1) The motion of a rigid body can be described by tracking the position of a reference trihedron attached to the body over time. 2) At each instant, the orientation of the mobile trihedron relative to a fixed reference frame is defined by a rotation dyadic with a rotation axis and angle. 3) The rotation dyadic transforms the fixed frame vectors into the mobile frame vectors, allowing the position and orientation of any point on the rigid body to be determined as a function of time.

1. THE RIGID BODY MOTION
REDUCED TO THE STUDY OF SPHERICAL CURVES.
by
Elysio R. F. Ruggeri
Ouro Preto, MG, Brazil
Abstract
In this paper one makes a brief exposition of the kinematics of the rigid body going over the
Polyadic Calculus concepts and basic operations, outstanding those connected with the rotation dyadic (of
each instant). The axis of this dyadic is the instantaneous rotation axes of the body; its angle of rotation is
the angle of rotation of the body. One introduces the concept of spherical indicator curve of the
instantaneous rotation axis and one interprets the body motion by the geometrical properties of this
spherical curve. One derives also: the dyadic general laws for displacement, velocities and acceleration
distributions using a dragging dyadic, and the corresponding classical vector form of these laws.
§ 01 – Rigid body: definition.
A rigid body (or solid) is a dense set of points1
whose mutual distances are invariant.
Four given non coplanar points of a rigid body – which we shall denote by 1, 2, 3 and 4  O (an
origin for vectors)– define in this body a quadrangle2
. Let us denote by e e e1 2 3, , the vectors with origin at
point O and ending points in 1, 2 and 3 respectively. As these three vectors are non coplanar by
hypothesis, they define a suit which we will consider positive (in the sequel e e e1 2 3, , ). This suit admit a
reciprocal suit (also positive) whose vectors we will denote by e e e1 2 3, , , vector e1 corresponding to e1
etc.. Each one of these suits may be considered a local reference system3
, that is, a system to refer events
occurring in the rigid body (Fig. 01).
One any fifth point of the body, say P, stays evidently defined in position in relation to a local
reference system (the local quadrangle) if, for example, are known the projections of its position vector
r0
(with origin at O and end at P) onto the supports of the vectors of the system; hence we write:
r P O r .e e r .e e0 0
i
i 0 i
i   ( ) ( ) , (i=1,2,3) (01).
1
The density of the set is a concept identical to that of the line: given two any points of the set, it will exist at least one point
between then.
2
One calls quadrangle the figure defined by for any points, of which they are the vertex, and the six segments they define, of
which they are the sides. When the points are coplanar the quadrangle is sad to be plane.
3
It is common to confuse "reference system" with "coordinate system" since one only defines the "reference vectors" with the
preliminary introduction of the concept of oriented axes.

2. 2
In general we can define geometrically the form of a rigid body when its generic point, P, is
defined by a uniform and continuous vector function of three numerical variables 1
, 2
, 3
whose varies
inside intervals well defined; and we write
r r0 0
1 2 3
 ( , , )   , (02).
To label points of the body it is enough to couple to the reference system { } { , , }e e e e  1 2 3 the
corresponding system of (rectilinear) Cartesian coordinates X1X2X3 marking in these axes the coordinates
of P; or, what is the same, jointing to the reciprocal system of the former the (new) coordinate system
X1X2X3
4
.
If in (02) two of the variables stay fixes and the third varies, P describes a space curve whose
points are points of the rigid body; if just one of the variables stay fixed, P describes a surface whose
points are points of the rigid body. Hence each point of the body is the common point to three of these
surfaces; and by each point of the body pass three of that space curves. Particularly, the three space
curves of a point are said to be the curvilinear coordinates of that point related to the local system of
rectilinear Cartesian coordinates coupled to { } { , , }e e e e  1 2 3 .
Analogous considerations and concepts are applicable when one considers the reference system
{ }e reciprocal of { }e . Now the new coordinates of the point will be reciprocal curvilinear coordinates of
the first. As by hypothesis the first ever exists, so it will exists the later ones.
§ 02- Kinematics.
The motion of a rigid body is of intuitive perception and related to each observer. The study of the
movement of a rigid body independently of the causes that produces it is the part of the Rational
Mechanics called Kinematics.
One says in Kinematics that the motion of a rigid body is determined to a certain observer when
this one may define in a uniform way the position of any one of its points in each time.
From the Cartesian point of view, the motion of a body means to an observer that he can to for-see
as varies the coordinates of all its points in each time when referred to a rigid and fixed global system of
coordinates, linked to him at some instant, and which sometimes we call fixed trihedral5
. This is the same
to say, in vector words, that the positional vectors of all points of the body might be determined from the
data necessary and sufficient to this.
The global system of reference and its reciprocal, or fixed trihedral systems, will be represented
with origin in O and base vectors e e e1 2 3
, , , and e e e1 2 3
, , , respectively; when it was necessary the
corresponding coordinate axes will be represented by X X X
1 2 3
, , and X X X1 2 3, , .
4
It is good to remember that all the axes must have the same unit segment, that is, the points with quote +1 of all these axes are
points of a same spherical surface of radius 1 centered in the origin.
5
The name trihedral is of common use in Mechanics as synonym of reference system.

3. 3
§ 03 – The motion of a rigid body.
We shall show at once, elucidating the reasoning by Fig. 02 a), that the motion of a trihedron – to
which we shall refer now on as mobile trihedron, with a mobile origin, O, mobile vectors e e e1 2 3, , and
axes X1X2X3 – will be determined if the observer:
1°) – consider that in a certain initial instant of time, t0, the mobile trihedron has origin
coincident with O and is isogonal with the fixed trihedron, that is, in that instant its vectors e01
,e02
and
e03
are congruent (not coincident but able to over-put) respectively with e e e1 2 3, and . Tel conditions
will be called the "initial conditions of the motion".
2°) – consider that from the prior position once, at each instant, the mobile trihedron is isogonal
with the fixed trihedron, that is, the directions (just the directions) of the vectors of the mobile trihedron
in relation to the fixed trihedron are different. Hence we can write for the instant t:
  i t ti i i
12 3, , : ( ) | |  ( )e e e , (01),
where  ( )ei
t represents the ei unit vector, and
    i | | constanti i i
12 3 0
, , : | | | |e e e , (011).
Hence, using classical notation for time derivative, we write
 | |

e
e
e
e
i
i
i
i
d
dt
d
dt
  , (012).
3°) – admit that, in the arbitrary instant t, it is possible to determine the position (of origin O and
vectors ei) of the mobile trihedron with respect to the fixed trihedron, that is, for the observer
t)(and)t(:1,2,3=it, iO er , (02),
are known functions of class C2
6
, given or able to determine, rO(t) been the position vector of the mobile
origin (O), the ei t( ) satisfying (011).
It runs from this third hypothesis that, if rO
t( ) is known, the trajectory, the velocity and the
acceleration of O are determined.
As we know from the Kinematics of the Point, the velocity vector v rO O
t) = t)(  ( , tangent to the
trajectory7
, may be written in the form v r tO O
t) = t) = v(  ( , where v is its modulus and t the unit vector of
the tangent. The acceleration vector a rO O
t) = t)(  ( - a vector of the osculating plan to the trajectory at
6
One says that a function is of class CN with respect to a one of its variables if this function and its N first derivatives with are
uniform and continuous function (of this variable).
7
Tangent to a curve in a point (of this curve) is the line that contains the point and its consecutive (on the curve).

4. 4
point8
O – is written in the form a r t n( )  ( )   t t v v
RO
2
   where R is the curvature radius (of flexion) 9
of
the trajectory at O and n the unit vector to the principal normal10
. The projection of a(t) onto the tangent,
v, is the tangential acceleration of O; the projection of a(t) onto the normal, v R2 / , is the normal
acceleration or centripetal acceleration of O.
It runs also from the third hypothesis (and from a Polyadic Calculus (PC) classical theorem) that
exist a dyadic11
(the complete12
changing base dyadic), function of time, (t), that transforms the fixed
trihedron vectors into the mobile trihedron vectors. If we put
e .ei i
t) t)( (  , then (t) (t)i
i
 e e , (03).
Calculating the principal dyadic13
of  we have: P
i
i
i j
i k j
k
(t) e e e .e e .e e e( )( ) . From the hypothesis of
instantaneous isogonallity of the trihedron we deduce: e .e e .ei k i k
 ; hence
 P
j i
i k j
k j
k j
k
k
j
j
k
j
j
    ( )( ) ( )e .e e .e e e e .e e e e e e e .
In accordance with a classical PC theorem a dyadic is equal to its principal if and only if this dyadic is a
rotation dyadic. Hence this instant t rotation dyadic, which now on we will denote by
)t(=t)(),ˆ(  j , or simply by ),ˆ( j , (031),
has a rotation axes with unit vector jˆ (t) passing through O(t), and a rotation angle (t) well determined; it
will be called the instantaneous dyadic rotation of the mobile trihedron. Then we write:
(t)and)t(ˆˆbeing,ˆsin+)ˆˆ-(cosˆˆ)t()t( i
i),ˆ(  jjjjjjjeej  , (04).
Denoting by E
e V
, respectively, the scalar and the vector of , we can write (in agreement
with the theory of rotation dyadics):
cos (t)
t tE i
i
 


 ( ) ( )1
2
1
2
e .e
, (041),
and





2sin
(t)
2sin
(t)
(t)ˆ
i
iV ee
j

, (  K ) (042),
noticing that jˆ , although expressed in function of de , is independent of . In Cartesian words, this
means also that  is as a function of three scalar parameters: the angle  and two of the three coordinates
of jˆ .
For the instant t0 we will write:
   00000
i
0i0),ˆ(
ˆsin+)ˆˆ-(cosˆˆ)t()t( jjjjjeej , (043).
8
The osculating plane of a curve in a point of this curve is the plane defined by the tangent to this curve in the point and the tangent
in the consecutive point.
9
The flexure curvature radius of a curve in one of its points is the radius of the circle that contains the point and two of its
consecutive points.
10
The principal normal to a curve in one of its points is a line of the osculating plane of this point perpendicular to the tangent.
11
A dyadic is a symbolic sum of dyads. Dyads are ordinate and juxtaposed products of two vectors, the like ab; a is the antecedent
and b is the consequent. Every dyadic can be represented in the form - called trinomial - of a sum of three dyads whose antecedent
(or consequent) be three non coplanar vectors.
12
The third of a dyadic, when this one is represented in trinomial form, is the mixed product of its antecedent by the mixed product
of its consequent. A Um dyadic is sad to be complete if its third is a non vanish number.
13
The principal of a complete dyadic (i.e., of non vanish third) is the dyadic obtained from the first changing the antecedent and the
consequent of any one of its trinomial representation by its corresponding reciprocals.

5. 5
In this instant there will exist an initial rotation axes whose unit vector is
o
ee
jj 





0
i
0i
0
V
00
2sin2sin
ˆ)(tˆ 
, (044).
Referred to O the positional vector of the current point P of the body for the instant t0 is 0r . For
the instant t, and referred to O, this positional vector is r; and referred to O is r( )t (Fig. 02,b)), needing to
notice that r0
and r do not vary with t. Denoting by rO(t) the vector position of O referred to O in the
instant t, we write:
r r r( ) ( )t tO  0),ˆ(O (t))t( r.r j   , (05).
Indeed, observing that from isogonallity of the trihedron we can write r.e r .ei i
 0 , we deduce, in
sequel: r r.e e r .e e e e .r  ( ) ( ) ( )i
i
i
i i
i
0 0 . Now, considering (03) e (031), it is immediate to confirm
(05).
Remembering that the addition of vectors is commutative, we see by (05) that the motion of each
point of the body between the initial instant t0 and any instant t, may be ideally intended as realized by a
instantaneous translation of vector rO t( ), followed by a instantaneous rotation of angle (t) - 0 about the
axes of unitary jˆ (t) applied at O(t). These motions in reverse order are also possible.
Indicator of an instantaneous rotation axis.
A simple useful manner to understand the motion consists to imagine the unit vector jˆ of the
instant t applied in the fixed origin O , and to understand the angle  of this instant as the angle of a
(imaginary and instantaneous) rotation of the mobile trihedron about this vector, departing from its initial
position (correspondent to  = 0 in t = t0 ). Under these conditions the couple ( jˆ ,t) defines a point K on
the spherical surface of unit radius centered in the fixed origin O . Imagined this operation realized in
each instant, one sees that (as t varies) K describes a curve (twisted in general) on this surface (a spherical
curve) which we will call the spherical indicator of the axes of rotation of the mobile trihedron; the point
K we will call the indicator of the axis and that of the instant t0, the initial indicator of the axes.
As to each t corresponds a rotation angle well determined, , we can cote the spherical indicator
showing to each of its points the corresponding couple (t,), the like in Fig. 03, a). This statement must be
understood with certain reserve because the angle  is determined by (042) unless fore an integer number
of  radians.
By other side we must consider that the indicator can show multiple points (say, a double point), or
points to which correspond times that implies to the dyadic one certain unit vector axes and angles of

6. 6
rotation that differ between then by 2K radians. This means that the solid, in various instants, may
acquire identical rotation conditions in different points in the space.
Another useful manner to geometrically visualize the same question (Fig. 03,b) consists in
representing, with the same origin, at each instant, the vector semi-tangent of rotation associated to ,
jq ˆ
2
tg

 , (06).
Under these conditions (as t varies), the geometrical loci of the ending points of these vectors are points
of a same twisted curve to which we will call the indicator semi-tangent of the rotation axes. As in the
later case, to the ending point of each vector q we can associate the couple (t,) which, with the direction
of q (the direction of the rotation axes) and the modulus of q, tg /2, define perfectly the rotation at each
instant.
Let  be the angle between the axes of the instants t0 and t, to which correspond the indicators K0
and K, respectively (Fig. 04,a)).
If iˆ is the tangent unit vector to the indicator pointing on the sense of increasing s, jˆ is the principal
normal unit vector pointing to the center of the sphere, then jik ˆˆˆ  is the binormal14
to the indicator;
we shall refer to the set { kji ˆ,ˆ,ˆ } as the Frenet-Serret trihedron of the point K(t) of the indicator.
The rotation motion of the axes in the instant t is that of the corresponding indicator K (on the
indicator curve); the vector velocity and the vector acceleration of K are
iiv ˆRˆ
dt
ds
  , since ds=Rd, (07),
and, using Frent-Serret's formulas,
)ˆˆR( 2
jia   , since )
ds
ˆd
dt
dsˆR(
i
ia   ,  R
dt
ds
and j
i ˆ
R
1
ds
ˆd
 (071).
Hence the rotation velocity vector and the acceleration rotation vector belongs both to the osculating
plane of P in the instant t (Fig. 04 a)).
14
The binormal to a curve in one of its point is the line orthogonal to the osculating plane at this point. In the case of the spherical
indicator the osculating plane in a point is a plane that passes through the center of the sphere (and contains the tangent and the
principal normal). The tangent plane to the curve in a point is the same as the tangent plane to the sphere at the point (which
contains the tangent and the binormal to the curve).

7. 7
In double points of the indicatrix (Fig. 04 b) to f)) the velocities and accelerations are not the same
in general, as in the case of crunode B15
; but they should be equal in case the multiple point be a
reversing point, point C, or opposed vectors in case of a cuspidal point16
, point D. There are rare cases
in which one of the tangents in a double point, like E (Fig. 04,e)), has a contact of the first order with a
branch of the curve and a contact of the third order with other branch (the point is a inflexion point in this
last branch); the double point is said to be a flecnoidal point. Rare also is the case in which the double
point is biflecnoidal point: both tangents have a third order contact with the respective branch (the
double point is a point of inflexion for the two branches simultaneously) as shown in Fig. 04,f).
An important indicator point is that to which correspond the null velocity (hence, also null
acceleration), like cuspidal indicator points to which we shall call translation indicators. This second
name derives from a mechanical view because the indicator needs to stop at the point on the indicator to
change the sense of its movement. This means that when t tends to a certain value, say ttra, the
corresponding indicator suffers a brake in its movement onto the indicator up to reach the cuspidal
position, to accelerate at once. The rotation motion of the mobile trihedron is annihilated at that cuspidal
instant 17
.
Differentiating the rotation dyadic.
In agreement with PC, the differential of the instantaneous rotation dyadic can be written in the
form
j.
j
ee jj
ˆd)
ˆ
(d)()t(d=)t(d ˆ
i
i),ˆ( 







 , (08),
the first and the second pieces of the last member representing the partial derivatives of  in the instant t,
respectively for fixed jˆ and . We have:
 

 jjjjjjj
ˆdsen)ˆˆ(d
2
sen2]ˆcos+)ˆˆ-(sen[d(t)d 2
),ˆ( , (081).
Putting       ii jj kk , the first two pieces of (081) can be written in the following different forms:



 ]ˆˆ[d])ˆ(cos)ˆˆ(sen[dd)( )2/,ˆ(ˆ jj jjjjj 

)ˆˆsenˆˆcosˆˆcosˆˆsen(d kkikkiii  (09).
Since kj ˆd
T
Rˆd  where T is the torsion radius of the indicator at point K, and in agreement with the PC
rules, we can write:






j
j
jkkjjjj.
j
jj
ˆ
ˆ
and)ˆˆ(d
T
R
)ˆˆ(dˆd
ˆ
)ˆˆ(
.
So, the last piece in the last member of (08), or the last two pieces in the second member of (081), can be
written in the form
kkjj.
j
ˆsend
T
Rˆˆ
2
send
T
R
2ˆd)
ˆ
( 2





 

,
or, ERRADO
)]ˆˆˆˆ()ˆˆˆˆ(
2
[tgsendˆd)
ˆ
( kjjkkiikj.
j







, (10).
15
A point of a curve is sad to be a crunode when in this point there are two different tangents (at least) to the curve.
16
A point of a curve is sad to be a reversion point when the unit vectors of the tangents in this point are, respectively equal or
opposed vectors.
17
As we shall see ahead, in the cuspidal instant the movement is a instantaneous translation.

8. 8
Finally, adding member by member (09) and (10) it comes
d d ( , ) ( , / )
[   ]k k
kk  
    2
+d sen [tg 

2
( ) (  )]ki ik kj jk   , (11),
To this (dyadic) differential is associated, in the instant t and base {, , }i j k , the matrix
[ ] ,
d
d sen d cos 2d sen
d cos d sen d sen
2d sen d sen
( )
2
2
 k ijk
    

     


 

 
 












2
2
0
, (111).
General law of displacement distribution.
Differentiating (05) in the instant t we have:
d t d t d d t d d t dO O O
T
r r r r .r r . .rk k k
( ) ( ) ( ) ( )( , ) ( , ) ( , )
         0
, (12).
Being   T.  , we deduce by differentiation: d dT T   . .  . As the differential of the
transpose is equal to the transpose of the differential, it comes: d dT T T   . . ( ) , that is, the dyadic
d T . is anti-symmetric. Considering (04) and remembering that e .e e .ei k i k (by hypothesis the
trihedrons are isogonal) we come to
d t t d t tT
i
i ( ) ( ) ( ) ( ). e e , (13).
Putting
d t d t t d tT T ( ) ( ) ( ) ( )   . , (131),
it comes, finally, from (12), the general law of the displacement distribution of the point of the solid,
d t d t d d t d tO Or r .r r .rk
( ) ( ) ( ) ( )( , )
     0
 , (14).
beeing
d d d tr .r .rk
 ( , )
( ) 0
 , (141).
Definition: (dragging dyadic)
The differential dyadic d, which permits the calculus of the instantaneous drag displacement
of the generic point of the body (with respect to O) will be called the dragging dyadic of the
point.
From (04) and (11) we can calculate the anti-symmetric matrix associated to the draging dyadic
(131) of the generic point with respect to the Frenet-Serret trihedron of that point in the instant t; we find
out, after some developments:
[ ( )]d t
d 2d sen
d d sen
d sen d sen
2
2
 ijk 














0
0
2
2
0
 


   


 
, (15).
Being d(t) an anti-symmetric dyadic, we can write it in the form
d (t) d t d t     c c( ) ( ) , (16),
or, if d(t)V is the vector of d(t):

9. 9
d (t) d (t) d sen d sen dV
2c i j k    1
2
2
2
   

   (17),
or, yet, in the form
d (t) d (t) d sen d com cos senVc u k u i j     1
2
2
2 2 2
 


  ,    (171).
Thus, the general law (14) assumes the vector differential form
d t d t d t d t d tO Or r c r r c P O( ) ( ) ( ) ( ) ( ) ( )       , (18).
The vector d tOr ( ) represents an infinitesimal instantaneous displacement of O in the direction of
the tangent to its trajectory. This displacement is, then, a displacement common to all the points of the
solid.
*
Exercício: To demonstrate the differential formulas:
ii
d)(d eec  , (19).
Solution:
From (13) and (131), we have: d di ie .e ( ) . Now, substituting the differential drag dyadic by its
expression (16) and applying classical properties of the operations between dyadics and vectors we find
out the thesis.
Rotation interpretation in fixed origin.
In agreement with (11) and (12), the vector d(  , )k
.r 0
presents two components; this means that
r0
can be transformed in two commutative stages. The vector
d = d01 0 0
r r .k k .rk
 
[ ( ) ]( , / )
  2
, (20),
can represent a first stage. The vector ( , / )k
.r  2 0
is the rotation of r0
about the axes k of the instant t
by an angle of   / 2 radians; the vector ( ) r .k k0
is the r0
component in the direction of k . Then, the
vector between brackets is the rotation, by the angle   / 2, of the orthogonal projection of r0
onto the
osculating plane of the instant t about the instantaneous rotation axes. The vector
d = d sen [tg02
r ki ik kj jk .r 

2 0
( ) (  )]   , (21),
represent a second stage. The dyadic that operates on r0
can be written in the form
d = d sen [ tg
2
tg
202  
 (  ) (  )]k i j i j k   , (22),
with respect to the Frenet-Serret's trihedron of the indicator curve, and its associated matrix is
[d ] d sen
tg
2
tg
2
02 ijk  












 


0 0
0 0 1
1 0
, (221).
This dyadic has a vanishing third; it second has associate matrix

10. 10
[(d ) ] (d sen
tg
2
tg
2
tg
2
0
02 2 ijk
2 2 













 

 
)2
1 0
0
0 0
, (23),
and can be written in the Cartesian form
( ) ( ) ( ) ( )d d sen tg
2
tg
202
2 2
2   
 
i j i j , (231).
As the scalar of 202)d(  is ( ) ( ) ( )d sen tg
2
d sen cos2 2 



2 21 2
2
  , this dyadic is linear in general,
being ortolinear in the instants in which   K / 2 (K=0,±1,±2, ...). Hence, the dyadic d 02 in study is
planar in general and, evidently, its scalar vanishes. The expression (22) of d 02 clearly shows its
planes, being possible to easily confirm that: 1) - the antecedent and the consequent of its second are,
respectively, perpendicular to the planes of its antecedent and consequent; 2) - that the rectifying and
normal planes are bisectors of the dihedral defined by its planes. The transformed of r0
through this
dyadic is then a vector that belongs to the plane of its antecedent (Fig. 05). We write, then,
d d d01 02( , )k
.r r r 0
  .
The dyadic d 02 is ortoplanar in the instant for which   K / 2. But as it has vanishing scalar,
in this instants it is anti-triangular.
§ 04 – The general law of distribution of velocities and acceleration.
From the general displacement distribution law ((14),§ 03), we deduce immediately the general
law of distribution of velocities of the points of the solid:
v v v v .r v .rkP O O Ot t t t t( ) ( ) ( )  ( ) ( )( , )
       0
 , (01),
where, in agreement with ((13) and (131),§ 03),
  ( ) ( )  ( ) ( ) ( ) t t t t tT T
i
i     . e e , (011),
or
  ( ) ( ) ( ) ( )t dt d t t dt dT
i
i    e e , (012).
Definition:

11. 11
The dyadic (t) - being the derivative with respect to time of the dragging dyadic of the generic
point of the body (with respect to O) - will be called instantaneous dragging rotation velocity
dyadic of the body.
With respect to the Frenet-Serret's trihedron of the spherical indicator curve of the instant t, the
associate matrix to the instantaneous dragging rotation velocity dyadic is
[ ( )] [( )]
 
 
 
 t t
2 sen
sen
sen sen
2
2
ijk ijk 















0
0
2
2
0
 


  


 
, (02).
As (t) is anti-symmetric, we can write it in the form
(t) t t    w w( ) ( ) , (03)
where, in different representations,
w i j k(t) (t) (t) sen senV V
2      1
2
1
2
2
2
          

 (04),
or
w u k i j k(t) sen sen tg    2
2 2
      )  

  

 (041),
(t)V being the vector of (t). Then, the general law (01) assumes the classical form
v v w r v w P OP O Ot t t t t( ) ( ) ( ) ( ) ( ) ( )       , (05).
Notices:
1) -We should call e w ei i  Poisson's formulas (generalized) since when the base
{ , , }e e e1 2 3 is normalized such formulas coincide with the classics (the Poisson's)  i w i  etc.
2) - If one pretend to deduce the expression of w as a function of the vectors of the mobile base
and its derivatives with respect to time, it is enough to consider   2w e ei
i, to substitute
each ei by its (classic) expression as a function of the vectors of the base { , , }e e e1 2 3 and to
develop the calculations; one finds:
w e e e e .e e e .e e e .e e  ( )[( ) ( ) ( ) ]1 2 3
2 3 1 3 1 2 1 2 3 , (041).
The classical formula corresponding to (041) is
w j .k i k . i j i . j k  ( ) ( ) ( ) .
*
Differentiating (01), it comes:
d t d t d d t d d t d t t dP O O Ov v v v .r v .r . rk
( ) ( ) ( )  ( ) ( ) ( )( , )
         0
 
or, remembering ((141),§ 03), and considering (012):
d d d dtP Ov v .r  ( ) 2 , (06).
Expression (06) gives the general differential law of the velocity distribution of the points of the solid
in the interval of time dt. Dividing both members of (06) by the correspondent interval of time we obtain
the general law of acceleration of the points of the solid:
a a .rP O  ( ) 2 , (061).
Considering ((16),§ 03) and its derivative with respect to time, that is,

12. 12
  )t(and)t((t) ww  , (07),
we deduce:
2(t)      ( ) ( ) ( )w . w w w   ,
whence, applying PC classical formulas:
2 2(t)  ww w , (08)..
The substitution of (07) and (08) in (061) gives:
a a w r r.w w w rP O     ( ) 2 , (10),
which is the classical acceleration formula, equivalent to (061).