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

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choose

()

cos /2αθ= and

()

sin /2βθ= , where θ, 0 θπ≤≤ is uniquely

determined by

α and β; the angles ϕ and ψ are introduced as functions of the arguments

of the complex numbers

α and β in the form

()

arg /2αψϕ=+ ,

()

arg /2βψϕπ=− − − , wherefrom we obtain arg arg /2ϕαβπ=+−,

arg arg /2ψαβπ=−+. Because

0arg 2απ≤<

and

0arg 2βπ≤<

, the

domain of variation of the parameters

ϕ and ψ is specified by /2 7 /2πϕπ−≤< ,

3/2 5/2πψπ−<< , ϕ and ψ being defined till a multiple of 4π. In this case, the

matrix

Q is of the form

ii 10 i

ii i

ρνμλ ν λμ

μλ ρν λμ ν

−+ + −

⎡

⎤⎡⎤

⎡⎤

==−+

⎢

⎥⎢⎥

⎢⎥

−+ −− + −

⎣⎦

⎣

⎦⎣⎦

() ()

+−−

−−+

⎡⎤

⎢⎥

⎣⎦

i/2 i/2

cos e i sin e

isin e cos e

ψϕ ψϕ

θθ

(14.1.13')

If the parameters

α and β are given, then one can find an infinity of systems

(,,)ψθϕ; but taking into account the intervals of definition of these angles, the solution

is unique. Conversely, a given system

(,,)ψθϕ determines uniquely the parameters α

and

β, because between these angles and the Cayley-Klein parameters there exists a

one-to-one correspondence; hence, the latter parameters determine, also, univocally, the

rotation of the rigid solid with respect to a point of it.

We notice that a matrix

Q with α, β, γ, δ arbitrary complex numbers can be

univocally represented in the form

()

() ()

()

=+++ +− +−QE

123

11 i 1

22 2 2

αδ βγ βγ αδσσσ

(14.1.14)

⎡⎤

≡

⎢⎥

⎣⎦

E ,

⎡⎤

≡

⎢⎥

⎣⎦

−

⎡

⎤

≡

⎢

⎥

⎣

⎦

⎡

⎤

≡

⎢

⎥

−

⎣

⎦

(14.1.14')

The matrix

E is the unit matrix, while

1, 2, 3j = , are the Pauli spin matrices

(Hermitian and unitary matrices, the traces of which vanish). If the trace of the matrix

vanishes (

0αδ+=

), then the first term of the sum (14.1.14) disappears, while if the

matrix

Q is Hermitian (α, δ real numbers, γβ= ), then the coefficients of Pauli’s

matrices are real. We notice that Pauli’s matrices verify the relations

222

123

===σσσE

, i

kjkll

=∈σσ σ,

j k≠

,1,2,3j k =

. Analogously, the

matrices

=−τσ, 1,2, 3j = , verify the multiplication rules of the quaternion units.

The matrix (14.1.13') will be thus represented in the form

203

14 Dynamics of the Rigid Solid

iρ=− +QEP,

123

λμν=++σσσP . (14.1.13'')

We notice that, through the relation

()

123

xxx x= σP

, each point

()

123

,,xxx

defines a matrix P and, reciprocally, each Hermitian matrix of zero trace defines a point

()

123

,,xxx . Let be a unitary matrix U, non-Hermitian (

≠UU

), in general, by

means of which we define the matrix

=PUPU

, which is Hermitian and of null trace;

we obtain thus a transformation

()

123 123

,, ,,xxx xxx

∗∗∗

→ of the space

E in itself.

Because

=UU E , we have det det 1

=UU , so that det det

∗

=PP and we may

write

jjj

xx xx

∗∗

= ; as well, the relation dd

∗+

=PUPU leads to dd dd

jjj

xx xx

∗∗

= ,

the transformation being thus a rigid rotation about the origin.

Let be the unitary matrices

()

() ()

cos /2 i sin /2

χχ χ=+σUE

, 1, 2, 3j = , χ

real.

The transformation

∗∗ ∗ ∗ +

⎡

⎤

=++= = +

⎣

⎦

PUPUE

11 22 33 3 3 3

cos i sin

xxx

χχ

σσσ σ

()

⎡

⎤

×++ −

⎣

⎦

11 22 33 3

cos i sin

xxx

χχ

σσσ σ

leads to

∗

11 2

cos sinxx xχχ,

∗

=− +

21 2

sin cosxx xχχ,

∗

xx,

(14.1.2')

where we took into account the relations verified by Pauli’s matrices; this

transformation (relative to a fixed frame of reference) corresponds to a rotation of angle

χ about the axis

Ox and is characterized by the matrix

∗

given by (14.1.2). Because

of symmetry reasons, the matrices

()χ and U

()χ lead, analogously, to rotations

about the axes

and

, respectively; each Pauli’s spin matrix is thus associated to

a rotation about a co-ordinate axis.

Let us consider the matrices

i/2

cos i sin

ψψ

−

⎡⎤

⎢⎥

≡=+

⎢⎥

⎣⎦

QEσ

⎡⎤

⎢⎥

≡=+

⎢⎥

⎣⎦

cos isin

cos i sin

isin cos

θθ

−

⎡⎤

⎢⎥

≡=+

⎢⎥

⎣⎦

i/2

cos i sin

ϕϕ

(14.1.13''')

204

MECHANICAL SYSTEMS, CLASSICAL MODELS

corresponding to rotations by Euler’s angles

ψ, θ, ϕ about the axes

′

Ox , ON and

Ox ,

respectively (Fig. 14.1); the motion of rotation of the rigid solid will be thus

characterized by the matrix

=QQQQ

, (14.1.13

)

corresponding to the formulae (14.1.13') and (14.1.13'').

One observes that the Cayley-Klein parameters (the matrices

Q too) are

characterized by the semi-angles of rotation, unlike the matrices where appear just

these angles. Thus, for

= 0χ or for = 2χπ, the matrix

∗

specified by (14.1.2) is

reduced to the unit matrix; in exchange, for

= 0χ , e.g., the matrix Q

is reduced to

the unit matrix

E, while for = 2χπ it becomes the matrix −E. In general, to a matrix

corresponds a couple of matrices

{ }

−QQ, , the matrix Q being thus a bivalent function

of the matrix ; as a matter of fact, we have made an arbitrary choice of sign in the

relations established between various representations. We mention that the relations

between the matrices and

Q correspond to the relations which are established

between the real space

E and a two-dimensional space corresponding to a matrix of Q

type; a two-dimensional complex vector will be called spinor, the corresponding space

being a spinor space. This space is more adequate to the physical reality in the quantum

model of mechanics, the wave function of a part of it having a spinorial character;

indeed, the semi-angles and the property of bivalence are closely connected to the fact

that the electron spin is a semi-integer.

α and

β are the matrices corresponding to two finite rotations, then we notice that

≠αβ βα

, because the product of matrices is not commutative; hence, the sum of two

finite rotations depends on the order in which they are effected. In case of infinitesimal

rotations, there correspond infinitesimal orthogonal transformations of matrix

=+Eαε, where the product of two infinitesimal matrices of type is neglected with

respect to such a matrix; in this case,

()()

++=+++=++EE EEE E

12 1212 12

εε εεεε εε,

so that the product of two such matrices is commutative. Hence, in case of infinitesimal

rotations the order of their application is immaterial. We notice also that to a rotation of

angle d

χ about an axis of rotation there corresponds the matrix

⎡⎤⎡⎤

⎢⎥⎢⎥

=+=− =+ −

⎢⎥⎢⎥

⎣⎦⎣⎦

1d0

010

d10 d100

001 000

χχαε

(14.1.2'')

14.1.1.4 Kinematic Considerations

205

14 Dynamics of the Rigid Solid

Starting from these considerations, we can find again the results obtained in Chap. 5, §2

concerning the kinematics of the rigid solid. We mention, especially, the Theorem 5.2.4

which characterizes the general motion of the rigid solid with the aid of the fixed and

movable axoids.

The rotation velocity of the movable frame of reference

R, rigidly linked to the rigid

solid and with the pole at O, with respect to the movable frame

R and to the fixed

frame

′

, is characterized by the angular velocity vector , which is expressed as

function of Euler’s angles in the matric form (5.2.34) and, in components, with respect

to the frame

R or to the frame

′

, by means of the kinematic relations of Euler

(5.2.35) or (5.2.35'), respectively. Starting from the latter relations, we may express the

angular velocities corresponding to Euler’s angles

ψ, θ, ϕ in the form (see Chap. 3,

Sect. 2.2.3 too) (Fig. 14.1)

()

sin cos cosecψωϕω ϕ θ



=−

cos sinθω ϕω ϕ



()

31 2

sin cos cotϕω ω ϕω ϕ θ=− +



(14.1.15)

or in the form

()

31 2

sin cos cotψω ω ψω ψ θ

′′ ′

=− −



′′

cos sinθω ψω ψ



()

′′

=−

sin cos cosecϕω ψω ψ θ



(14.1.15')

respectively. We remark also some interesting differential relations, e.g.,

, =−

, +=

, = 1,2k .

(14.1.15'')

Analogously, Euler’s angles which specify the position of the frame

R or of the

frame

′

with respect to the frame R are

′

=−ψϕ

′

=−θθ

′

=−ϕψ

; as well,

′

=−ωω

is the rotation angular velocity vector of the inertial frame with respect to the

non-inertial one. We can thus pass from the relations (5.2.35) to the relations (5.2.35')

or from the relations (14.1.15) to the relations (14.1.15').

To pass from

()

tω ,

= 1,2, 3i

, to Euler’s angles ()tψ , ()tθ and ()tϕ , hence to

integrate the system (14.1.15) with respect to the latter unknown functions, one can

introduce the intermediate unknown functions

()

tα ,

= 1,2, 3

, which are the

direction cosines of the

′

Ox -axis with respect to the movable frame R; one obtains

thus the relations (5.2.36) which lead to

cosθα

, =

tan

(14.1.16)

206

MECHANICAL SYSTEMS, CLASSICAL MODELS

the angle

ψ resulting by a quadrature from the first relation (14.1.15). The functions

α ,

= 1, 2, 3

, are connected by the differential relations (5.2.37').

The velocities distribution is given by Euler’s formula (5.2.3), written in the form

′′

=+×

vv r

ω ,

(14.1.17)

and the accelerations distribution by the formula (5.2.6), i.e.

′′

=+×+××

aa r r()



ωωω,

(14.1.17')

where we have put in evidence the quantities related to the frames

′

and R,

respectively. Differentiating the rigidity condition (14.1.1) with respect to time, we get

()()( ) ( )

⎡⎤

′′ ′′ ′ ′ ′ ′

−=−⋅−=⋅−=

⎣⎦

rr rr v v v v

220

ji ji ji ij ji





;

(14.1.18)

noting that the relation (14.1.17) can be written in the form

′′

−=×

iij





ω , it

results that Euler’s formula verifies this condition. The compatibility condition of

velocities may be thus written in the form

′′

⋅=⋅

ij i ij

PP PP



 

(14.1.18')

too, corresponding to the relation (5.2.4). Starting from the relation (14.1.18'), written

successively for the couples of points

()

,PP ,

()

,PP and

()

,PP , as well as for

()

,PP ,

()

,PP and

()

,PP (the point P non-coplanar with the points

123

,,PPP),

R. Voinaroski and L. Livovschi have found again the relation (14.1.17). Because it can

be stated from hypotheses of rigidity, it results that the respective relation has an

intrinsic character; on the other hand, the angular velocity vector can be obtained as

an axial vector associated to an antisymmetric tensor of second order, defined by means

of the velocities of three non-collinear points of the rigid solid (hence, independent on

the movable frame

R ). The condition of compatibility of the accelerations is of the

form

() ()

()

′′ ′′

−⋅ =− =×

aa vv

ijij ij ij

PP PP



 

ω ,

(14.1.18'')

corresponding to the relation (5.2.10). As a matter of fact, one can use all the results

contained in Chap. 5, §2.

With the aid of the results obtained in Chap. 5, Sec. 3.2 concerning the relative

motion of the rigid solid, one can state the group character of the rigid motions.

Corresponding to the Theorem 5.3.3, we can thus state that the set of translations of a

free rigid solid forms an Abelian group, while from the Theorem 5.3.4 it results that the

set of rotations of a free rigid solid about concurrent axes of rotation form an Abelian

group too; as well, the Theorem 5.3.5 allows to state that the set of rotations of a free

207

14 Dynamics of the Rigid Solid

rigid solid about parallel axes of rotation, so that the resultant angular velocity vector is

non-zero, constitutes also an Abelian group.

Applying the divergence and curl operators to the velocities (14.1.17), taking into

account the formulae (A.2.31'), (A.2.31'') and observing that the vectors

′

and are

constant with respect to the vector

r and that =rdiv 3 , curl =r0, ⋅=r()ω∇ ω, we

may write

′

vdiv 0 ,

′

vcurl 2ω ,

(14.1.19)

in the frame

R ; replacing

′′

=−

rr r

, we can make the same affirmation for the

frame

′

too. We state thus

Theorem 14.1.3 The velocities’ field of a free rigid solid with respect to an inertial

frame of reference is a solenoidal one (a field of curls), its curl being equal to the

double of the rotation angular velocity vector with respect to this frame.

In the case in which the non-inertial frame of reference

R is not rigidly linked to the

rigid solid, intervenes a supplementary rotation vector , which characterizes the

motion of this frame with respect to the rigid solid; obviously, the results which are

obtained are more complicated, intervening the difference

−ωΩ, but they are not of a

particular practical interest.

mentum in Case of a Rigid Solid

Various mechanical quantities which appear in the study of continuous mechanical

systems get particular forms in case of a rigid solid; we will express these quantities

with respect to an inertial frame of reference

′

with the pole at a point

′

and with

respect to a non-inertial frame

R with the pole at a point O, rigidly linked to the

considered rigid solid, the connection between the two frames being specified by

′′

rr r

(14.1.20)

for a particle P of the rigid solid. We introduce, as well, the frame

R too, with the

pole at the same point O and with the axes parallel to the axes of the frame

′

; the

connection between the frames

′

and R is of the same form (14.1.20). Defining

the momentum of the rigid solid in the form

′′

∫∫∫

Hrv() d

Vμ ,

(14.1.21)

where V is the volume, and taking into account the expression (14.1.17) of the velocity

and the relations

∫∫∫

Mr()d

Vμ , =

∫∫∫

rr()d

MVμρ ,

(14.1.22)

where is the position vector of the mass centre in the frame

R, we obtain the formula

14.1.1.5 Momentum and Moment of Mo

208

MECHANICAL SYSTEMS, CLASSICAL MODELS

′′

=+×

Hv()

M ωρ,

(14.1.21')

which is a particular case of the formula (11.2.11) (for

=H0); observing that the

velocity of the mass centre is given by

′′

=+×

ωρ,

(14.1.17'')

we can write

′′

(14.1.21'')

too, and we state

Theorem 14.1.4 The momentum of a rigid solid with respect to a given frame of

reference is equal to the momentum of its mass centre with respect to the same frame,

assuming that the whole mass of it is concentrated at this centre.

The moment of momentum of the rigid solid is defined in the form

[]

′

′′′

=×

∫∫∫

Krrv() d

Vμ .

(14.1.23)

Taking into account (14.1.20), (14.1.17), we may assume

()( )

′

′′′ ′′

=+×+×=×

∫∫∫ ∫∫∫

Krrrvrrvr() d ()d

OO OO

VVμμω

⎡⎤

′′

+× × − × + × ×

⎣⎦

∫∫∫ ∫∫∫ ∫∫∫

r rr v rr rr r()d ()d () ( )d

VVV

VV Vμμμωω,

the relations (14.1.22) leading thus to

()

′

′′′′

=+ ×−×

KK rvv

OC O

M ρ ,

(14.1.23')

where the pseudomoment of momentum is given by

=××=×

∫∫∫ ∫∫∫

Krrr rr

() ( )d () d

μμω

(14.1.24)

We find thus again the formulae (11.2.16), (11.2.16'), where

=v0

. To λ , λ scalar,

corresponds the homologous vector

λ , while if

ω and

ω have as homologous

vectors

and

, respectively, then to the vector

ωω

corresponds the vector

+KK, due to the distributivity of the vector product with respect to the addition of

vectors; hence, the linear transformation (14.1.24) is a vector homographic

transformation, called inertial homography. Noting that

[]

() ( )

rVμ=−⋅

∫∫∫

Kr rrωω

introducing the moment of inertia tensor

of components (3.1.81) and taking into

account the relation (3.1.83), we obtain the remarkable relation

209

14 Dynamics of the Rigid Solid

==KKI

ω,

(14.1.24')

which corresponds to the relations (11.2.17), (11.2.17') where we make

=K0

; K

represents the moment of momentum with respect to the pole O in the non-inertial

frame and – in case of the rigid solid – is reduced to the pseudomoment of momentum

considered above. We state thus

Theorem 14.1.5 The moment of momentum of a rigid solid with respect to a pole

′

of a given inertial frame of reference

′

, in this frame, is equal to the sum of the

pseudomoment of momentum of the rigid solid with respect to an arbitrary pole O,

rigidly linked to the rigid solid (the contracted product of the moment of inertia tensor

with respect to the same pole by the rotation angular velocity vector of a non-inertial

frame

R with the pole at O, rigidly linked to the rigid solid, with respect to the inertial

frame), the moment of momentum of the centre of mass, translated at the pole O, where

the whole mass of the rigid solid is considered to be concentrated, taken with respect to

the pole

′

, in the frame

′

, and the moment of momentum of the pole O, translated

at the centre of mass, where it is assumed that the whole mass of the rigid solid is

concentrated, calculated with respect to the pole O, in the inertial frame

′

too.

In the particular case in which the frame of reference

R is of Koenig type ( = 0ω ,

hence

=K0

), there results the formula

() ()

′

′′ ′ ′

=× +×

Kr v v

OC O

MMρ .

(14.1.25)

Taking into account that this frame is rigidly linked to the rigid solid, it results that the

latter one will have a motion of translation (we are in a particular case of motion).

If the pole of the non-inertial frame coincides with the centre of mass of the rigid

solid (

≡OC, = 0ρ ), then we obtain a formula of Koenig type (in which, instead of

the moment of momentum with respect to the centre of mass, in the frame

R, which is

equal to zero, appears the corresponding pseudomoment of momentum)

()

′

′′′

=+×

KI v

Mωρ ,

(14.1.25')

where

is the central moment of inertia tensor; thus, the motion is not particularized

and has – further – a general character. If we have also

= 0ω , then the non-inertial

frame

R is a Koenig frame and we get

()

′

′′ ′

=×

Mρ .

(14.1.25'')

Hence, the moment of momentum of a rigid solid in motion of translation, with respect

to a given frame of reference, is equal to the moment of momentum of its centre of mass

with respect to this frame, assuming that its whole mass is concentrated at this centre.

()

′

′′′ ′

=−×

KKr v

OOC

M ,

we obtain

()

′′

=+×

KI v

OO O

Mωρ ,

(14.1.26)

As in Sect. 11.2.2.1, starting from (14.1.23') and noting that

210

MECHANICAL SYSTEMS, CLASSICAL MODELS

a formula analogue to (11.2.23), the moment of momentum

′

being calculated with

respect to the inertial frame

′

; in particular, if ≡OC, then we can write

′

KI()

ωS ,

(14.1.26')

corresponding to the relation (11.2.23'), where we make

=K0

. We state thus

Theorem 14.1.6 The moment of momentum of a rigid solid with respect to the centre of

mass, in an inertial frame of reference, is equal to the contracted product of the central

moment of inertia tensor by the rotation angular velocity vector of a non-inertial frame

rigidly linked to the rigid solid, with the pole at the centre of mass too, with respect to

the inertial frame.

This result has a general character and takes place for any motion of the rigid solid; it

represents the most simple formula of kinetic nature corresponding to such a motion. It

can be obtained also directly, starting from the formula (14.1.24) and making

≡OC; it

results

′′ ′′

=×=−×−

∫∫∫ ∫∫∫

Ir rrvv

() d ()( ) ( )d

C C

CP V V

μμ









ωρ

[]

′

′′ ′ ′ ′′ ′

=−× +×−=

∫∫∫ ∫∫∫

rrvvr K()()d ()d

VVμρρ,

because the last integral represents the statical moment of the rigid solid with respect to

the mass centre, so that it vanishes.

It is thus put in evidence the necessity to introduce the moment of inertia tensor,

studied thoroughly in Chap. 3, Sec. 1.2.

We can define the dynamic resultant of the rigid solid in the form

′

∫∫∫

() d

μ .

(14.1.27)

Taking into account (14.1.17') and (14.1.22), we obtain

′′

=+×+××

Aa ()

MM M



ωρ ω ωρ,

(14.1.27')

corresponding to the relation (11.2.11') for

=H0 and ∂=r0/ t∂ . As well, the

dynamic moment of the rigid solid, defined in the form

′

⎡⎤

′′

=×

⎢⎥

⎣⎦

∫∫∫

Drr

() d

μ ,

(14.1.28)

leads to

()

′

′′

=+×

DI a

()

ωρ ,

(14.1.28')

corresponding to the relation (11.2.25), where we take into account (14.1.26) and

(14.1.17').

211

14 Dynamics of the Rigid Solid

The kinetic energy of the rigid solid is defined in the form

′′

∫∫∫

() d

TvVμ .

(14.1.29)

With the aid of the relation (14.1.17), we can write

′′

⎡⎤

′′

=+⋅×+×

⎣⎦

∫∫∫ ∫∫∫ ∫∫∫

rv rr rr

2 2

()d ()d ()( )d

VVV

Tv V V Vμμμωω

wherefrom, using the relations (14.1.22), we obtain

()

′′′ ′′′

=+ + =− + ⋅

vvv

OO OOC

T T Mv M T Mv Mωρ ,

(14.1.29')

corresponding to the relations (11.2.28), where we make

=v0

. We have introduced

the pseudokinetic energy of the rigid solid

()

=×=

∫∫∫ ∫∫∫

rr r

11d

()( )d () d

22d

TVV

μμω

=⋅ ×× =⋅ ×

∫∫∫ ∫∫∫

rr r rr

11d

() ( )d () d

22d

μμωωω .

(14.1.30)

Taking into account (14.1.24) and (14.1.24'), we can also write

()

==⋅=⋅KI

TT ωωω,

(14.1.30')

corresponding to the relations (11.2.28''), (11.2.29), where we make

=K0

and

= 0T ; T represents the kinetic energy with respect to the non-inertial frame R ,

which – in case of the rigid solid – is reduced to the pseudokinetic energy considered

above. We may thus state

Theorem 14.1.7 The kinetic energy of a rigid solid with respect to a given inertial

frame of reference

′

is equal to the sum of the pseudokinetic energy of this solid

with respect to an arbitrary non-inertial frame

R with the pole at O, rigidly linked to

the rigid solid (the semi-scalar product of the rotation angular velocity vector by the

contracted product of the moment of inertia tensor with respect to the pole O by the

rotation angular velocity vector of the frame

R with respect to the frame

′

) and the

scalar product of the velocity of the pole O with respect to the frame

′

by the

momentum of the rigid solid with respect to the same frame

R, from which is

subtracted the kinetic energy of the pole O at which is considered to be concentrated

the whole mass of the rigid solid, in the frame

′

14.1.1.6 Kinetic energy and work in case of a rigid solid

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MECHANICAL SYSTEMS, CLASSICAL MODELS