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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
Lagrangian Mechanics explains the subtleties of analytical mechanics and its applications in rigid body mechanics. The authors demonstrate the primordial role of parameterization, which conditions the equations and thus the information obtained; the essential notions of virtual kinematics, such as the virtual derivative and the dependence of the virtual quantities with respect to a reference frame; and the key concept of perfect joints and their intrinsic character, namely the invariance of the fields of compatible virtual velocities with respect to the parameterization. Throughout the book, any demonstrated results are stated with the respective hypotheses, clearly indicating the applicability conditions for the results to be ready for use. Numerous examples accompany the text, facilitating the understanding of the calculation mechanisms. The book is mainly intended for Bachelor's, Master's or engineering students who are interested in an in-depth study of analytical mechanics and its applications.
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Veröffentlichungsjahr: 2019
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Table of Contents
Cover
Preface
1 Kinematics
1.1. Observer – Reference frame
1.2. Time
1.3. Space
1.4. Derivative of a vector with respect to a reference frame
1.5. Velocity of a particle
1.6. Angular velocity
1.7. Reference frame defined by a rigid body: Rigid body defined by a reference frame
1.8. Point attached to a rigid body: Vector attached to a rigid body
1.9. Velocities in a rigid body
1.10. Velocities in a mechanical system
1.11. Acceleration
1.12. Composition of velocities and accelerations
1.13. Angular momentum: Dynamic moment
2 Parameterization and Parameterized Kinematics
2.1. Position parameters
2.2. Mechanical joints
2.3. Constraint equations
2.4. Parameterization
2.5. Dependence of the rotation tensor of the reference frame on the retained parameters
2.6. Velocity of a particle
2.7. Angular velocity
2.8. Velocities in a rigid body
2.9. Velocities in a mechanical system
2.10. Parameterized velocity of a particle
2.11. Parameterized velocities in a rigid body
2.12. Parameterized velocities in a mechanical system
2.13. Lagrange’s kinematic formula
2.14. Parameterized kinetic energy
3 Efforts
3.1. Forces
3.2. Torque
3.3. Efforts
3.4. External and internal efforts
3.5. Given efforts and constraint efforts
3.6. Moment field
4 Virtual Kinematics
4.1. Virtual derivative of a vector with respect to a reference frame
4.2. Virtual velocity of a particle
4.3. Virtual angular velocity
4.4. Virtual velocities in a rigid body
4.5. Virtual velocities in a system
4.6. Composition of virtual velocities
4.7. Method of calculating the virtual velocity at a point
5 Virtual Powers
5.1. Principle of virtual powers
5.2. VP of efforts internal to each rigid body
5.3. VP of efforts
5.4. VP of efforts exerted on a rigid body
5.5. VP of efforts exerted on a system of rigid bodies
5.6. Summary of the cases where the VV and VP are independent of the reference frame
5.7. VP of efforts expressed as a linear form of the
5.8. Potential
5.9. VP of the quantities of acceleration
6 Lagrange’s Equations
6.1. Choice of the common reference frame
R
0
6.2. Lagrange’s equations
6.3. Review and the need to model joints
6.4. Existence and uniqueness of the solution
6.5. Equations of motion
6.6. Example 1
6.7. Example 2
6.8. Example 3
6.9. Working in a non-Galilean reference frame
7 Perfect Joints
7.1. VFs compatible with a mechanical joint
7.2. Invariance of the compatible VVFs with respect to the choice of the primitive parameters
7.3. Invariance of the compatible VVFs with respect to the choice of the retained parameters
7.4. Invariance of the compatible VVFs with respect to the choice of the parameterization
7.5. Perfect joints
7.6. Example: a perfect compound joint
8 Lagrange’s Equations in the Case of Perfect Joints
8.1. Lagrange’s equations in the case of perfect joints and an independent parameterization
8.2. Lagrange’s equations in the case of perfect joints and in the presence of complementary constraint equations
8.3. Example: particle on a rotating hoop
8.4. Example: rigid body connected to a rotating rod by a spherical joint (no. 1)
8.5. Example: rigid body connected to a rotating rod by a spherical joint (no. 2)
8.6. Example: rigid body subjected to a double contact
9 First Integrals
9.1. Painlevé’s first integral
9.2. The energy integral: conservative systems
9.3. Example: disk rolling on a suspended rod
9.4. Example: particle on a rotating hoop
9.5. Example: a rigid body connected to a rotating rod by a spherical joint (no. 1)
9.6. Example: rigid body connected to a rotating rod by a spherical joint (no. 2)
9.7. Example: rigid body subjected to a double contact
10 Equilibrium
10.1. Definitions
10.2. Equilibrium equations
10.3. Equilibrium equations in the case of perfect joints and independent parameterization
10.4. Equilibrium equations in the case of perfect joints and in the presence of complementary constraint equations
10.5. Stability of an equilibrium
10.6. Example: equilibrium of a jack
10.7. Example: equilibrium of a lifting platform
10.8. Example: equilibrium of a rod in a gutter
10.9. Example: existence of ranges of equilibrium positions
10.10. Example: relative equilibrium with respect to a rotating reference frame
10.11. Example: equilibrium in the presence of contact inequalities
10.12. Calculating internal efforts
10.13. Example: internal efforts in a truss
10.14. Example: internal efforts in a tripod
11 Revision Problems
11.1. Equilibrium of two rods
11.2. Equilibrium of an elastic chair
11.3. Equilibrium of a dump truck
11.4. Equilibrium of a set square
11.5. Motion of a metronome
11.6. Analysis of a hemispherical envelope
11.7. A block rolling on a cylinder
11.8. Disk welded to a rod
11.9. Motion of two rods
11.10. System with a perfect wire joint
11.11. Rotating disk–rod system
11.12. Dumbbell
11.13. Dumbbell under engine torque
11.14. Rigid body with a non-perfect joint
Appendix 1: Tensors
Appendix 2: Typical Perfect Joints
A2.1. Point contact between two rigid bodies
A2.2. Ball-and-socket joint (or spherical joint)
A2.3. Cylindrical joint
A2.4. Pivot (or hinged joint)
A2.5. Prismatic or sliding joint
A2.6. Helical joint (or screw joint)
Bibliography
Index
End User License Agreement
List of Illustrations
Chapter 1
Figure 1.1. Reference solid and observation instrument
Figure 1.2. The position in R
i
of the physical coordinate system for R
i
Figure 1.3. Positions of a particle, observed at the same instant by two differe...
Figure 1.4. Multiple positions in the presence of multiple reference frames
Figure 1.5. Representation of the problem in Figure 1.4 using the common referen...
Figure 1.6. (a) Disc rolling on a plane support; (b) positions of the systems in...
Figure 1.7. Coordinate system of reference frame R
i
Figure 1.8. Composite derivative of a vector
Figure 1.9. Velocity at the contact point between two rigid bodies
Chapter 2
Figure 2.1. Particle moving in a plane
Figure 2.2. The Euler angles. Global view. For a color version of this figure, s...
Figure 2.3. Euler angles. Decomposition in three successive rotations
Figure 2.4. Disc moving in a plane
Figure 2.5. Disc in contact with a plane
Figure 2.6. Example of parameters
Figure 2.7. Disc rolling on an axis
Chapter 3
Figure 3.1. Examples of forces
Figure 3.2. Example of torque
Chapter 5
Figure 5.1. Force exerted by a spring on a rigid body
Figure 5.2. Constraint inter-efforts due to a spring
Chapter 6
Figure 6.1. Disk rolling along an axis
Figure 6.2. Cart connected to a particle
Figure 6.3. External efforts on the cart and particle
Chapter 7
Figure 7.1. Particle moving along a planar curve: parameterization using Cartesi...
Figure 7.2. Particle moving along a planar curve: parameterization using polar c...
Figure 7.3. Particle on a hoop rotating about a fixed axis
Figure 7.4. The two bijections [7.20] and [7.36]
Figure 7.5. Particule on a hoop
Figure 7.6. Mechanical joints in a system
Figure 7.7. Perfect joint between a particle and a planar curve – parameterizati...
Figure 7.8. Perfect joint between a particle and a planar curve – parameterizati...
Figure 7.9. Perfect joint between a particle and a hoop
Figure 7.10. The relative virtual velocity of p with respect to (C)
Figure 7.11. Perfect joint between a disc and an axis
Figure 7.12. Combination of a ball-and-socket joint and a point contact
Figure 7.13. Only the spherical joint at A
Figure 7.14. Only the point contact at I
Chapter 8
Figure 8.1. Particle moving around a hoop
Figure 8.2. Rigid body connected to a rotating rod by a spherical joint
Figure 8.3. Rigid body in contact with a rotating plate and the vertical axis
Figure 8.4. Contact at point K
Figure 8.5. Contact at point I
Chapter 9
Figure 9.1. Disk rolling along a suspended rod
Figure 9.2. Rigid body connected to a rotating rod by a spherical joint
Chapter 10
Figure 10.1. Example of parametric equilibrium
Figure 10.2. Equilibrium of a jack
Figure 10.3. Equilibrium of a lifting platform
Figure 10.4. Equilibrium of a rod in a gutter
Figure 10.5. Equilibrium position of the rod as a function of ratio l/R
Figure 10.6. Equilibrium of a disk connected to a spring
Figure 10.7. Relative equilibrium with respect to a rotating reference frame
Figure 10.8. Equilibrium of a rod connected to a spring in the presence of a wal...
Figure 10.9. The function H(θ) used in this example
Figure 10.10. Variation of vs. θ
Figure 10.11. Hexagonal system with eight bars
Figure 10.12. New system with the bar A′ A cut
Figure 10.13. New system with the bar B′ B cut
Figure 10.14. Tripod
Figure 10.15. New system with the degree of freedom θ
Chapter 11
Figure 11.1. Equilibrium of two rods
Figure 11.2. Equilibrium of an elastic chair
Figure 11.3. Force and second derivative of the potential versus angle α.
Figure 11.4. Equilibrium of a dump truck
Figure 11.5. Ratio F/mg versus θ
Figure 11.6. Equilibrium of a set square
Figure 11.7. Motion of a metronome
Figure 11.8. Equilibrium and motion of a hemispherical envelope
Figure 11.9. A block rolling on a cylinder
Figure 11.10. The primitive parameters of the block
Figure 11.11. Disk welded to a rod
Figure 11.12. Disk welded to a rod
Figure 11.13. Movement of two rods
Figure 11.14. System with a perfect wire joint
Figure 11.15. Rotating disk–rod system
Figure 11.16. Dumbbell
Figure 11.17. Dumbbell under engine torque
Figure 11.18. Rigid body with a non-perfect joint
Appendix
Figure A2.1. Point contact
Figure A2.2. Ball-and-socket joint
Figure A2.3. Cylindrical joint
Figure A2.4. Pivot joint
Figure A2.5. Prismatic joint
Figure A2.6. Helical joint
Guide
Cover
Table of Contents
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To our parents; to Nicole and Younnik, to Mai; to Yemma Nounou, Shehrazed, Rayane, Redwan, Elyas
Series Editor
Noël Challamel
Lagrangian Mechanics
An Advanced Analytical Approach
Anh Le van
Rabah Bouzidi
First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd
27-37 St George’s Road
London SW19 4EU
UK
www.iste.co.uk
John Wiley & Sons, Inc.
111 River Street
Hoboken, NJ 07030
USA
www.wiley.com
© ISTE Ltd 2019
The rights of Anh Le van and Rabah Bouzidi to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2019937362
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-436-0
Preface
There are two distinct yet equivalent approaches to solving a problem in rigid body mechanics: the Newtonian approach, based on Newton’s laws, and Lagrange’s approach, based on the postulate called the principle of virtual powers, and which lead to Lagrangian or analytical mechanics.
Although both approaches yield equivalent results, they differ on a number of points both in terms of conception as well as formulation. In addition to the usual ingredients – velocity, acceleration, mass and forces – analytical mechanics involves a new concept that does not exist in Newtonian mechanics, which is given the enigmatic name “virtual velocity”. It is based on this concept that virtual powers are defined. While Newton’s laws state relationships between vector quantities (force and acceleration), the principle of virtual powers, is written in terms of virtual powers, which are scalar quantities. Analytical mechanics is also distinguished by the fact that parameterization plays a primordial role here: given the same mechanical problem, it is possible to choose different parameterizations and the resulting equations – and thus, the information they yield – differ based on the chosen parameterization. Another salient feature demonstrated in analytical mechanics is that once the parameterization is chosen, the kinematical behavior of the system, a vector description in essence, is condensed into a scalar function, called the parameterized kinetic energy.
While Newtonian mechanics brings into play physical concepts that are easy to apprehend, Lagrange theory appears more complicated because of the virtual velocity and the very statement of the principle of virtual powers. However, two technical advantages compensate for this conceptual difficulty:
(i) It is seen that the physicist’s task is practically reduced to choosing the appropriate parameterization for the system under study. Once the parameterization is chosen, the Lagrange equations systematically lead to as many equations as there are unknowns (if the problem is well-posed). Of course, it is also possible to obtain a sufficient number of equations using the Newtonian approach, but there is no systematic way of doing so. One must first carry out an analysis of the applied forces and must often subdivide the mechanical system being studied in an adequate manner and then write the equations for the subsystems.
(ii) The operations carried out in analytical mechanics – especially the calculation of the parameterized kinetic energy and its derivatives – are purely algebraic and, therefore,
programmable
. This explains the success of analytical mechanics in the study of complex systems containing a large number of kinematic parameters, where it is more difficult to obtain the equations of motion using the Newtonian approach.
This book strives to explain the subtleties of analytical mechanics and to help the reader master the techniques to obtain Lagrange’s equations in order to fully use the potential of this elegant and efficient formulation. It is meant for students doing their bachelors or masters degree in Engineering, who are interested in a comprehensive study of analytical mechanics and its applications. It is also meant for those who teach mechanics, engineers and anyone else who wishes to review the fundamentals of this field. Although the content does not require any prior knowledge of mechanics, it is preferable for the reader to be familiar with the Newtonian approach.
The format adopted in this book
When writing this book, the authors laid out the objective of revisiting analytical mechanics and presenting it from a different angle both in form and style. This was done:
– by adopting a more axiomatic and formal framework than a conventional course,
– and by taking special efforts concerning notations to arrive at mathematical expressions that are both precise and concise.
By “axiomatic framework”, we mean that all through this book, the chapters are constructed in a manner that is similar to a mathematical discussion, where the ingredients are presented in the following order:
the definitions to establish the vocabulary used,
the theorems, where results are proven and where we specify the hypotheses, clearly indicating the conditions of applicability for this or that result,
and finally, examples to illustrate the nuances of the theory and the mechanisms of the calculation.
While the theory is constructed in a deductive manner and forms a monolithic block, each theorem is written in a self-contained and condensed form – that is, hypotheses followed by results – in order to make it “ready to use”.
Synopsis
This book contains 11 chapters and two appendices:
Chapter 1 reviews the basic ingredients of kinematics: time, space and the observer (or reference frame). We present here the key concept of the derivative of a vector with respect a reference frame and introduce the concept of a “common reference frame”, which is used to connect or relate two different reference frames.
Chapter 2 focuses on an important operation in analytical mechanics, namely the parameterization of the mechanical system being studied. This operation consists of choosing a certain number of primitive parameters of the system, expressing all existing constraints in terms of these parameters and, finally, classifying the constraint equations into two categories, called “primitive” and “complementary” equations. This task, incumbent on the physicist, is specific to analytical mechanics and has no equivalent in Newtonian mechanics. It is important because, as we will see, the Lagrange equations that are obtained (and, consequently, the information that may be extracted from them) are essentially dependent on the choice of parameterization.
The parameterization of the system leads to the definition of the parameterized velocities and the parameterized kinetic energy, the concept on which the Lagrange kinematic formula is based.
Chapter 3 reviews the conventional concept of efforts that includes forces and torques. These can be classified as either internal efforts and external efforts, or given efforts and constraint efforts. The virtual powers of efforts are calculated in Chapter 5 depending on how the efforts are categorized.
Chapter 4, dedicated to virtual kinematics, introduces new kinematic quantities that are the counterparts of those introduced in Chapter 1 and are labeled “virtual”: the virtual derivative of a vector with respect to a reference frame, the virtual velocity of a particle, the virtual velocity fields in a rigid body or a system of rigid bodies, and, finally, the virtual angular velocity of a rigid body. This chapter provides formulae to calculate these quantities and, notably, the formulae for the composition of virtual velocities.
Chapter 5 deals with virtual power, which is, grosso modo, the product of an effort, seen in Chapter 3, and a virtual velocity, seen in Chapter 4. The presentation closely follows the conventional presentation of real powers in Newtonian mechanics and we obtain several results that are analogous to those obtained for real powers. Two results are, nevertheless, specific to analytical mechanics: the virtual power expressed as a linear form and the power of the quantities of acceleration.
Chapter 6 shows how to exploit the principle of virtual powers using the results obtained in the previous chapters in order to arrive at the final product of analytical mechanics, namely the famous Lagrange equations. In this chapter, we also see several examples which illustrate how important the choice of parameterization is and what consequences it has on the obtained results. This chapter concludes with the statement of Lagrange equations in a non-Galilean reference frame.
Chapters 7 and 8 are concerned with perfect joints. The chief advantage of these joints is that the generalized forces present in the right-hand side of the Lagrange equations are then zero or may be easily calculated using Lagrange multipliers. The concept of the perfect joint also exists in Newtonian mechanics, but they are defined there in a simpler manner, with more obvious consequences. In analytical mechanics, the definition of a perfect joint is less natural inasmuch as it involves the parameterization and the virtual velocities that are compatible with the complementary constraint equations. It is, therefore, important to verify that the perfect character of a joint is intrinsic, i.e. it does not depend on the chosen parameterization. For this reason, a large section is dedicated to the invariance of virtual velocity fields with respect to the parameterization.
Chapter 9 is dedicated to the first integrals, which offer the advantage of yielding first-order differential equations that are easier to solve. The first integral called “Painlevé’s first integral” has no equivalent in Newtonian mechanics and presents the unique feature of being able to exist for systems that receive energy from the exterior. The energy integral resembles that of Newtonian mechanics. However, the conditions for application are slightly different.
Chapter 10 shows how the Lagrange equations are simplified in the particular case of equilibrium. The chapter also contains a brief discussion on the question of the stability of an equilibrium position.
Chapter 11 contains several examples to revise all the concepts presented in the book.
The book concludes with two appendices. Appendix1 provides a few basic concepts related to second-order tensors, which are necessary for studying kinematics.
Appendix 2 complements Chapter 7 and establishes the necessary and sufficient conditions of perfectness for joints that are usually encountered in mechanics.
By introducing the concept of virtual quantities, Lagrangian mechanics is more abstract than Newtonian mechanics. Nonetheless, it proves to be more fertile in that it extends beyond the mechanics of rigid bodies in order to lead to more elegant and systematic formulations in the field of mechanics of deformable media, to say nothing of physics in general. This is why analytical mechanics is one of the fundamental subjects taught in mechanics. The authors of this book hope to offer the reader a comprehensive view of and perfect mastery over this beautiful formulation.
Acknowledgments
The first author wishes to thank Mrs. Annick Molinaro, a lecturer of mechanics in the Faculty of Sciences at Nantes and the Ecole Nationale Supérieure de Mécanique (E.N.S.M.) in the 1970s and 1980s, who inculcated in him the systematic use of the so-called “double index notation”, which is both concise and precise and which has been used throughout this book. He also wishes to thank his former colleagues in the teaching team at E.N.S.M. Certain examples in this book, taken from the team’s tutorial archive, are the result of long reflection and discussions between several colleagues, some of whom are no longer with us.
The second author wishes to thanks Professor Pierre Jouve for having supported him in his early years of teaching. Some of the examples that appear in this book, even if they have been rewritten, were originally his creations. May he be forever gratefully remembered for his significant contribution to the teaching of mechanics at the Faculty of Science and Technology of Nantes.
ANH LE VAN AND RABAH BOUZIDINantes, France, March 2019
1Kinematics
Mechanics, the science that studies the motion of bodies, generally comprises three parts: (i) kinematics, where motions are described regardless of the causes that provoked the motions; (ii) kinetics, where we combine kinematics with the concept of mass; and (iii) dynamics, where we add the concept of the forces acting on the body.
This chapter will focus on kinematics and will review the essential concepts of classical mechanics: the observer or reference frame, time, space, as well as velocity and acceleration. Chapter 2 will focus on the parameterization, the parameterized kinematics as well as the parameterized kinetic energy, concepts that are essential for analytical mechanics. Forces will be studied in Chapter 3.
Apart from the above-mentioned real quantities, analytical mechanics also brings in virtual quantities (virtual velocity, virtual power), which will be discussed in Chapters 4 and 5.
1.1. Observer – Reference frame
We admit the existence of observers, real or fictional, located in areas that may or may not be accessible to humans. An observer is denoted by , the integer index i making it possible to distinguish from the different observers.
The observer needs an instrument called a clock to note the start, end or duration of an event (the notions of event and time will be seen hereafter).
They also need an observation instrument placed on a support called a reference solid, to observe mechanical systems and their positions in physical space and at each instant (Figure 1.1; the notions of mechanical system and position will be seen in section 1.3).
Figure 1.1.Reference solid and observation instrument
Definition. A reference frame is an observer equipped with a clock and an observation instrument placed on a reference solid. The reference frame associated with the observer will be denoted as Ri.
Thus, the term “reference frame” is almost synonymous with “observer”, while having a slightly more precise sense. Furthermore, the term “observer” makes one think of the presence of a human in the study, whereas “reference frame” is more impersonal. For notational convenience, the notation Ri is preferred to in mathematical relationships.
1.2. Time
Each observer possesses the following concepts with respect to time: (1) the consciousness in the moment when a given instantaneous event takes place; (2) the perception of the chronological order, of anteriority or posteriority, and, consequently, of the simultaneity of two instantaneous events; (3) and finally, the perception of the duration of an event.
In order to carry out calculations, the observer must transcribe the data from the clock into a mathematical set. It is decided that this set is a one-dimensional affine space called (mathematical) “time”, which is the set ℝ of all real numbers equipped with (1) the partial order ≤ in order to account for the chronological order (anteriority or posteriority) and (3) a structure of vector space or affine space, which makes it possible to represent the duration by a scalar.
1.2.1. Date postulate
The following postulate shows how an observer passes from the set of instantaneous events to the time set:
Date postulate.
An observer possesses a clock which enables them to match each instantaneous event with one and only one scalar, t(i), called the instant of an event observed by the observer (or with respect to the reference frame Ri).
In short, the observer is able to mark an instantaneous event with an instant.
The upper index (i) in t(i) reminds us that the involved instants are observed with respect to the reference frame Ri.
1.2.2. Date change postulate
In a mechanical problem with multiple observers, each observer has his own clock that allows him to mark an instantaneous event by an instant t(i) in ℝ. The question which then arises is that of communication between these observers and more precisely, that of the correspondence between their different observed instants. The date change postulate makes it possible to establish a relation between simultaneously observed instants with respect to different reference frames:
Date change postulate. Let and be two arbitrary observers. According to the date postulate [1.1], it is possible to know the instants t(1) and t(2) marked by and , respectively, and corresponding to the same instantaneous event.
When we consider different instantaneous events, we obtain different corresponding couples (t(1), t(2)). It is assumed that there exists a continuous and strictly increasing mapping, χ21: ℝ→ℝ, that gives the date t(2) as a function of the date t(1):
Using the mapping χ21, we know the correspondence between the instants noted by the two observers. If we know the instant t(1), noted by the observer , we may deduce the instant t(2) noted simultaneously by the observer , by making , and vice versa. The term “simultaneously” signifies that the two dates correspond to the same instantaneous event.
Since the application χ21 is continuous and strictly increasing, it is a homeomorphism (i.e. bicontinuous bijection) from ℝ to ℝ. The most natural choice is to take χ21 equal to an affine function or, even more simply, to take it equal to the identity function. In other words, it is assumed that all observers mark the same instantaneous event with the same scalar:
An important consequence of this choice is
This is why we will only encounter a single derivative with respect to time from now on.
Eventually, a single clock is enough for all observers and this is what we will assume in the sequel.
1.3. Space
1.3.1. Physical space
The physical space (or the material world) is composed of vacuum and matter. It is common to all observers (who are embedded in the same physical space). It is intrinsic in the sense that it exists even in the absence of observers.
1.3.1.1. Mechanical system
A mechanical system is loosely defined as an invariant collection of matter. This definition is an intuitive one, but is not rigorous as the word matter has not been defined.
As with physical space, the concept of a mechanical system is intrinsic and has been defined independently of the observer.
1.3.1.2. Particle
It is assumed that with the help of an observation instrument, the observer is able to distinguish, within the physical space, mechanical systems (or mechanical subsystems) that they consider to be small. This kind of system is called a particle (or material point) for the observer .
This definition is not precise because we do not know how the observer may evaluate that a system is small. The particle is a model, that is a choice made by the physicist of how to represent the system under consideration, with respect to the nature of the problem being studied and the objective that is fixed, and it is the simplest model in mechanics.
With the concept of the particle having been defined for a given observer, it can be assumed that the concept of particle is, in fact, invariant for all observers, i.e. in a given problem, a system perceived to be a particle will be a particle for any other observer.
We consider that any mechanical system is a
union, which may or may not be finite, of particles
. As the family of particles in question is always the same, it is assured that we have an invariant collection of matter.
In this book, a mechanical system is made up of a finite number of rigid bodies (of which some may be reduced to particles).
1.3.2. Mathematical space
In order to carry out calculations, the observer must transcribe the results of their observation over time into mathematical data. To prepare for this task, we introduce a new mathematical structure, apart from time, which is defined as follows:
Definition. The mathematical space, denoted by ε, is a three-dimensional real affine space, given beforehand.
[1.5]
As will be seen in the position postulate [1.10], all observers use the mathematical space ε in order to enter their observations from the physical space. The following terms associated with the space ε are well known in mathematics.
Definition.
[1.6]
An element
A
∈
ε
is called
a point
.
A bi-point
or
a vector
is the difference between two points
A, B
∈
ε
.
The vector space
is the vector space
E
associated with
ε
.
A coordinate system
of
ε
is defined by a point
O
∈
ε
and a basis of
E
that is made up of three independent vectors. We denote it by (
O
;
e
) or . The point
O
, which is arbitrarily chosen in
ε
, is called the
origin
of
ε
or
of the coordinate system
.
If , then (
x
1
, x
2
, x
3
) are called
the components
of the vector in the basis , or
the coordinates
of the point
A
in the coordinate system .
By choosing, beforehand, an origin O in ε and a basis in E, in other words, by choosing, beforehand, a coordinate system in ε, we have the following bijections:
1.3.3. Position postulate
Definition.
[1.8]
A physical coordinate system for the observer (or reference frame Ri) is the quadruplet made up of four non-coplanar particles (real or fictitious, i.e., materialized or non-materialized) taken in the reference solid (Figure 1.2(a)).
Keep in mind that the order of the particles, oi, ai, bi, ci, is important: the first particle oi is, by definition, the origin of the physical coordinate system, the particles ai, bi, ci will correspond, respectively, to the three vector of the basis E, as will be seen in [1.14b].
The physical coordinate system (oi; ai, bi, ci) should be distinguished from the mathematical coordinate system in ε.
Any couple formed by two particles will be called a physical segment and [pq] will denote the segment formed by two particles p, q. It is assumed that by means of their observation instrument, the observer is able to measure the distance between any two particles p, q and this distance is called the physical length of the segment [pq], and also that they are able to measure the angle between any two segments. Two segments are said to be physically orthogonal when the angle between them is 90◦ (the right angle may be detected by using, for example, a plumb line and the water level at the same location or by using a protractor).
Definition.The physical coordinate system (oi;ai, bi, ci) is said to be physically orthonormal if the segments [oiai], [oibi], [oici] are of unit length and mutually physically orthogonal.
[1.9]
The following postulate stipulates how an observer, at every instant, carries out the passage from the physical space to the mathematical space:
Position postulate in Ri
(a) The observer possesses an observation instrument placed on the reference solid, by means of which they are able, at each instant, to locate a particle in the physical space with a point in the space
ε
. More precisely, at each fixed instant
t
, they are able to establish a correspondence between each particle
p
in the physical space and a single point
P
(
i
)
in mathematical space
ε
. This point is denoted by and is read as
the position of the particle
p
with respect to
R
i
(or in
R
i
) at instant
t
. The upper index (
i
) reminds us that this involves results coming from the observer
R
i
.
The statement is then written as
(b) Conversely, given a fixed instant
t
, any point in the mathematical space
ε
is the position in
R
i
of at least one particle, real or fictitious (that is, materialized or not):
(c)
Convention on the physical coordinate system of
R
i
[1.13]
The following clauses concern the physical coordinate system and include the hypotheses that are part of the postulate as well as the conventions that aim to simplify the exposition.
Let be a physical coordinate system for the reference frame Ri (see definition [1.8]), and let us denote the respective positions of the particles oi, ai, bi and ci in Ri at instant t by and :
The observer chooses their physical coordinate system (
o
i
;
a
i
, b
i
, c
i
), physically orthonormal in the sense of definition
[1.9]
.
The positions in
R
i
of the four particles
o
i
,
a
i
,
b
i
,
c
i
are points in
ε
that are
fixed
over time. To simplify, we will make them equal to the fixed points that make up the mathematical space
ε
as follows:
○ ∀
t
, let us take the position of the origin
o
i
, in
R
i
, of the physical coordinate system equal to the origin
O
in
ε
(
Figure 1.2
):
○ It is assumed that it is possible to take
where are the vectors of the basis of E.
Figure 1.2.The position in Ri of the physical coordinate system for Ri
The definition of the position of a body or a mechanical system is based on that of a particle:
Definition.
[1.15]
The position of a body S, in the reference frame Ri and at an instant t, denoted , is, by definition, the set of positions P(i), at t, of all the particles in S. This is a subset of ε.
If S is a finite union of particles, posRi(S, t) is a discrete subset of ε made up of a finite number of points. If not, posRi(S, t) may be a volume, a surface or a curve in ε and we then say that the body S is a volumetric, surface or line body.
Definition.
[1.16]
Let us consider a mechanical system S, made up of a finite number of bodies (some of which may be reduced to particles). The position, in a reference frame Ri and at an instant t, denoted by , is, by definition, the set of positions P(i), at t, of all particles in S, in other words, the set of positions in Ri at t of all the constituent bodies. This is a subset of ε.
If S is a finite union of particles, posR0(S, t) is a discrete subset of ε. Otherwise, posR0(S, t) may be a union of volumes, surfaces or curves in ε and we then say that the mechanical system S is a volumetric, surface or line system.
1.3.4. Typical operations on the mathematical spaceε
We will review here the mathematical operations that are typically carried out on the mathematical space ε, together with the interpretations and the corresponding physical operations.
Endowing
E
with a structure of Euclidean space
Let be a basis of E. It can be easily verified that the following bilinear mapping defined on E is a scalar product (that is, a bilinear form, that is symmetric and positive definite):
The space E equipped with this scalar product is a Euclidean space. The definition [1.17] implies that
that is, the basis is orthonormal.
We can now understand why one had better choose a physical coordinate system that is physically orthonormal, as was done in convention [1.13]. The coordinate system of ε is the image of the physically orthonormal physical coordinate system and this is consistent with the fact that the basis in E is orthonormal.
The scalar product [1.17] makes it possible to define the following norm in E denoted by ǁ.ǁ:
The scalar is called the norm or the magnitude of vector
Orienting
E
and
ε
The observer names the four particles oi, ai, bi, ci of their physical coordinate system according to the right-hand rule, i.e. in such a way that when they are placed along oiai (their feet on oi and their head at ai) and when they are looking toward bi, they have ci on their left. The observer orients E and ε by deciding that is a right-handed orthonormal basis. Such a right-handed orthonormal basis is represented in Figure 1.2.
Throughout the sequel, we will work only with right-handed orthonormal bases.
1.3.5. Position change postulate
The difficulty when several observers come into play is establishing the relationship between their different observation results. Indeed, as the observers choose their reference solids independently of each other, it turns out that even if they observe the same physical space, there is, a priori, no relationship between the positions observed by various observers.
To illustrate this fact, let us consider two observers and , or two reference frames R1 and R2. Let us fix an instantt (this has a sense according to hypothesis [1.3]) and let us consider a particle p in the physical space (Figure 1.3(a)).
The observers learn the same mathematics and they use the same mathematical space ε where they write down the results of their observation from the physical space. According to the position postulate [1.11], the observer may mark the position ofpinR1, att, as P(1)posR1(p, t), this is a point in ε (Figure 1.3(b)). The observer may, in turn, mark the position ofpinR2, att, as P(2)posR2(p, t), that is, a priori, another point in ε (Figure 1.3(b)).
There arises the following question: what is the relationship between the two points P(1) and P(2) in ε? In other words, what is the relationship that gives the position of a particle in R2 as a function of the position of the same particle in R1 at the same instant?
The following postulate, called the position change postulate, makes it possible to connect the observed positions in different reference frames for the same particle at a given instant.
Figure 1.3.Positions of a particle, observed at the same instant by two different observers
Position change postulate. ∀ reference frames R1 and R2, ∀ instant t, ∃ a positive isometryQ21(t): ε → ε such that ∀ particle p, located, respectively, at P(1)≡posR1 (p, t), P(2)≡posR2 (p, t) in R1 and R2, we have
where the right-hand side is the image of point P(1) under Q21(t).
In other words, at every instant t, the positions of the same particle, observed in the two reference frames R1 and R2, are connected by a positive isometry denoted by Q21(t).
The bijection Q21(t) constitutes a “dictionary”, with the help of which the two observers are able to establish correspondence between their observed positions. This dictionary varies with time.
REMARK. According to [1.19], if the position of a particle p is fixed in R1, that is, if the point P(1)posR1 (p, t) is a fixed point in ε, then the position (P(2)) of this particle in R2 will be, a priori, a point that varies with time and vice versa.
■
We will need the following terminology:
Definition.A biposition is the difference between two positions of two particles (just as a bipoint is the different between two points).
[1.20]
Using the position change postulate [1.19], we can establish correspondence between the bipositions observed in two different reference frames. In order to do this, let us use the following mathematical result, which is well known for a point isometry:
Theorem and definition.Q21(t) is a point isometry, is affine and its linear part denoted by is a vector isometry.
We then have
where the right-hand side is the product of and vector .
is, by definition, the rotation tensor of R1with respect to R2at instant t or the reference frame change tensor (see Appendix 1 for a brief review of tensor algebra).
From this, we have the following result, which is the vector version of the point relationship [1.19] :
Theorem of biposition change. At a given instant t,
let P(1), P(2) be the positions of a particle, observed, respectively, in R1, R2, and let Q(1), Q(2) be the positions of another particle, observed, respectively, in R1, R2. We have
PROOF. It follows immediately from the position change postulate [1.19] and from theorem [1.21] that
The following theorem brings together three very useful properties that isometries possess:
Theorem. ∀ reference frames R1, R2, R3, ∀t,
where I is the identity function and the symbol T denotes the transpose.
These equalities are also valid when we replace the point isometries Qij with the rotation tensors .
1.3.6. The common reference frameR0
A single event can be observed in different reference frames (R1, R2, R3,…) at different instants (t(1), t(2), t(3),…), connected by the date change postulate [1.2]. There is no difficulty even if there are many of these instants as we have assumed, in [1.3], that they are equal: t(1)t(2)t(3)···. This makes it possible to use the same symbol t to denote all of them.
With regard to the space, however, the situation becomes a little more complex. At a given instant t, a particle p is observed at a multitude of positions P(1), P(2), P(3),…, which are related through the position change postulate [1.19]. All these points P(1), P(2), P(3),… are elements of the (mathematical) space ε, and they are, a priori, distinct and cannot be confused (contrary to what is done with instants).
Since the notations multiply rapidly with an increase in the number of the involved particles and reference frames, it is practically impossible to represent, in space ε, all the observation results in the different reference frames. For example, if there are two particles p, q and three reference frames R1, R2, R3, we have, at a given instant, six positions represented in Figure 1.4. Thus, the figures quickly become indecipherable and too tedious to make.
Fortunately, it can be seen that there is no need to represent the observations in all the existing reference frames. For example, the observations in R1, R2 give rise to two different sets of points; however, they are, in fact, identical within an isometry, which is the isometry Q21 introduced in [1.19]. As a result, regardless of the number of reference frames in play, we can content ourselves with using the observation results from one single reference frame that is arbitrarily chosen from all the reference frames.
Figure 1.4.Multiple positions in the presence of multiple reference frames
Definition and notational convention.
[1.24]
We arbitrarily choose one among the existing reference frames. This is called the common reference frame and denoted by . As it is enough to represent the observed positions in a single reference frame, we will choose to do this in the common reference frame, R0.
We agree to give R0 a special status by simplifying the notation in R0 as follows: the position P(0)posR0(p, t) of a particle p at an instant t, observed in the common reference frame R0, will be denoted by P, without the index (0) for the reference frame:
(while the position P(i) in another reference frame Ri must include the index (i)).
Simply put, we will represent only positions inR0and these positions will not include the index (0) for the reference frame.
In the example in Figure 1.4, by choosing R0R1 we obtain Figure 1.5, which is more legible.
Figure 1.5.Representation of the problem in Figure 1.4 using the common reference frameR0
In a mechanical problem that involves only a single reference frame, the problem of choosing a common reference frame does not arise at all. The reference frame R0 is the only available one. On the other hand, if several reference frames come into play, we may wonder which one is to be chosen as the common reference frame R0. While the choice is arbitrary in principle, in practice the nature of the problem often causes a particular reference frame to come up and naturally impose itself as the most convenient reference frame for the role of R0: this is the reference frame – i.e. the observer – with which the physicist identifies from the start and this is the reference frame that will be chosen as the common reference frame R0.
The reference frame R0 is common in the sense that it is shared by everyone. It may also be referred to as the script reference frame in the sense that, in general, the positions which appear in the mathematical relationships are the positions in R0.
By applying relationship
[1.22]
, with
R
1
R
i
, R
2
R
0
and by taking into consideration the notation convention
[1.24]
, we obtain:
By making in this relationship, and by taking into account [1.14], , we obtain
Using the notation convention
[1.24]
, the convention on the physical coordinate system
[1.13]
is written in
R
0
as follows:
Convention on the physical coordinate system of R0.
[1.27]
Let (o0;a0, b0, c0) be a physical coordinate system for the reference frame R0. Let O0, A0, B0 and C0 denote the respective positions of the particles o0, a0, b0 and c0, in R0 and at the current instant t:
The observer
R
0
chooses their physical coordinate system (
o
0
;
a
0
, b
0
, c
0
), physically orthonormal as per definition
[1.9]
.
The positions of the four particles
o
0
,
a
0
,
b
0
,
c
0
in
R
0
are the points in
ε
that are
fixed
over time. For simplicity, they will be equaled to the fixed points forming the coordinate system of
ε
as follows:
○ ∀
t
, it is agreed that the position in
R
0
of the origin
o
0
of the physical coordinate system is equal to origin
O
of
ε
:
○ It is assumed that it is possible to take
where are the vectors of the basis of E.
EXAMPLE. Let us consider the bidimensional problem of a disc S2, of unit radius, rolling on a plane support S0, as shown in Figure 1.6(a). The common reference frame R0 is chosen as the frame whose reference solid is the support S0 (this seems the most natural thing to do here). The physical coordinate system of R0 is defined by three particles [o0;a0, b0] as shown in Figure 1.6(a).
We also define the reference frame R2 whose reference solid is the disc S2. The physical coordinate system in R2 is defined by three particles [o2;a2, b2], and the particle o2 is located at the center of the disc S2 and the particles a2, b2 are on the edge of the disc.
Figure 1.6.(a) Disc rolling on a plane support; (b) positions of the systems inR0
According to convention [1.13], the physical coordinate systems are chosen such that they are physically orthonormal in the sense of definition [1.9]. The (mathematical) space ε, in which the positions in different reference frames are written, as well as the associated vector space E are two dimensional. The (mathematical) coordinate system of E is denoted by .
According to the notation convention [1.24], we will represent the positions of the mechanical systems in R0. By applying [1.24] and then [1.14]a−b, we find that the positions in R0 of particles o0, a0 are:
Similarly, we find that the position in R0 of particle b0 is . This problem is represented in Figure 1.6(b).
■
1.3.7. Coordinate system of a reference frame
Let us introduce another terminology that is commonly used in mechanics:
Definition.
[1.29]
Let us consider a given reference frame Ri endowed with its physical coordinate system (oi;ai, bi, ci) and let
,
A
i
,
B
i
and
C
i
denote the respective positions of the particles
o
i
a
i
,
b
i
and
c
i
, in
R
0
and at the current instant
t
:
and
Ri is said to be endowed with the coordinate system or, put another way, is the coordinate system ofRi. These expressions are convenient contractions in language that make it possible to designate the physical coordinate system of a reference frame using one point and three vectors, rather than four particles:
– “
R
i
is endowed with the coordinate system ” is a contraction of “
R
i
is endowed with the physical coordinate system (
o
i
;
a
i
, b
i
, c
i
) whose position in
R
0
is (
O
i
;
A
i
, B
i
, C
i
)”.
– “ is the coordinate system of
R
i
” is the contraction of “(
O
i
;
A
i
, B
i
, C
i
) is the position in
R
0
of the physical coordinate system (
o
i
;
a
i
, b
i
, c
i
) of
R
i
”.
Note that the definition of a coordinate system of Ri involves the positions of the particles oi, ai, bi, ci in R0.
When making figures, we usually plot the coordinate system (more precisely, its position in R0) to represent or visualize the reference frame Ri (see Figure 1.7).
Figure 1.7.Coordinate system of reference frame Ri
In the 2D example in Figure 1.6, is the coordinate system of R2.
According to [1.28]–[1.28b], the coordinate system of the common reference frame R0 is identified with the coordinate system of the affine space ε:
xs
For any reference frame Ri, one has to be careful to avoid confusing the “coordinate system of Ri” and the “(mathematical) coordinate system”:
– The coordinate system of
R
i
is indeed a coordinate system in the mathematical sense of the term (set of a point in
ε
and of three vectors of
E
) and thus it can be given all the classical terminology relative to a mathematical coordinate system: the point
O
i
is called the
origin of the coordinate system of
R
i
and
the basis of the coordinate system of
R
i
, the coordinate system of
R
i
is said to be
a right-handed orthonormal basis
if the basis is right-handed orthonormal. We also speak of coordinates of a point in the coordinate system of
R
i
.
– Conversely, a mathematical coordinate system is not necessarily a coordinate system for
R
i
in the sense of the above definition. We will see, in
[1.37]
, a condition required for the mathematical coordinate system to be the coordinate system of
R
i
.
Theorem. Let be the coordinate system of a reference frame Ri, defined in [1.29]. The image of vectors under are the vectors of the basis of E:
PROOF. Let us carry out the proof for , the proof for the two other vectors being similar. With the notations in definition [1.29] for the coordinate system of Ri, we have:
In the case RiR0, the relationship [1.31] gives
and we once again arrive at [1.30].
1.3.8. Fixed point and fixed vector in a reference frame
Let A be a point in ε. The expression “the point A is fixed in ε” is perfectly well defined. It signifies simply that at any instant A is the same point in ε. The following definition is a new definition:
Definition. Let A be a point in E (which does or does not vary with t). We say that A is fixed inRi or attached toRi if
The following theorem ensures that definition [1.33] effectively renders the intuitive idea of the fixity of a point with respect to a reference frame.
Theorem. Let A be a point in ε (which does or does not vary with t).
A ∈ ε is fixed in , the (fictitious) particle of position A in R0 at any instant is at a fixed position in Ri.
[1.34]
PROOF. Let a denote the (fictitious) particle whose position in R0 at any instant is A: ∀t, AposR0(a, t) (there exists such a particle according to the position postulate [1.12]). The following equivalences hold:
Let us introduce the following definition, similar to [1.33]:
Definition. Let be a vector in E (a vector that may or may not be variable with t). We say that is constant inRi (or fixed inRi, or attached toRi) if is a constant vector in E.
[1.35]
The following theorem gives the physical interpretation for a constant vector in a reference frame:
Theorem. Let be a vector in E (which does or does not vary with t).
is constant in and the biparticle of biposition in R0, at any instant, has a fixed biposition in Ri.
PROOF. Let us write , with A, B ∈ ε, a (respectively, b) the (fictitious) particle of position A (respectively, B)in R0 at any instant: ∀t, AposR0(a, t), BposR0(b, t).
According to definition [1.35], ∈ E is fixed in is a constant vector in E. Now,
REMARK. The fixity of a point and the constancy of a vector are concepts that have a sense only with respect to a reference frame. Moreover, one should not confuse the constancy of a vector with that of its norm.
■
The following theorem can be proved:
Theorem.
(i) The coordinate system of a reference frame
R
i
, defined in
[1.29]
, is
fixed in
R
i
, that is
the point
O
i
is a fixed point in
R
i
,
and the vectors are fixed vectors in Ri.
[1.36]
(ii) Conversely, if a (mathematical) coordinate system and
E
) is fixed in a reference frame
R
i
, then it is a coordinate system of the reference frame
R
i
as defined in
[1.29]
.
[1.37]
1.4. Derivative of a vector with respect to a reference frame
Consider a vector quantity (for example, the position vector of a particle) whose observation result in the common reference frame R0 is a vector that is function of t (in R0 we write rather than ).
The vector is the classical derivative of with respect to time, a derivative that any observer can calculate by using universally known mathematics. One way of calculating is to differentiate the components of in a fixed basis of E.
Let us now introduce a new concept, more complicated than the previous classical derivative. It is called the time derivative of the vector with respect to a given reference frameR1.
Definition.The time derivative of a vector with respect to a reference frame R1, denoted by is, by definition
The computation of is made up of three elementary steps summarized in the flowchart below:
This flowchart may be interpreted in figurative terms as follows. Let us imagine that the reference frames R0 and R1 are two infinite, transparent sheets of paper laid one atop the other, and sliding against each other, such that the observations in a reference frame may then be “read through transparency and traced to the other reference frame”.
Assume that one can draw the vector under consideration, , on the sheet R0 at different instants. At each instant t, one can take the vector drawn on the sheet R0 and “trace it onto the sheetR1” to obtain the vector associated with in . By repeating this tracing operation at different instants, it becomes possible to construct, on the sheet R1, the family of vectors
which is, a priori, different from the family of vectors drawn on the sheet R0 because the two sheets R0 and R1 move with respect to one another. The mapping [1.40] represents the transfer onto R1 of the evolution over time of the vector . This is operation (1) in flowchart [1.39].
Knowing the mapping [1.40], one calculates the derivative of vector , which is simply the classical derivative with respect to time. Like , the derivative is drawn on the sheet R1. This is operation (2) in flowchart [1.39].
The third and final operation, number (3) in the flowchart (i.e. calculating ), consists of “tracing backwards onto the sheetR0” to obtain vector defined in [1.38]. This operation is required in rigid bodies mechanics with the objective of simplifying the formulae obtained in kinematics.
For a vector such that depends on time and other space variables, the partial derivative of with respect to time in the reference frame
R
1
is defined in a similar manner:
Definition.The time partial derivative of a vectorwith respect to a reference frameR1, denoted by , is, by definition
Theorem.
DEMONSTRATION.
We also have the following classical relationships (whose demonstration can be found in books on Newtonian mechanics):
Theorem. ∀ reference frame
Note that the derivative on the left-hand side of [1.45] is the derivative of a scalar function and does not depend on the reference frame, while the derivatives on the right-hand side relate to vectors and carry the index for the reference frame R1.
Theorem. Consider , where is a vector basis in E, fixed inR1. Then
PROOF. Apply relationships [1.42]–[1.44].
■
1.5. Velocity of a particle
Definition. By definition, the velocity, with respect to the reference frameR1and at the instantt, of a particlep whose position is P is
whereO1 is a fixed point inR1.
1.6. Angular velocity
The following theorem is a classic result in Newtonian mechanics of rigid bodies:
Theorem and definition. Composite derivative of a vector. Let R1, R2 be two reference frames. We have: ∀ vector :
where is an orthonormal basis made of vectors in E, fixed inR2 (Figure 1.8). The vector is called the angular velocity vector ofR2with respect toR1(at instantt).
The skew-symmetric tensor associated with is called the angular velocity tensor ofR2with respect toR1(at instantt). It is related to the rotation tensors and through
Figure 1.8.Composite derivative of a vector
To obtain an explicit expression for in [1.48], let us write instead of . We then have
The following theorem is an important particular case of [1.48] :
Theorem. Derivative formula with respect toR1of a vector constant inR2.
PROOF. One has just to apply [1.48]
