier for the differential system (15.1.24), then

−

a multiplier for the differential system (15.1.32).

By a direct calculation, one can also show that

()

∂

∂∂

−

∑∑

(15.1.34)

which justifies once more the above statement, if we take into account the equation

(15.1.29) of the multiplier.

Let us suppose that the differential system (15.1.24) admits the independent first

integrals

, 1,2,...,ik= , kn< , and let us make the change of variable

yx= , 1,2,...,ink=−,

nk j

−+

= , 1,2,...,jk= . (15.1.35)

In this case, the considered differential system takes the form

294

15 Dynamics of the Rigid Solid with a Fixed Point

...

YY Y

−

=== ,

(15.1.36)

where the functions

Y , 1,2,...,ink=−, are obtained from the corresponding

functions

X , where we take into account the transformation (15.1.35) (

x is replaced

y for 1,2,...,ink=−; then, for

1ink=−+

2nk−+

,…,n, the second

group of relations (15.1.35) allows to replace

x as functions of

xy= ,

xy= ,…,

nk nk

−−

= and of

nk j nk j

−+ −+

= ,

1,2,...,jk=

nk j

−+

being constants). In this

case, the functional determinant (15.1.31') becomes

11 1

22 2

11 1

22 2

100 0

010 0

000 1

000 0

nk nk nk

nn n

xx x

∂

∂∂

∂∂ ∂

∂

∂∂

∂∂ ∂

∂

∂∂

∂∂ ∂

∂

∂∂

∂∂ ∂

∂

∂∂

∂∂ ∂

∂

∂∂

∂∂ ∂

−− −

−+ −+ −+

……

…………… … … … …

……

…………… … … … …

……

()

[]

, ,...,

det

, ,...,

nk nk

xx x

∂

−+ −+

⎡⎤

⎢⎥

⎣⎦

;

(15.1.37)

in the notation introduced above, the index

k puts in evidence a section in the set

{

}

1,2,...,n . Taking into account the properties mentioned above, it results that

[]

−

[] []

−

= , is a multiplier for the differential system (15.1.36) if M is

a multiplier for the differential system (15.1.24); this multiplier verifies the equation

[]

()

MD Y

y∂

−

∂

∑

(15.1.38)

The knowledge of

k independent first integrals reduces thus the number of differential

equations which remain to be integrated with

k units (we remain with a sequence of

nk− equal ratios), eliminating k variables. In particular, if

is a first integral of the

differential system (15.1.24) (

1k = ), then we eliminate the variable

(we remain

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MECHANICAL SYSTEMS, CLASSICAL MODELS

with a sequence of

1n − equal ratios); if M is a multiplier for the differential system

(15.1.24), then

()

Mf x∂∂

−

is a multiplier for the new differential system thus

obtained.

A particularly important case is that in which

2kn=−; if we would also know a

first integral

−

independent on the first 2n − first integrals

, 1,2,..., 2in=−,

then we could express

1n − variables as functions of the nth variable and of 1n −

constants of integration, obtaining thus a general integral of the differential system

(15.1.24). But this system is reduced to the system

ddyy

(15.1.39)

which is, in fact, an ordinary differential equation in the Pfaff form

21 12

dd0Yy Yy−=. (15.1.39')

Multiplying by the integrant factor

μ, we obtain an exact differential of the form

()

d, 0yyϕ = , hence

()

,constyyϕ = , which represents the first integral which

was still necessary to integrate the differential system; the equation which must be

verified by the integrant factor is written in the form

()()

∂μ ∂μ

∂∂

+=.

(15.1.38')

Comparing with the equation (15.1.38), we can state that the multiplier

[]

−

called the last multiplier, is an integrant factor for the differential equation (15.1.39').

We can state

Theorem 15.1.6 If we know a multiplier M for the differential system (15.1.24), then

2n −

first integrals are sufficient to integrate it.

In particular (the system (15.1.24) has a multiplier equal to unity), it results

Theorem 15.1.6' There are necessary 2n − first integrals to obtain the general

integral of a differential system (15.1.24), which verifies the condition (15.1.30).

15.1.1.6 Cases of Integrability

Let us assume that the rigid solid with a fixed point

O is acted upon only by its own

weight

M=Gg, applied at the mass centre C. In this case, the problem is governed by

the dynamic equations (15.1.21) and by the geometric equations (14.1.54). In the

particular case in which

OC≡

, the dynamic equations read

()

11 3 2 23

0IIIωωω+− =



()

22 1 3 31

0IIIωωω+− =



()

33 2 1 12

0IIIωωω+− =



(15.1.40)

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15 Dynamics of the Rigid Solid with a Fixed Point

no more containing the unknown functions

()

tαα= ,

1, 2, 3i =

; starting from this

system, we can determine firstly the unknown functions

()

tωω= , passing then to

the system (14.1.54) to obtain the position of the rigid solid.

Introducing the notations

x ω= ,

x α

= ,

()

iji

ijk Oj k k

XKMgIωαρ

′

=∈ +

(without summation with respect to

i),

ijk k

X αω

=∈ , 1, 2, 3i = , where, taking

into account (15.1.7), we have

KIω

′

KIω

′

KIω

′

, we may

write the system (15.1.21), (14.1.54) in the form

12 6

dd d

... d

xx x

XX X

====

(15.1.41)

considered in the preceding subsection. Let us assume now that for the system

12 6

dd d

...

xx x

XX X

===

(15.1.41')

which does not contain the time explicitly, we succeeded to determine the independent

first integrals

()

12 6

, ,...,

xx x C= , 1,2,...,5k = , const

C = , which form a

fundamental system of first integrals (the rank of the matrix

[

]

x∂∂,

1,2,...,5k = , 1,2,...,6j = , is 5); we can thus express five of the variables as a

function of the sixth one (e.g.,

()

612

, , ,...

xxxCCC= , 1,2,...,5k = ), so that the

system (15.1.41) be reduced to the differential equation with separate variables

()

66612

d , , ,..., dxXxCC Ct= . By a quadrature, we get

()

xtτ=+,

constτ =

; noting that

666

d/d d/d 1/ 0

xtx X==≠, the theorem of implicit

functions leads to

()xxtτ=+, obtaining then

()

, , ,...,

xxt CC Cτ=+ ,

1,2,...,5k = , too. Hence, to integrate the system of differential equations (15.1.21),

(14.1.54) it is sufficient to determine five independent first integrals, not depending on

time. Noting that

∂

∑

(15.1.41'')

and using the theory of the last multiplier, it results that it is sufficient to know four

independent first integrals

1234

,,,

fff of the considered differential system to can

determine a fifth first integral too, independent on the other first integrals; a

corresponding integrant factor is

[]

−

, where we use the notation of the preceding

subsection.

Euler’s system (15.1.21) can be written in the vector form

() ()

OO O

′

=+⋅ = ⋅



ωωωω

II I i.

(15.1.21'')

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MECHANICAL SYSTEMS, CLASSICAL MODELS

A scalar product by

′

i leads to

()

[

]

d/d 0

′

⋅=ωIi or

()

[

]

d/d0

′

⋅=ωIi , so

that

()

′

′′

⋅=ω

Ii ,

(15.1.42)

where, taking into account (15.1.7), we notice that

′

represents the constant

projection of the moment of momentum on the fixed axis

′

(distinct from the

component

′

of the moment of momentum

′

K along the

Ox -axis). We obtain

thus a scalar first integral of the moment of momentum (conservation of the component

of the moment of momentum along the local vertical) in the form

111 222 333

II I Kωα ωα ωα

′

++ =

(15.1.42')

By a scalar product of the equation (15.1.21'') by

ω, we get

()

()()

()

333

Mg Mg Mg Mg

∂

′

⋅

′′′

⋅= = × ⋅=− ⋅=−



ωω ω ρ ω ρ ρ

Iiii

the differentiation taking place with respect to the movable frame of reference; by

integration, we obtain

()

Mg h

′

⋅=− ⋅+ωω

Ii,

(15.1.43)

where

h represents the energy constant. It results thus the first integral of the

mechanical energy (it can be obtained also from the theorem of kinetic energy

(15.1.12)) in the form

()

222

11 22 33 11 22 33

22III Mg hωωω ραραρα++=− ++ +.

(15.1.43')

The third first integral will be

222

123

1ααα++=,

(15.1.44)

which is justified because

′

i is a unit vector.

Taking into account the above results, we can state that the problem of integration of

the system of equations (15.1.21), (14.1.54) is reduced to the problem of finding a

fourth first integral of this system. This problem has been studied by H. Poincaré, Ed.

Husson, P. Burgatti, F. Quatela and others. In the same period (the last two decades of

the XIXth century), H. Burns dealt with algebraic first integrals for the problem of

particles. Starting from this problem, H. Poincaré passed to the existence of uniform

solutions of the equations of motion which contain a small parameter

ν ; in the case of

the rigid solid with a fixed point

O one takes Mgνρ= , the small parameter being thus

the product of the weight

Mg of the solid by the distance OC ρ= from the fixed point

to the centre of mass (a quantity of the nature of an energy). H. Poincaré showed thus

that, in case of a small parameter

ν and of some arbitrary initial conditions, it is

298

15 Dynamics of the Rigid Solid with a Fixed Point

necessary that the ellipsoid of inertia at the fixed point be of rotation so that a fourth

algebraic first integral does exist. In several papers at the beginning of XXth century

(including his doctor thesis from 1906), Ed. Husson showed that this result remains

valid for an arbitrary parameter

ν ; moreover, if the ellipsoid of inertia corresponding to

the fixed point is not of rotation, then, for arbitrary initial conditions, it exist a new

algebraic first integral of the problem only if

0ν = (hence, if 0ρ = or 0M = ,

neglecting the own weight of the rigid solid). We may state

Theorem 15.1.7 (Husson) In the problem of the heavy rigid solid with a fixed point O,

governed by the geometric equations (14.1.54) and by the dynamic equations (15.1.21),

in case of arbitrary initial conditions (15.1.19), (15.1.19'), besides the three first

integrals (15.1.42–15.1.44), there exists a fourth first integral, algebraic function of

123123

,,,,,ωωωααα, which does not depend explicitly on t, if and only if the fixed

point

O is just the centre of mass C (OC≡ , hence 0ρ = , the Euler-Poinsot case) or

if the ellipsoid of inertia is of rotation (

II= and

0ρρ==, the Lagrange-

Poisson case;

12 3

2II I== and

0ρ = , the Sonya-Kovalevsky case).

If there exists an algebraic first integral in the hypothesis

II= , then – as it has

been shown by R. Liouville – there exists also a first integral in the form of a

homogeneous polynomial of first degree in

123

,,ωωω and of second degree in

123

,,ααα; consequently, in the considered case, any algebraic first integral is an

algebraic combination of homogeneous polynomials. Using this result, P. Burgatti gave

an elementary demonstration to Husson’s theorem.

As it can be easily seen, the Euler-Poinsot case differs somewhat from the other two

cases of integrability. Indeed, in this case Euler’s equations are of the form (15.1.21'),

so that the moment of momentum

′

= ω

KI is constant with respect to the inertial

frame of reference

′

R , its magnitude being conserved with respect to the inertial

frame

R too; we can write the first integral

()

222

2222

123

CCC

IIIKωωω

′

=++=ω

I ,

(15.1.45)

to which we associate the first integral (15.1.43') in the form

() () ()

222

123

CCC

III hωωω++=.

(15.1.43'')

Thus, the integration of the system of differential equations (15.1.21') (and,

analogously, the integration of the system (15.1.40)) is reduced to a quadrature, so that

this system may be separately considered. We pass then to the integration of the

differential system (14.1.54) in the form

456

XXX

(15.1.41''')

where we know the variable coefficients

()

tωω= , 1, 2, 3i = , and for which we

have at our disposal only the first integral (15.1.44). We cannot apply the theory of the

299

MECHANICAL SYSTEMS, CLASSICAL MODELS

last multiplier because the time appears explicitly in the functions

,,XXX; but we

can write other two first integrals, starting from the conservation of the moment of

momentum with respect to a fixed frame of reference. One can show that the integration

of the system of equations (14.1.15), which gives Euler’s angles, is reduced to the

integration of an equation of Riccati type; to integrate this equation (by reducing to

quadratures) one can determine two particular integrals, which can be considered to be

the searched first integrals.

If we give up the generality concerning the initial conditions and if we allow some

particular initial conditions, then we can get also other cases of integrability (by

quadratures), e.g.: the Hess’s case, the Goryachev-Chaplygin case, the Bobylev-Steklov

case etc.

15.1.2 The Euler-Poinsot case

L. Euler considered in 1758 the case in which the moment of the given external forces

with respect to the fixed pole vanishes (

=M0) (e.g., the case in which the given

external forces are reduced to a vanishing resultant or to a resultant passing through the

fixed point); the case mentioned above (the case in which the rigid solid is fixed at the

centre of mass (

OC≡ ), being acted upon only by its own weight) is thus a particular

case of that with which we will deal in what follows.

One can imagine also other cases of loading of the rigid solid leading to the same

system of differential equations. Let us suppose, for instance, that each point of position

vector

r and of mass dm of the rigid solid S is acted upon by a system of elastic forces

(it is attracted or repulsed by a discrete mechanical system

S of p fixed particles of

masses

m and of position vectors

r , 1,2,...,jp= , the magnitudes of the attractive

or repulsive forces being in direct proportion to the masses and to the distances,

respectively); hence, these forces are of the form (the coefficient of proportionality

K is

positive for attraction and negative for repulsion)

()

Km m KM m

−= −

∑

ρrr r ,

where

M is the mass of the system S , while

is the position vector of the mass

centre

(the influence of the system S is replaced by the influence of the mass

centre

C , at which the mass M is considered to be concentrated). The resultant of

these forces is (

()μ r is the density of the rigid solid of volume V)

() ()

() d

KM V KMMμ=−=−

∫

ρρρ

Rrr .

In particular, if

OC≡

, then we have

KMM=−

. As well, the moment of the

respective forces with respect to the fixed point is given by

()

() d ()d

KM V KM V KMMμμ=⋅−=−⋅=−⋅

∫∫

ρρ ρρ

Mrrr rr .

300

15 Dynamics of the Rigid Solid with a Fixed Point

In the same case we have

=M0 and we are in the Euler case. Analogously, if

OC≡

we are in the same case (

=M0

, even if the point C is mobile with respect

to an inertial frame of reference); in case of forces of attraction (

0K > ), the mass

centre

C describes an ellipse of centre C (e.g., the case of the elliptic oscillator), the

motion of rotation about this centre being governed by Euler’s equations.

In 1834, L. Poinsot has made a profound synthetic study of the case considered by

Euler, obtaining a particularly elegant geometric representation of the motion; this case

of integrability (considered to be the Ist case of integrability) is thus called the Euler-

Poinsot case. If

OC≡

, then the motion corresponding to this case is an inertial one,

because – as we have seen in Sect. 14.1.1.9 – it takes place (excepting the translation)

also in the case of the free rigid solid for which the given forces are in equilibrium

(their torsor vanishes at any point of the space) if

≠ω 0 . If =R0, then the motion

remains inertial about a point

O distinct from C too. Introducing the elliptic functions,

C.G.J. Jacobi gave a final form to the solution of this problem, expressing the direction

cosines of the axes of the frame of reference

R with respect to the frame

′

uniform functions of time, while – in 1883 – Ch. Hermite reduced the determination of

these cosines to the integration of an equation of Lamé, determining analytically all the

elements of Poinsot’s solution.

After a general study of motion of the rigid solid with a fixed point in the considered

case, one passes to the determination of its position with respect to the fixed frame of

reference

′

R ; the geometric study of the motion made by Poinsot and MacCullagh is

then presented, as well as some complementary results.

15.1.2.1 Kinematic Solution of the Motion

We begin the study of the motion by the dynamical equations (15.1.40), corresponding

to an arbitrary fixed point

O, which specify the motion of the rigid solid fixed at the

above mentioned point (the angular velocity

ω about this fixed point); the principal axes

of inertia of the rigid solid at

O have been chosen as co-ordinate axes of the non-inertial

frame of reference

We notice that, multiplying the first equation (15.1.40) by

I ω , the second one by

I ω and the third one by

I ω and summing, we obtain a first integral of the form

(15.1.45); analogously, multiplying the first equation (15.1.40) by

ω , the second one

ω and the third one by

ω and summing, it results a first integral of the form

(15.1.43''). The theorem of kinetic energy (15.1.12) leads to

d/d 0Tt

′

= , hence to

constTh

′

== (the elementary work of the given forces vanishes); one obtains a new

interpretation of the first integral of the mechanical energy, which becomes a first

integral of the kinetic energy. The constants

′

and T

′

which intervene in these first

integrals, are, obviously, positive; we agree to denote them in the form

KIΩ

′

= ,

2TIΩ

′

= , where I is a quantity of the nature of a moment of inertia, while Ω is a

quantity of the nature of an angular velocity (we have

IK T

′′

= , 2/

TKΩ

′′

= ).

301

MECHANICAL SYSTEMS, CLASSICAL MODELS

Because the moment of momentum

′

= ωKI

is constant with respect to the fixed

frame of reference

′

R , it results that its direction is constant in time with respect to

this frame. Taking into account the relation (15.1.8), we notice that

()

2cos,

OOO

TKω

′′ ′ ′

=⋅=ωωKK, hence

()

cos ,

Ωω

′

= ωK ,

(15.1.46)

and we can state

Theorem 15.1.8 (Lagrange, Poinsot) In the Euler-Poinsot motion, the projection of the

instantaneous angular velocity on the invariant direction of the moment of momentum

is constant in time.

As a matter of fact, one obtains thus an interesting interpretation for the constant

Ω.

The relation (15.1.8''), where we have introduced the instantaneous axis of rotation

too, allows to write

Ωω= , so that

()

cos ,

′

= ωK

(15.1.46')

One obtains thus a relation which gives a remarkable interpretation for the constant

and puts in evidence the variation of the moment of inertia

. Using the vector

associated to the moment of inertia tensor

I and defined by (15.1.17), one gets the

relation

IJ= too. Obviously, the constants I and Ω are specified by the initial

conditions.

In this case, the motion is governed by the dynamical system

22 22 22 2 2

11 22 33

III IωωωΩ++=,

2222

11 22 33

III IωωωΩ++=,

(15.1.47)

()

22 1 3 13

0IIIωωω+− =



(15.1.47')

the equation (15.1.47') being one of the three equations (15.1.40). We associate the last

three initial conditions (15.1.19) to these equations. The ratio of the two finite relations

(15.1.47) is written in the form

22 22 22

11 22 33

222

11 22 33

III

ωωω

(15.1.48)

Assuming that the principal moments of inertia are ordered in the form

123

III≥≥

and noting that

112123233

1 12 123 23 3

a aa aaa aa a

bbb bbb bbb

++++

≥≥ ≥≥

++++

302