2 Kinematics of the material point. Simple motionsKinematics is a branch of mechanics that describes the motion of mechanical systems

as function of time, without considering cause of the motion, that is without taking into account mass and forces acting upon systems. In this chapter, we consider the kinematics of an ideal particle referred to as material point (MP), which has no mass and dimension. Though choice of coordinate system does not influence the final answer regarding the characteristics of motion, a wise choice of it may considerably simplify the calculus. The intrinsic coordinates, the curvilinear orthogonal coordinates are such possible alternative to the Cartesian coordinate system. In some cases, motion is easier to be described by introducing relative motion of two coordinate systems. In this chapter, we introduce rotation vector, and discuss trajectory, velocity and acceleration of the MP for several usual coordinate systems. Several applications are considered at the end of the chapter.

2.1 Trajectory, velocity, and acceleration in fixed Cartesian coordinate system

Trajectory is the curve described by MP in motion. Mathematically it is a function described as a parametric equation, for example, r(t), when time t is the parameter or as an implicit scalar function of coordinates, for example, f(r)=0.

In a Cartesian coordinate, MP moves between moments t and t', and changes position from r(t) to r(t'). Velocity at moment t is defined as time derivative of the position vector, that is,

(2.1.1)

or by components,

. (2.1.2)

Unit of measure for velocity, is

Acceleration at moment t is defined as time derivative of velocity, that is,

(2.1.3)

or by components,

. (2.1.4)

Acceleration is measured in,

.

2.2 Rotation vectorP1 An infinitesimal rotation can be represented by a vector.Proof P1

Let us consider a practical problem: make two successive counter-clockwise rotations of 90o of a six-sided die about first x and second about z axis, and reverse, first about z and second about x axis. We obtain a different final orientation of the die, that is, such a composition of two finite rotation is not commutative.

Now, suppose that we take a general vector c and rotate it about z axis by a small angle (this is called active rotation). This is equivalent with rotating the coordinate axes about the z axis (this is called passive rotation) by . Then, with eq. (1.5.11) relation between the rotated and initial coordinate system is,

, (2.2.1)

where , were used. Then, the rotated by vector c is expressed by,

, (2.2.2)

that is, in the first order approximation,. (2.2.3)

The above equation can easily be generalized to allow infinitesimal rotations about x and y axes by and , respectively. Infinitesimal rotation matrix about x axis is

. (2.2.4)

The previous composed rotation is commutative for the infinitesimal rotation angles and , that is,

, (2.2.5)

in the first approximation (the infinitesimal product vanishes). Thus the infinitesimal rotation matrix group is commutative. On the other hand, for a general infinitesimal rotation we can write, , (2.2.6)and thus we associate to the rotation matrix the quantity,

. (2.2.7)which is a vector [see, e.g., ref.1 p. 163]. Then, , which defines the infinitesimal rotation as a vector, automatically satisfies vector commutation relation. Concluding, since the infinitesimal rotation matrix group is commutative it can be associated to the vector group (which is commutative), while for finite rotation matrix group such a correspondence with the vector group does not exist.

For kinematics duration of this variation rotation is relevant. Thus, the time derivative,

, (2.2.8)

defines the angular velocity. From relation , with , we can write,

, (2.2.9)

which is the time derivative of a vector of constant modulus.

P2 Vector c of constant modulus and its time derivative are perpendicular to each other.Proof P2

.

Time derivative gives,

,

that is . A geometrical version of the proof is suggested in Fig. 2.

If we identify by r a rotating position vector of constant modulus, which is definition of circular motion, we have,

. (2.2.10)For a general motion is generally not perpendicular to r. Its characteristics are discussed in the following section.

2.3 Velocity and acceleration in intrinsic coordinatesFrenet frame (sometimes called Frenet–Serret) is an intrinsic moving orthogonal

frame whose origin and orientation is function of the current position of MP. To define it, we first introduce the tangent to trajectory vector. As an application in kinematics, we end this section by introducing the tangential and normal acceleration.

P3 Velocity is tangent to the trajectory.Proof P3To prove P3, we consider the arc length, s(t), as a time dependent coordinate of MP, with the direction given by the displacement direction of MP along the trajectory. Then,

(2.3.1)

We identify as the unit vector tangent to the trajectory (see below) and the modulus of the velocity or the speed. Thus, t and v have the same direction, and algebraically,

. (2.3.2)

z

y

x

c'c

Δc

Δθz

O

Fig. 1 Infinitesimal rotation vector about z axis of a constant modulus vector.

Fig. 2 Geometrical construction for relation .

c

θ

c(t)

c(t+Δt)

Δc

Δθ

θ(t)

A geometrical description might help a easier understanding of the notions introduced. Thus, in Fig. 3, we represented some trajectory of point P in the three dimensional Euclidean space (trajectory is not restricted to an in-plane curve). An infinitesimal displacement along arc AB, between s(t) and s(t+Δt), which is a portion of the osculating circle, is drawn. The osculating circle is the circle of maximum contact with the trajectory at any point. The osculating circle coincides with the circular trajectory in a circular motion. Its radius is the radius of curvature. The inverse of the radius of curvature is called curvature. With the geometry of Fig. 3, we can write,

, (2.3.3)

and

, (2.3.4)

where R is the radius of curvature. To make connection with the tangential and normal acceleration, we consider a positively defined quantity. By inspection, we observe,

, (2.3.5)

where is the tangent vector of magnitude unity. On the other hand,

, (2.3.6)

z

yx

t

t(t+Δt)

t+Δt

Δt

r+Δr

r

Δϑ/2

Δϑ

A: s(t)

B: s(t+Δt)Δr

O

R

n

b

Fig. 3 Frenet frame. t is the tangent unit vector, n is the normal unit vector, and b is the binormal unit vector. The osculating circle of radius of curvature R is the plane of t and n.

P

which is positively defined (arc length is positive quantity which increases in time) and we obtain,

(2.3.7)Next, we introduce Frenet frame. The motion of point P along a three-dimensional

path is considered in Fig. 3. The tangent direction is defined by the unit tangent vector t. By differentiating with respect to s coordinate, we obtain

,

that is is a vector perpendicular to the vector t. Geometrically, for the tangent vector derivative with respect to , we can write (see Fig. 3),

, ` (2.3.8)

Regarding direction, we observe that tends to be an inward curve vector (oriented to the interior of the curve) perpendicular to t (this holds since is a positively defined quantity). The unit vector with this orientation is referred to as the unit normal vector, and is defined by relation,

. (2.3.9)

Then we can write,

, (2.3.10a)

, (2.3.10a)

where definition for the radius of curvature is introduced by

, (2.3.11)

and is curvature. Eq. (2.3.10a) is the first Frenet formula. As ds and are positive R is positively defined. Such positive R characterizes the so-called regular curves. A more general treatment, which uses both positive and negative (or zero) curvatures is a subject of differential geometry. As connection with kinematics is much more involving when complete geometrical description of the curve is important, we only give some references on this subject.

In addition to vectors t, the unit binormal vector b is introduced by relation (see Fig. 3),

. (2.3.12)Frenet frame is moving along the curve and has a rotating motion. The instantaneous rotation vector associated to the time variation of the unit vectors t, b, n is (sometimes) called Darboux vector. It can be obtained in terms of the Frenet characteristics of motion by using eq. (2.2.9) as follows. Rotation vector is generally written as,

(2.3.13)and with eq. (2.3.10b)

(2.3.14)

that is and .

, (2.3.15)

where T is the radius of torsion, and is torsion, and .

(2.3.16)

By inspection of eqs. (2.1.13-16) one obtains

. (2.3.17)

For a plane curve, and

bbω

R

s, (2.3.18)

and as both and R are positive has direction of b. Intuitively, curvature measures the failure of a curve to be a straight line, while torsion measures the failure of a curve to be plane.

Tangential and normal acceleration

P4 The acceleration vector is sum of tangential, , and normal, ,

accelerations,. (2.3.19)

Proof P4To prove P4, we take the time derivative of eq. (2.3.7),

(2.3.20)Time derivative of the second term, with the chain rule, can be written as follows,

, (2.3.21)

If we want to correlate with the trigonometric counter-clockwise measured angle from abscissa (denoted by in Fig. 4) as usually happens in applications (for example, for an in-plane curve, , and ), we need to observe that as function of velocity direction of P along the curve. Then, to respect sign adopted convention we redefine radius of curvature by,

. (2.3.22)

Exercise (2.3.1)Prove the following relations (Frenet formulae):

i) ;

ii) ;

iii) .

Exercise (2.3.2)Prove the following relations:

i) ; ii) ; iii) ;

iv) ; v) .

Concluding, for eq. (2.3.19), we obtain proposition P4,

. (2.3.23)

In Fig. 5, the tangential and normal accelerations are represented for a three dimensional Euclidean space trajectory. Tangential acceleration is parallel to the velocity, in same direction for increasing speed, and opposite direction for decreasing speed. It characterizes speed change. Normal acceleration has direction of the normal vector, that is, inward oriented (to the interior of the trajectory) and perpendicular to the trajectory tangent vector. It characterizes velocity orientation change and is not vanishing even the speed is constant but velocity orientation changes.

y

x

Δϑ

θ(t)

θ'= θ(t+Δt)

R

Fig. 4 For in-plane curve, and Δs, , R are positively defined. is equal to as function of velocity direction on the curve (here ).

Δs

Exercise (2.3.3)Show that for an in-plane curve , the radius of curvature R is given by

P5 The normal acceleration can be written as , where is the Darboux vector associated to the instantaneous rotation of the intrinsic coordinate system.Proof P5From eqs. (2.3.19, 20), by replacing c by t in eq. (2.2.9), we have,

2.4 Velocity and acceleration in curvilinear orthogonal coordinate systemsConsider MP moving in three-dimensional space. The position vector of MP is written

in terms of an orthogonal parameterization of space as in eq. (1.6.1). The velocity vector is computed with the help of the chain rule for derivatives,

. (2.4.1)

The expression for the acceleration vector can be obtained by time derivative of eq. (2.4.1) as follows,

. (2.4.2)

In-plane polar coordinateIn Fig. 6, we represented the velocity and the in-plane polar unit vectors for an in-plane trajectory. For velocity, from eqs. (2.4.1), we have,

, (2.4.3)and for acceleration

. (2.4.4)By direct calculation from eqs. (1.6.1.4), we obtain,

, (2.4.5)

and acceleration is. (2.4.6)

z

y

x

an

at

a

Fig. 5 Normal and tangential accelerations.

n t

b

For circular motion, we set , and obtain,(2.4.7)

(2.4.8)

Spherical coordinatesFor velocity, from eqs. (2.4.1), we have,

, (2.4.9)For acceleration, from eq. (2.4.2), we have,

, (2.4.10)

The partial derivative of the spherical unit vectors are calculated with formulae from Exercise 1.6.3 as follows.

; ; ;

; ; ;

; ; . (2.4.11)

By replacement from eq. (2.4.11) in eq. (2.4.10), one obtains,

. (2.4.12)

For we obtain the velocity and acceleration for in-plane polar coordinates.

2.5 Kinematics of relative motion

i

j

θer

eθ

v

y

x

Fig. 6 Position vector r, and velocity vin the in-plane Cartesian and polar coordinates.

rP

Exercise (2.4.1)Calculate velocity and acceleration in cylindrical coordinates.

Next, we consider motion of MP with respect to two reference systems, one fixed which is usually called laboratory frame that we denote it by S, and the other mobile that denote it by S'. Origin O' of S' has position vector r0, velocity v0, and S' is rotating with angular velocity ω and angular acceleration ε respect to S. We denote by {ik} and by {ek} the set of unit vectors of S and S', respectively. We denote the MP by P, and consider it has position vector r with respect to S and r' with respect to S'. This type of description of the motion is useful, for example, in the study of the motion of a particle on the surface of the Earth. The motions are classified as:

i) absolute motion, which is the motion of the particle with respect to S; ii) relative motion, which is the motion of MP with respect to S'; iii) transport motion, which is the motion of S' with respect to S.

Composition of velocitiesWe take the time derivative of position vectors

, (2.5.1)and obtain,

, (2.5.2)where,

(2.5.3)is the velocity of O' with respect to O, and is the absolute velocity of P (with respect to S), and is the relative velocity of P (with respect to S'). In section 2.2, angular velocity associated to the time variation of a vector of constant modulus was defined by eq. (2.2.9). Next, angular velocity associated to rotation of S' will be introduced.

v

Fig. 7 Reference frames for absolute and relative motion of a material point.

r

P

e1

e3

e2

i2

x1

i1θ

x3

x2

x'3

x'2

i3

O

O'

r'

r0

ω

v0

P6 The quantity

, with (2.5.4)

is the i component of a pseudovector (see section 2.7 for definition).Proof P6The time derivative can be expressed as a linear combination of {ek},

(2.5.5)Then, the cross product of eq. (2.5.4) by ek yields

. (2.5.6)From the ortonormality condition, we have

. (2.5.7)The time derivative gives

, (2.5.8)that is is an antisimetric tensor of second order. Then, according to the tensor algebra, to any antisymmetric tensor one can associate a pseudovector by relation,

,

P7 The pseudovector introduced by eq. (2.5.4) is the angular velocity of S'.Proof P7We multiply eq. (2.5.4) by the Levi-Civita symbol,

(2.5.9)

Then,(2.5.10)

The dot product of eq. (2.5.9) by gives, (2.5.11)

On the other hand,, (2.5.12)

that is,. (2.5.13)

According to eq. (2.2.9), is the angular velocity, which, in this case characterizes rotation of S'. Once we have introduced angular velocity of S', we can write eq. (2.5.2) as follows.

. (2.5.14) is the transport velocity, that is P velocity as it were fixed (r'=ct) with respect

to S'. If r' is constant,(2.5.15)

Composition of accelerationsFrom time derivative of eq. (2.5.13), we obtain,

, (2.5.16)

where,, (2.5.17)

is the acceleration of O' with respect to O,

, (2.5.18)

is the relative acceleration of P with respect to S', and

, (2.5.19)

Concluding,, (2.5.20)

where(2.5.21)

is the transport acceleration, and(2.5.22)

is the Coriolis acceleration.

ExampleEnd A of a bar of length L is moving to the right with velocity vA. Both ends are in

permanent contact with the ground and the vertical wall (see Fig. ). Find velocity vB and angular velocity of the bar as function of .

SolutionWe consider eq. (2.5.15) with , , , and . Then, with ,

, , , , , , we have,,

that is,

From components equality of the above equation, we obtain,

,

that is has direction of k unit vector, and.

θvA

B

A

y

x

L

j

i

vB

Exercise (2.5.1)Draw schematically , , and , , , when the material point has a uniform circular motion about vector (axis) . is the angular velocity associated to the mobile system of reference. Consider the origin of the mobile system of reference fixed with respect to the laboratory frame ( ). If necessary you can use ref..

2.6 Applications

2.6.1 Linear motionLinear motion (also called rectilinear motion) is a motion along a straight line, that is a one-dimensional motion. The linear motion can be: i) uniform - linear motion with constant velocity (zero acceleration), ii) non uniform - linear motion with variable velocity (non-zero acceleration). The kinematics characteristics of the motion along x axis are obtained as follows. The acceleration is

, (2.6.1.1)

and by integration, the velocity is

, (2.6.1.2)

where .The velocity is

. (2.6.1.3)

From eq. (2.6.1.3) by integration we have,

, (2.6.1.4)

and if ,

, (2.6.1.5)

where . The trajectory is is a parabola if . By eliminating the time from eq. (2.6.1.2) and (2.6.1.5), one obtains

, (2.6.1.6)where is denoted by v.

2.6.2 Circular motionCircular motion is the movement of material point along the circumference of a circle, that is

. The circular motion can be: i) uniform rotation with constant rotation vector, ii) non uniform rotation with variable rotation vector. The kinematics characteristics of the motion

Exercise (2.5.2)Draw schematically , , and , , , when the material point uniformly moves parallel to the vector (axis) . is the angular velocity associated to the mobile system of reference. Consider the origin of the mobile system of reference fixed with respect to the laboratory frame ( ). If necessary you can use ref.2. See also section 2.6.3.

Exercise (2.5.3)Draw schematically , , and , , , when the material point has a radial uniform motion, that is perpendicular to the vector (axis) . is the angular velocity associated to the mobile system of reference. Consider the origin of the mobile system of reference fixed with respect to the laboratory frame ( ). If necessary you can use ref.2. See also section 2.6.5.

have been obtained for the in-plane polar coordinates through eqs. (2.4.7, 8). On the other hand, by using the intrinsic coordinates for in-plane motion, we find velocity and acceleration are given by eqs. (2.3.7, 23), respectively. Since circular motion is a simple motion, we can easily make connections between notions we introduced for intrinsic and in-plane polar coordinates. By comparing the two coordinate systems, we find that,

, (2.6.2.1)Thus, by using and eq. (2.3.18), we ca write,

, (2.6.2.2)where is the angular velocity (see Fig. 4 for definitions of and ). Eq. (2.6.2.2) verifies that v and t have the same direction, since and r are both positive. Rotation. In addition, by comparing eq. (2.6.2.2) with eq. (2.4.7), we obtain, (counter-clockwise rotation) if

, and (clockwise rotation) if . Then, from eq. (2.4.8), we ca write,

, (2.6.2.3)and

. (2.6.2.4)

Since in the circular motion b is constant, we can introduce the angular acceleration by,, (2.6.2.5)

and write the tangential acceleration as. (2.6.2.6)

The vectors are shown in Fig. 7

For uniform rotation, and by time integration we have,. (2.6.2.7)

For and by time integration we have, , (2.6.2.8)and with again by time integration we have,

Fig. 7 Circular motion: case . according to convention used for intrinsic coordinates. Velocity and angular velocity decrease in time.

at

a

an

ω

ε

v

b

r

n

t

. (2.6.2.9)

In eqs. (2.6.2.7-9), , , and is the initial moment. Angular velocity is measured in,

. (2.6.2.10)

Angular acceleration is measured in,

. (2.6.2.11)

2.6.3 Helical motionIn this application we describe the helix motion. The helix is given by the following t (time) parameterization:

, , , (2.6.3.5)with r, , and v constant. For , r=3, and v=0.5 the trajectory is shown in Fig. 8 for

. Time derivative of the vectors are,

, , ,, , . (2.6.3.6)

The velocity and acceleration are computed as, (2.6.3.7)

, (2.6.3.8)and they both are constant. From i) Exercise (2.3.2), we obtain the unit tangent vector,

. (2.6.3.9)

From ii) Exercise (2.3.2), we obtain the unit normal vector,. (2.6.3.10)

For this case of constant speed, v, we also can use, . From iii) Exercise (2.3.2), we have the unit binormal vector,

. (2.6.3.11)

From iv) Exercise (2.3.2), we obtain the curvature,

. (2.6.3.12)

From v) Exercise (2.3.2), we obtain the torsion,

. (2.6.3.13)

We find that both curvature and torsion are constant. Parametrization by the arc length s is also used by as v=ct.

2.6.4 Cycloidal motionA cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage. Next, i) we find the cycloid equation, and ii) we describe motion in Frenet frame of the point on the rim by considering the wheel is rolling with constant speed.i) We observe that arc length PI is equal to OI (initially, the tangent point I is in the origin O and an imaginary rope wrapped on the wheel is left on the straight line when the wheel rolls, see Fig. 9) Then, the parametric equations of position of point P are,

, (2.6.4.1a), (2.6.4.1b)

where r is the wheel radius and is shown in Fig. 9.ii) From the uniform motion of the wheel with speed , we observe and consequently and .

20

2

4

2

0

2

0

5

10

4 2

02

4 2

0

2

0

5

10

a) b)Fig. 8 Circular helix with r=3, v=0.5, , and: a) ; b) . Frenet unit vectors are represented, t-blue arrow, n-red arrow, b-green arrow at t=4.5 and 12.2 for case a), and at t=2 and 11.5 for case b).

Cartesian components of velocity are,, , (2.6.4.2)

and the speed of P is

. (2.6.4.3)

We observe , and consequently,

, (2.6.4.4)

that is I is the instantaneous center of velocity (see Chapter 3). In addition, from eqs. (2.6.4.1) and (2.6.4.2), we have,

. (2.6.4.5)

Since

, (2.6.4.6)

we obtain from eq. (2.6.4.5) and (2.6.4.6) that,

, (2.6.4.7)

that is velocity of P always has direction PA, which is the tangent to the cycloid curve.Cartesian components of acceleration are,

, , (2.6.4.8)

and the magnitude of acceleration is

. (2.6.4.9)

In Frenet frame, we have the tangential and normal accelerations,

, (2.6.4.10)

Fig. 9 Cycloidal motion.

P

O

O1

v0

v

I

θ/2

θan

at

Q

B

A

. (2.6.4.11)

Matching expression of the acceleration in Cartesian and intrinsic coordinates, we have,

, (2.6.4.12)

and obtain the radius of curvature is

. (2.6.4.13)

Thus, at the end, as it should, we obtain the radius of curvature of cycloid only depends on the geometry of the curve and not on the kinematics characteristics of P.

2.6.5 Coriolis acceleration visualization This application is suggested by https://en.wikipedia.org/?title=Coriolis_effect, where

the animation of motion we analyze is presented. Laboratory frame, S, has unitary vectors i, j, and S', has unitary vectors e1, e2. The body moves uniformly with speed v (<0) in opposite direction to j and S' is rotating uniformly with angular velocity ω >0. In S trajectory is the line in opposite direction to j. We want to find, kinematics characteristics, that is, velocity, acceleration, and curvature in S'.

Fig. 10 Uniform motion of a point with respect to a rotating frame.

From eq. (2.5.14), we have,(2.6.5.1)

and

Problem 1Consider a circular wheel rolls along a straight line without slippage such as the point on the rim has constant speed. Find velocity of the center of the wheel. Prove that projection of acceleration on the perpendicular to the straight line is constant, and reciprocal, if a material point moves with constant speed on a planar curve and projection of acceleration on some straight line in the plane is constant, then the curve is a cycloid.

(2.6.5.2a)(2.6.5.2b)

From eq. (1.6.1.6), we have,(2.6.5.3)

(2.6.5.4)

where . Thus, with , eq. (2.6.5.1) gives the equations,

. (2.6.5.5)

With the initial conditions, , solutions is

(2.6.5.6)

or by eliminating t,

, (2.6.5.7)

Trajectory is the Archimedes' spiral, which for , is drawn in Fig. 11 for (that corresponds to trajectory drawn in Fig. 10), and in Fig. 12 for

. For the parametric form from eq. (2.6.5.7), and .

1.5 1.0 0.5x '

0.5

0.4

0.3

0.2

0.1

y '

Fig. 11 Spiral trajectory in S' for

10 5 5 10x'

10

5

5

y'

Fig. 12 Spiral trajectory in S' for

Velocity in S' is

(2.6.5.8)

and acceleration is

(2.6.5.9)A correct calculus should yield the same expression of relative acceleration as that computed by using eq. (2.5.20). Thus,

, (2.6.5.10)

and, (2.6.5.11)

.(2.6.5.12)

By adding the above two equations, we have,

(2.6.5.13)By substituting we obtain indeed the same expressions as in eqs. (2.6.5.9). For the spiral we obtained, can calculate the curvature with eq. iv) from Exercise (2.3.2) as follows,

(2.6.5.14)

2.7 Special topicsAn important classification of vectors with respect to reflection (symmetry operation)

can be introduced easier now. Thus, a vector is called polar, if on reflection with respect to a

plane it matches its mirror image. For example, the position and velocity are polar vectors. On the other hand, a vector is called axial (or pseudovector), if geometrically it is of equal magnitude but of the opposite direction to its mirror image. It is obtained as vector product of two polar vectors. For example, angular velocity is an axial vector, and its geometrical description is shown in Fig.12. A simple picture, is as follows. If , then

. If a and b are polar vectors, by inversion (reflection with respect to a plane

perpendicular to the vector), , , consequently, , that is, c is not changed. It is an axial vector.

Fig. 12 and under inversion v (velocity) and r (position vector) change sign, but angular velocity is invariant as .

Mathematically, a vector is called axial if it transforms under rotation R as [1],x′ = [R]x, for polar vector, (2.7.1)

andx′ = det([R])[R]x, for axial vector. (2.7.2)

If the rotation is proper, polar and pseudo-vectors transform the same way. If the rotation is improper, polar vectors transform as regular vectors, but pseudovectors flip direction.

References[1] K-.T. Tang, Mathematical Methods for Engineers and Scientists 2: Vector Analysis, Ordinary Differential Equations and Laplace Transforms, Springer 2007.

Further reading1. J.J. Stoker, Differential geometry, Wiley, 1989. Discussion of the curvature and torsion for plane a space curves in chapters II and III.2. http://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/mathe.html. An electronic book useful for algorithms in differential geometry.3. P.P. Teodorescu, Mechanical systems, Classical models. Discussion of the relative motion in chapter 5.

1 H. Goldstein, C. Poole, J. Safko, Classical mechanics, 3rd edition, Addison Wesley, 2000.

r

-v

ω

v

-r