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 0i

i 0 ii ( ) ( ) , (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

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

X1X2X34.

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 X1 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

§ 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 30

, , : | | | |e e e , (011).

Hence, using classical notation for time derivative, we write

| |

ee

ee

i

i

i

id

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=i t, iO er , (02),

are known functions of class C26, 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 r

Ot( ) 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 t

O Ot) = 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

point8 O – is written in the form a r t n( ) ( ) t t v

vRO

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

ii

i ji 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 ii k j

k jk j

k

kj

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 j (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(ˆˆ being ,ˆsin+)ˆˆ-(cosˆˆ)t()t(i

i),ˆ(

jjjjjjjee

j , (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

iiV ee

j

, ( K ) (042),

noticing that j , 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 j .

For the instant t0 we will write:

00000i

0i0),ˆ(ˆ sin+)ˆˆ-(cosˆˆ)t()t( jjjjjee

j, (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

In this instant there will exist an initial rotation axes whose unit vector is

oee

jj

0

i0i

0

V00

2sin2sinˆ)(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.rj

, (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

i0 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 j (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 j 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 ( j ,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

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 ˆ2

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 i is the tangent unit vector to the indicator pointing on the sense of increasing s, j 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 ˆ Rˆdt

ds , since ds=Rd, (07),

and, using Frent-Serret's formulas,

)ˆ R( 2jia , since )

ds

ˆd

dt

dsˆ R(i

ia , Rdt

ds and j

i ˆR

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

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

eejj

ˆd)ˆ

(d)()t(d=)t(d ˆi

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 j and . We have:

jjjjjjj

ˆd sen)ˆˆ(d 2

sen 2]ˆ cos+)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

)ˆˆ senˆˆ cosˆ cosˆ sen (d kkikkiii (09).

Since kj ˆ dT

Rˆd where T is the torsion radius of the indicator at point K, and in agreement with the PC

rules, we can write:

j

jjkkjjjj.

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

ˆ sendT

Rˆˆ

2sen d

T

R2ˆd)

ˆ( 2

,

or, ERRADO

)]ˆˆˆˆ()ˆˆˆˆ( 2

[tg sen dˆd)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

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

20

, (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 OTr r r r .r r . .r

k 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 tTi

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

22

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

d (t) d (t) d sen d sen dV2c i j k

12

22

(17),

or, yet, in the form

d (t) d (t) d sen d com cos senVc u k u i j 12

22 2 2

, (171).

Thus, the general law (14) assumes the vector differential form

d t d t d t d t d t O 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 d i 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 = d 01 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 [tg 02

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 [ tg2

tg202

( ) ( ) ]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

tg2

tg2

02 ijk

0 0

0 0 1

1 0

, (221).

This dyadic has a vanishing third; it second has associate matrix

10

[(d ) ] (d sen

tg2

tg2

tg2

0

02 2 ijk2 2

)2

1 0

0

0 0

, (23),

and can be written in the Cartesian form

( ) ( ) ( ) ( )d d sen tg2

tg202

2 22

i j i j , (231).

As the scalar of 202)d( is ( ) ( ) ( )d sen tg2

d sen cos2 2

2 21 22

, 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 Ti

i . e e , (011),

or

( ) ( ) ( ) ( )t dt d t t dt d Ti

i e e , (012).

Definition:

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

22

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 V2

12

12

22

(04),

or

w u k i j k(t) sen sen tg 22 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 ei

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 32 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 O v 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

)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).