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An organization of classical particle mechanics
An Organization of Classical Particle Mechanics by R.M.ROSENBERG
Department of Mechanical CA 94720, U. S. A.
Engineering,
University
of California,
Berkeley,
Certain techniques, concepts and methods used to model and analyze the motion of massive bodies under forces are not admissible under the Newtonian axioms. Among them are: forces acting on massless components rather than on particles, using the so-called free-body diagram technique, and the existence of infinite forces of bounded impulse. Moreover, modeling physical space by Euclidean 3-space introduces some basic dificulties which are usually silently ignored. We propose here an organization of classical particle mechanics which provides the additional axioms and defined terms needed to legitimize the mathematical models and methods presently used in the analysis of Newtonian systems.
ABSTRACT:
I. Introduction In what Destouches (1) calls “the experimental domain of its applicability”, classical particle mechanics has had an unparalleled history of success in predicting observed motions of massive bodies under forces. It has only failed in very large systems involving velocities near that of light where the mass of a particle can no longer be considered constant, and in very small systems in which matter can no longer be considered infinitely divisible. Although axiomatization of mechanics is not essential for its comprehension or application, many attempts have been made to axiomatize it. David Hilbert once said (2) “I think that everything that can be an object of scientific thought at all, as soon as it is ripe for a theory falls into the lap of the axiomatic method . . .“. Here, Hilbert echoed Newton because the Principia was modeled on Euclid’s Elements of Geometry. Both treatises are divided into “books”, both begin with “definitions” followed by “axioms” and then by “propositions” which are subdivided in both into “theorems” and “problems”. Opinions differ as to the need for, or value of axiomatizing classical particle mechanics. Truesdell (3), among others, considers it “essentially trivial” except for “. . . the precision of the concept of force”; Newton, Kirchoff, Hertz, Lagrange, Mach and others, and more recently Bunge, Hamel, Hermes, Marcolongo, McKinsey, Pendse, Simon, Sugar, Suppes and many more have thought it worthwhile. One of the earliest significant modern attempts to lay a foundation for classical particle mechanics is that by Hamel (4) who discusses also the Lagrangean construction of the foundations of the motion of perfect fluids and of some elastic systems.
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149
R. M. Rosenberg
A paper by McKinsey, Sugar and Suppes (3), published in 1953, injected new life into the problem of axiomatizing classical particle mechanics. However, to quote from that paper, the authors incorporate a number of “idealizations . . . on pragmatic grounds . . .” and for reasons of “ . . . mathematical . . . convenience and elegance . . .“. This has resulted in an axiomatic which does some violence to their intended model and it fails to provide a foundation for some of the techniques used in the application of that discipline. As an illustration of the former, these authors omit an axiom of impenetrability because “. . . it does nothing but complicate proofs”. However, the failure to postulate such an axiom admits resistance-free motions of particles through each other; such a motion violates physics whether particles are taken to be mass centers or models of small bodies. (While there exist special cases in which mass centers can pass resistancefree through each other, this is not true in general, and it is never possible if particles are models of small bodies.) To illustrate the latter, the admission of infinite forces having bounded impulse is a widely used technique for modeling impacts, but such forces are not admissible under the axiomatic of McKinsey et al. (3). As Destouches (1)has observed, in “physic0-logical problems”, both logical conditions and physical interpretations arise. The latter remain at the level of intuitive semantics, and this presents difficulties in creating and testing an axiomatization. To date, no axiomatization of Newtonian particle mechanics exists whose elements comprise primitives, defined terms, definitions, axioms, theorems and proofs, in which the primitives and axioms are examined for independence, consistency and completeness, and which is isomorphic to present-day practice in the solution of dynamics problems within the compass of Newtonian particle mechanics. This has led Bunge to observe (5) that “no generally accepted axiomatization of elementary [particle] mechanics seems to exist. . . the best of them (McKinsey et al.) has unjustifiably failed to attract the attention of physicists”. This paper does not have the ambitious aim of constructing an axiomatic of Newtonian particle mechanics which satisfies the criteria listed above. Instead, it is our purpose to suggest an organization of classical particle mechanics which legitimizes and clarifies the methods and concepts utilized in present-day applications of solving dynamics problems whether they are or are not admissible under “strictly Newtonian” viewpoints. II. The Models
The practitioner constructs a “mathematical model”, and the axiomaticist deals with a so-called “intended model”. In discussing these, our language remains of necessity at the colloquial level of intuitive semantics. An observable will be called a prototype. Then, we shall say: The mathematical model of a prototype classical particle mechanics.
The process of mapping the prototype
150
is its image
in the domain
into its mathematical
of
model is called
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An Organization of Classical Particle Mechanics
an idealization. It involves ignoring properties of the prototype which are thought by the practitioner to have either no influence on the phenomenon under study (for instance, the color or temperature of a massive object), or only a negligible influence (for instance, quantities which are “very small” compared to others). The mathematical model comprises components which have (probably) no counterpart in the world of observables. Thus, particles are objects having zero volume but positive mass and hence, infinite density; forces are taken to act at points thus producing infinite bearing stresses; infinite forces of bounded impulse are admitted, as are massless connections which may or may not be infinitely rigid; matter is replaced by mass; and mass and time are taken to be infinitely divisible. The construction of a successful mathematical model requires insight, experience, intuition and good luck; it is non-axiomatic in character and is more nearly an art than a science. An axiomatization of a fairly complete body of knowledge may be regarded as a form of description (2), and what is being described is the “intended model”. We shall say: The intended model of our organization of classical particle mechanics is the collection of concepts and methods used presently to solve problems which deal with the motion of massive bodies acted on by forces.
If our organization of classical particle mechanics is isomorphic to the intended model, there exists strong evidence to believe that it is consistent (i.e. no statement can be demonstrated to be both true and false) because proper application of Newtonian particle mechanics has a long, varied, and unblemished history of consistency on the domain of its applicability. However, that statement should not be confused with a proof of consistency. We shall say: Our organization of classical particle mechanics is adequate if the catalog of its concepts and methods is isomorphic to present-day practice in the application of this discipline.
In organizing classical particle mechanics, the elements of logic, arithmetic, analysis including vector algebra and calculus, and the elements of topology and set theory are tacitly adopted. Thus, the primitive notions of these disciplines are incorporated here without special mention. III. The Spaces
Consider the Euclidean reference frames
3-space
X3 and the class
93 = {bk = (6:. G;, 6:);
&=1,2,...}
where the Sr, (i = 1,2,3) are three linearly independent X3, and
I
X3 = X: X = f: Xicf Vol. 313, No. 3, pp. 149-164, Printed in Great Britain
i=l
March
1981
’
(Xl, X2*X3);
of global bases
xiE@‘,
or (1)
unit vectors spanning
(i=1,2,3)
I
(2)
151
R. M. Rosenberg where (5;’is the metrized line of reals. It is well known that X3 is homogeneous and isotropic. We may regard X3 as the mathematical model of physical 3-space, and a reference frame bk as the mathematical model of an observer. Next, we introduce two one-dimensional metric spaces. One is E = {t: t E @?}
(3)
called the “time space”. The other is the half space SJ2={m: mECE’; O
(4)
called the “mass space”. The names assigned to these spaces suggest their interpretation. It is implied that mass and time are bounded scalar continua which possess the additivity property. Moreover, non-positive masses do not exist. Mass may be regarded as the mathematical model of matter. IV. Particles In colloquial language, a particle is an object having zero volume and finite constant mass, i.e. a “point-mass” which has at every instant of time a unique position relative to an observer. Since “position relative to an observer” and “mass” are essential properties of every particle we introduce them jointly by means of the 4-space which is the Cartesian product
L24=x3xm={(x,m):
xEX3; mExR},
(5)
where x is the position relative to a b” E ‘23. We can now define a particle. D.l. A point P, = (Y, m,) is a particle having position xr E X3 and mass m, E 1Dzif and only if (i) there exists a mapping SE+, U4, namely x’(t), m,(t), such thatt (a) x’(t) E X3 is a single-valued vector function for all t ES, and every xi(t), (i = 1,2,3) is continuous on 5X,piecewise of class C’ on any compact subset of &, and piecewise of class C2 on any subset of 5E on which it is of class C’; (b) m,(t) = m,(t,,) b m,Vt E SC; to E &; (ii) For two distinct particles P, and Pr, x’(t) = x’(t) if there exists a to E & such that x’(tJ = x’(t,J; (iii) for two distinct particles P, and Pf, x’(t) # x’(t) Vr E a if there exists a to E % such that x’(tO) # x’(t,,). The primitives are m, and x’. The verbal interpretation of D.l is as follows: (i) ensures (a) that a particle can and does have one and only one position relative to an observer at any tBy a common
152
abuse of notation
we denote a function
as well as its value by the same symbol.
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An Organization of Classical Particle Mechanics
instant of time, but it can have isolated velocity and acceleration discontinuities, and (b) that the mass of every particle is positive and constant for all time; (ii) and (iii) constitute the axiom of impenetrability; (ii) says that if two particles have the same position at some instant of time they have the same position for all time and are thus equivalent to a single particle of mass equal to the summed masses and (iii) says that particles having different positions at some instant have different positions for all time. Thus, one and only one particle can occupy a given position at any instant of time. It is part of D. 1 that the particle velocity i’(t) k dx’(t)/dt may have isolated discontinuities. The general mapping from E into 114 is illustrated in Fig. 1, and the particular mapping m,(t) = m,(t,J of D.l (i) (b) is illustrated in Fig. 2. Of course, these illustrations suffer from the defect that they are at best projections onto a 2-dimensional space (the page) of a 4-dimensional space, but it is hoped that they will nevertheless clarify the procedure. The impenetrability principle is illustrated in Fig. 3 where (a) refers to D.l (ii) and (b) to D.l (iii). We denote by SE, = {tj: xr(tj)
not defined;
tj E E;
j = 1,2,.
. .)
the set of instants (empty, finite, or denumerably infinite) at which the motion of P, has velocity discontinuities. The particle P, = (mr, x,(t)) EU4. If we project that point “down the maxis” onto X3 we obtain the “particle projection” x’(t) E X3; it is permanently associated with the r’th particle of mass m, The set of all particle projections onto X3 at t E 5X is P.l.
q’(t)
=(x’(t):
r = 1,2,. . ., s (00)
C X3.
This proposition asserts that particles exist and the number of particles in the universe is finite. The new primitive is s.
FIG. 1. The mapping from E into U4. Vol. 313. No. 3. pp. 149-164, March Printed in Great Britain
1981
153
R. M. Rosenberg Particle T
trajectory
in II4
d
Plone
m&t
1~ m&t,1
I I I Trajectory
FIG. 2. The mapping
V. Massless
m,(t) = m,(Q
in x3
from & into !?JL
Connections
In colloquial language, the term “massless connection” denotes an element having negligible mass to which one or more than one body (which can be modeled as a particle) is attached permanently; the length and shape of such connections may be fixed (as in rigid systems of particles for instance), or variable (as in “massless” coil or leaf springs, for instance). The concept of the massless connection arises naturally in prototypes in which some components have “very small” mass compared to that of other components. Then, in the construction of the mathematical model, it is natural to delete the mass of those components whose mass is regarded as negligible. The massless connection is particularly troublesome from the standpoint of axiomatics because in Newtonian particle mechanics forces can only act on massive elements. Some of these difficulties were pointed out by Stadler (6) who resolved them in a way different from that suggested here by postulating a
m
m
I
9 mr’
pr’
Xr’ =xr’/
(0)
FIG. 3. Visualization
154
(b) of the impenetrability
principle
of D.l (ii) and (iii).
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An Organization of Classical Particle Mechanics
separate axiom of angular momentum not derivable from the linear momentum axiom. We shall use the term “line segment” to mean a set in X3 which is homeomorphic to a straight line segment, and we denote such a set by Ea. If there are particle projections which coincide at some instant to E E with the line segment Ga, we denote the set of these coincidences by D.2.
L(G)
to EE.
i Ea f-l @3(tO);
(6)
If that set is not empty, we call Ga a structure Ga, i.e., EU = Ea(to) iff 3,(&J f 4.
D.3.
(7)
Then we have the proposition P.2.
G,(t)
= E&(&l) vt
E E.
(8)
The verbalization of P.2 is that if particle projections lie in a line segment at some instant of time they remain in that line segment for all time. Finally, we define the massless component G,2: of a structure G, by stating D.4.
6: = G,\S,:
x” = (x,O1,xzn, x37
x2 = fz(P),
x3 = f3(p);
E
G,;
Xl
=
fl(P),
p a parameter.
Thus G,*, is the arc of a curve in the 3-space X3. The manner in which structures are generated is illustrated in Fig. 4 in which some particle projections coincide with the line segment while others do not. VI. Systems of Particles
When the term “system of particles” is not clearly defined, difficulties are likely to arise in the application of Newton’s third law and in the treatment of those variable-mass dynamics problems which are within the compass of classical particle mechanics. For instance, McKinsey et al. (3) suggest that variable-mass dynamics problems such as rocket flight (where the rocket mass decreases as fuel is expended) require the introduction of a time-dependent particle mass. However, the concept of variable particle mass is totally foreign to Newtonian dynamics, as stated in D.l (i) (b). Thus if timedependent particle mass were admitted one would not have
I
2
I
m,
‘0
dx
dr
r
dr = m,.? + const.
These difficulties disappear with the introduction Vol. 313. No. 3. pp. 149-164, Printed in Great Britain
March
1981
of a boundary 2X. 155
R. M. Rosenberg
FIG. 4.
D.5.
The boundary
A structure 6,: C, = 2.
% is a surface homeomorphic
G = x E 3E3: i
to the spherical surface
x2= R*;
(9)
i=l
We can now define the configuration space by stating D.6. The configuration solid sphere
space B is the set of points homeomorphic
XEX~: $x:
to the
(10)
. I
Verbalized, D.6 states that the configuration space is the set of points within and on the boundary. Then, the particles of the system are defined as follows: D.7.
The particles of the system are the members of the set ‘$3(t) = {x’(t):
(11)
x’(t) E B; r = 1,2,. . . , n(t); t E 5X).
This definition states that the particles of the system at the time t are those whose projections at the time t are within or on the boundary of the configurations space (5. We can now state the axiom P.3.
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An Organization of Classical Particle Mechanics
whose verbal interpretation is that the configuration space always contains at least one particle. The new primitives are !?I and n(t). We note the following properties: (i) The total membership n(t) of the “population” of the configuration space is a diophantine time function, i.e., particle projections may pass in either direction through the boundary and may thus be acquired or lost by the system; (ii) the total mass of the particles of the system at the time t is n 1)
M(t)=
m, h r=l
(13)
VII. The Mass Center
One of the very useful concepts of Newtonian particle dynamics is the of the system. We state D.8. The mass center of the particles of the system at the time t E 5Xis the point g(t) E X3 such that mass center of the particles
E(t)
t$m, = g
m$(t).
(14)
Under certain circumstances, and in some respects to be discussed later, mass centers behave like projections of particles of mass M(t). This has led McKinsey et al. (3) to observe that “. . . physicists are not quite agreed . . . as to what is meant by a particle. . ., should small bodies serve. . . or should we use centers of mass?” To clarify this issue we state: Mass centers are not particles.
To prove this theorem we note that
because of D.l (i) (b) which requires that the particle mass be constant, and because of D.l (ii) and (iii) which require that one and only one particle can occupy a given position. As pointed out by McKinsey et al. (3) the latter is violated, for instance, when a bullet is shot through a doughnut where two mass centers may coincide at one instant without coinciding for all time. Finally, we define the system as the set
m =Pmt). a.
(15)
In words, the system is the set of particle projections ‘$3C B together with the configuration space B. The notion of the system is important if we wish to study several systems and their interaction. Vol. 313, No. 3, pp. 14%162. March Printed in Great Britain
1981
157
R. M. Rosenberg VIII. Remarks
on Newton’s
Second Law
The dynamical axioms describe the interaction of particles and structures with forces. They define, in fact, the mapping X ++ U4 of D.l (i). Undoubtedly, the most far-reaching of all axioms of particle mechanics is Newton’s second law. It is usually given in a form equivalent to the following statement: There exists a reference frame and a time (called by Newton “absolute space” and “absolute time”) such that with respect to that reference frame and that time, the acceleration a(t) of a particle is permanently proportional to the force F which acts on the particle, or mu(t) = F ‘it. For this equation to have meaning, Now, under present-day practice, are admitted, i.e. forces such that
F must have finite magnitude for all t. infinitely large forces of bounded impulse
I
f
I = lim F(T) dT# 0 r+rj $ is bounded. Moreover, as already stated, it is present-day practice to admit forces which “act on” massless components of structures. To legitimize the first of these, Newton’s second law will be given in its integrated form called the “axiom of bounded momentum” and to legitimize the latter, a new axiom will be required. Moreover, the phrase “the force acts on (or at) a point” needs clarification. IX
Force
The interaction between forces on the one hand and particles and/or structures on the other is conventionally described in physical 3-space. However, the incorporation of this interaction into X3 meets with the difficulty that the set X3 is not a physical space, i.e. it contains neither particles nor forces; instead, its members are only points occupied by particle projections, points occupied by massless components of structures, and points not occupied by either. Thus, for forces to “act on” particle projections and on massless components of structures, a dynamically meaningful mapping of the force into X3 must be devised. We introduce a 3-dimensional 0space with origins 0 (Q = 0) called the “force space” and spanned by the base b” E 23. Consider now the directed line segment dQ = (Q,, QZ, Q3) E 0 and a point x = (x,, x2, x3) E X3 relative to the same base b”. Then we map Q = 0 +x and Q-R, x, R E X3 such that
F = (F!, F2, F3) = R -x where
158
Fi = Qj, (i = 1,2,3).
Then
E X3
we say: the force
(16)
F acts on (or at) the point Journal
of The Franklin Institute Pergamon Press Ltd.
An Organization of Classical Particle Mechanics x. In general, Q = Q(t), and we require that Jk Q(7) d7 exist on any interval rto, t1 E SE. The process described here provides for the treatment of force-particle structure interaction in X3 and it defines precisely the meaning of the phrase “a force acts on (or at) a point”. This mapping is illus’trated in Fig. 5. The new primitive is Q.
X. The Dynamical Axioms We introduce the integrated form of Newton’s axiom of bounded momentum (7).
second law called here the
P.4. There exists a set of reference frames @l= {g} C 58 and a time space E* C i3, and for every r = 1,2,. . . , n(t) a vector-valued integrable function F’ =
F’(xl, x2,. . .
)
xs; i’, i*, . . . ,i’“;
t)
such that for every t E X* and for every g E @, (i) the vectors T(t) are bounded for all t E E*, (ii) the vectors i’(t) satisfy for all t E SE* and for every r = 1,2,. . . , n(t) the equation i’(t)
=
$
I
et,
F’(xl, x2,. . . , xs; i’, si.‘,. . . , 1”; 7) d7 + i’(to),
(17)
fll
where x’(t) is the particle projection relative to g, and the ir(to) are arbitrary constant vectors; (iii) the vectors i-‘(t) are single-valued and continuous for all t E E:r =
FIG. 5. The mapping of a force into X3-space. Vol. 313, No. 3. pp. 149-164, March Printed in Great Britain
1981
159
R. M. Rosenberg &*\E, where F’dTf
0, 00; j = 1,2,. . . .
(18)
I
The new primitives are the set of “Galilean” frames @, and the “absolute” time space E*. The function F’ is called the resultant force acting on x’. The quantity m,_?(t) is called the linear momentum of the particle P,, and the integral Ji F’ d7 is called the impulse of F’ during the time interval [to, t] E ST’*.
Verbalized, P.3 states: With respect to a Galilean frame and absolute time, the linear momentum of every particle is permanently bounded, and the change of linear momentum during any time interval is equal to the impulse of the resultant force acting on that particle over that same time interval. Let us introduce the notation n(t) Ej
=
U
n(t) SE,;
iy
=
r=l
u Q’,.
(19)
r=l
Then
is the set of instants continuities. Therefore,
09:
m,f’=F’(x’,x*
when no particle of the system we have the theorems
has velocity
,...,
r=1,2
x”; i’,i2 ,...,
3;
t)VtEE:;
,...,
dis-
n(t) (21)
and (SN):
If and only if Ej = #J
m,.Y = F’(x’, x2,. . . , x’; i’, 1*, . . . ,1”; t) Vt E X*;
r= 1,2 ,...,
n(t). (22)
When 5Zj# 4 we call the system “Newtonian” (N), and when 5Ej= 4 we call it “strictly Newtonian” (SN). Theorem (N) states that the motion of every particle of the system satisfies Newton’s second law almost always; (SN) states that in the absence of unbounded forces (of bounded momentum) the motion of every particle of the system satisfies Newton’s second law for all time. The proof of these theorems follows from differentiating (16) with respect to t E iE*. To deal with forces F’$ (y = 1,2,. . . , C,) at points x”, in the massless components of a structure we consider the set of these forces $Ja = {F;: 160
y = 1,2,. . . , Cm}. Journal
of The Franklin Institute Pergamon Press Ltd.
An Organization Then the massless connection
of Classical Particle Mechanics
axiom is
P.5.
(23)
It states that every massless component of every structure is permanently in static equilibrium. In the absence of such an axiom, massless components of structures acted on by nonvanishing forces would, by Newton’s second law, have infinite acceleration because FZ 0, m = 0 implies ial = 00 in ma = F. Stadler (6) points out that his independent axiom of angular momentum is equivalent to the statement that the constraint forces are in static equilibrium, and that observation is similar to the massless connection axiom; in fact, when the only forces acting on massless components of structures are constraint forces the two axioms are completely equivalent. In Fig. 6, we illustrate the meaning of P.5 on an example in which not all forces on massless connections are constraint forces. One of the most useful theorems of Newtonian mechanics deals with the motion of the mass center of a system, and one of the most useful techniques in solving dynamics problems utilizes the so-called free-body diagram. The former requires Newton’s third law in the strong form,? the latter requires a similar law in the weak form. Thus we introduce Newton’s third law in the strong form by means of P.6.
The resultant force F’ acting on x’ is of the form
FIG. 6. The spring-restrained
pendulum.
A and B are fixed points
tThe strong form of Nkwton’s third law requires not only that internal (the weak form) but that the forces of each pair be collinear.
Vol. 313. No. 3. pp. 149-164, Printed in Great Britain
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1981
forces
in X3. vanish
in pairs
161
R. M. Rosenberg where 0; _=O;(xr,
&
xk) = ?\Ir;((x’ - xk()
Vk, r;
k# r
is called the resultant of the internal forces, and *\I(Jx’- xkl) is a scalar function of the distance Ixr - xkl. The new primitive is 9’;. The quantity 0’ is called the resultant of the external forces acting on x’. The free-body diagram is a technique by means of which geometrical constraints are replaced by forces which have the same effect as the constraints, i.e., they enforce the constraints. These forces may be interaction forces between particles on the one hand and massless components on the other, or they may be interaction forces between portions of a massless component. Newton’s third law does not include these forces because it only speaks of interaction forces acting on particles of the system. Thus, a new axiom is required to legitimize the technique of the free-body diagram. That axiom bears a close resemblance to Hamel’s “Befreiungsprinzip” (4) (liberation principle) in which rigid connections are “liberated” from the rigidity requirement (i.e. they become deformable), and then, forces are introduced which restore the undeformed configuration. However, within our organization, the Befreiungsprinzip must be broadened to include non-rigid massless connections, and it must conform to the free-body diagram technique which provides for separating parts of massless components of structures either from each other or from particles which are on the structure. Thus we introduce the free-body axiom. P.7. A structure G,, of a system % may be replaced by two connected subsets GJ+) and Gap(-) with x;(+)EG~J+) and x;(--)EG,,,(-_) if and only if a force F;(+) acts at xE(+) and a force Fz(-) at xz(-) such that F”d(-) = -F;(+) and G,,,(+) U Gap(-) = Ga when x”,(+) = x%(-) = x; E G,X. It is noted that the forces F;(+) and F%(-) satisfy the weak form for internal forces, i.e. they vanish in pairs. This axiom permits subdividing the structure at any point x; so long as D.l (i) (a) is not violated, i.e. the structure cannot be subdivided so that a particle projection belongs to both Q,,(+) and Gap(-). The example of Fig. 6 is used in Fig. 7 to illustrate the technique of the free-body diagram.
XI
Mass Center Motion
We can now examine in what measure mass centers behave like particles. The well known theorem of the motion of the mass center may be stated in the form: If and only if n(t) = n(QVt E SE’*;to E SE*, the motion of the mass center F
162
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An Organization of Classical Particle Mechanics
FIG. I. A free-body
of a system
diagram
% of particles
of the spring-restrained mass-axis is now shown.)
pendulum
of Fig. 6. (The
satisfies (24)
where d(to) is an arbitrary vector. The proof of this theorem is found in nearly all elementary texts on dynamics and will not be repeated here; it makes appeal to Newton’s third law in the strong form. The theorem states that, if the number of particles of a system is fixed during some time interval [to, t] E SE*, the motion of the mass center is the same during that time interval as that of a particle of mass M(to), n(tij) located at the mass center g, and acted on by the resultant Cp= X @’ of the r=l external forces acting on the system. Let us now consider several constantmass systems !Jln,(t>={@B,(t),Bs: with mass centers ~~ and system overlapping” in the sense that
6=1,2
masses
!Xzs,n ‘& = C#IV’S’, 8” = 1,2,. Then,
there
exists
a boundary
,...,
M,(t,).
. . , A;
3, and a time interval
A} These
systems
are “non-
6’ # 6”. [to, tA] E E* such that
where %?Ibis the boundary of CIA. Expressed in words, if we consider several non-overlapping systems during a time interval in which the number of particles of each remains constant, we can find a boundary ‘Sz, large enough so that all boundaries of the non-overlapping systems are in the interior of, or on the new boundary %?l~.In that case, each mass center g,, (6 = 1,2,. . . , A) Vol. 313, No. 3, PP. 149-164. Printed in Great Britain
March
1981
163
R. M. Rosenberg satisfies (23). However, (ii) and (iii).
two distinct
mass centers
g;i and 8: need not satisfy
D.l
XIZ. Conclusion The preceding definitions and axioms constitute our organization of classical particle mechanics. This organization admits the analysis of rigid or non-rigid constant or variable-mass systems of particles under forces whose amplitude may be finite, or infinite with bounded impulse, and the forces may be acting on particles or on massless components of structures. The proof of its isomorphism with present-day practice in analyzing dynamical systems would require a complete catalog of every concept and method used to solve problems of the motion of massive bodies and systems of bodies under forces and a complete catalog of the concepts and methods admitted under our organization, and finally the demonstration of a one-to-one correspondence between them. This demonstration has not been made partly because it is
tedious and difficult, and partly because it is considered unnecessary. Under (15) and (16), all systems not comprising structures, as defined in (7), are admitted whether the forces are bounded, or unbounded with bounded impulse, and under (22) massless connections and forces acting on them are also admitted. Finally, the free-body diagram technique is admitted under P.7. Thus, while there is no formal proof of the adequacy of our organization of classical particle mechanics, all methods, techniques and concepts known to us to be in present-day use for solving problems within the domain of Newtonian particle mechanics are legitimate under that organization. Acknowledgements Many fruitful discussions with Professor George Leitmann significant influence on this paper; it is by his choice that co-author.
have had a he is not a
References (1) J. L. Destouches, “The Axiomatic Method” (Edited by L. Henkin, P. Suppes and A. Tarski), pp. 390-402, North Holland, Amsterdam, 1959. (2) Quoted from R. von Mises, “Kleines Lehrbuch des Positivsimus”, p. 114. W. P. van Stockum & Zoon, The Hague, 1939. (Quote translated by R.M.R.) (3) J. C. C. McKinsey, A. C. Sugar and P. Suppes, J. Rat. Mech. Anal., Vol. 2, p. 253, 1953. (4) G. Hamel, “Handbuch der Physik”, pp. l-42, J. Springer, Berlin, 1927. See also G. Hamel, “Theoretische Mechanik”, pp. 508-525, Springer-Verlag, Berlin, 1949. (5) M. Bunge, “Foundations of Physics”, p. 130. Springer-Verlag, New York, 1967. (6) W. Stadler, unpublished paper (personal communication, 1980). (7) R. M. Rosenberg, “Analytical Dynamics of Discrete Systems”, p. 10, Plenum Press, New York, 1980.
164
Journal
of The Franklin Institute Pergamon Press Ltd.
Department of Mechanical CA 94720, U. S. A.
Engineering,
University
of California,
Berkeley,
Certain techniques, concepts and methods used to model and analyze the motion of massive bodies under forces are not admissible under the Newtonian axioms. Among them are: forces acting on massless components rather than on particles, using the so-called free-body diagram technique, and the existence of infinite forces of bounded impulse. Moreover, modeling physical space by Euclidean 3-space introduces some basic dificulties which are usually silently ignored. We propose here an organization of classical particle mechanics which provides the additional axioms and defined terms needed to legitimize the mathematical models and methods presently used in the analysis of Newtonian systems.
ABSTRACT:
I. Introduction In what Destouches (1) calls “the experimental domain of its applicability”, classical particle mechanics has had an unparalleled history of success in predicting observed motions of massive bodies under forces. It has only failed in very large systems involving velocities near that of light where the mass of a particle can no longer be considered constant, and in very small systems in which matter can no longer be considered infinitely divisible. Although axiomatization of mechanics is not essential for its comprehension or application, many attempts have been made to axiomatize it. David Hilbert once said (2) “I think that everything that can be an object of scientific thought at all, as soon as it is ripe for a theory falls into the lap of the axiomatic method . . .“. Here, Hilbert echoed Newton because the Principia was modeled on Euclid’s Elements of Geometry. Both treatises are divided into “books”, both begin with “definitions” followed by “axioms” and then by “propositions” which are subdivided in both into “theorems” and “problems”. Opinions differ as to the need for, or value of axiomatizing classical particle mechanics. Truesdell (3), among others, considers it “essentially trivial” except for “. . . the precision of the concept of force”; Newton, Kirchoff, Hertz, Lagrange, Mach and others, and more recently Bunge, Hamel, Hermes, Marcolongo, McKinsey, Pendse, Simon, Sugar, Suppes and many more have thought it worthwhile. One of the earliest significant modern attempts to lay a foundation for classical particle mechanics is that by Hamel (4) who discusses also the Lagrangean construction of the foundations of the motion of perfect fluids and of some elastic systems.
@The
FranklinInstitute 001~32/82/030149-16$03.00/O
149
R. M. Rosenberg
A paper by McKinsey, Sugar and Suppes (3), published in 1953, injected new life into the problem of axiomatizing classical particle mechanics. However, to quote from that paper, the authors incorporate a number of “idealizations . . . on pragmatic grounds . . .” and for reasons of “ . . . mathematical . . . convenience and elegance . . .“. This has resulted in an axiomatic which does some violence to their intended model and it fails to provide a foundation for some of the techniques used in the application of that discipline. As an illustration of the former, these authors omit an axiom of impenetrability because “. . . it does nothing but complicate proofs”. However, the failure to postulate such an axiom admits resistance-free motions of particles through each other; such a motion violates physics whether particles are taken to be mass centers or models of small bodies. (While there exist special cases in which mass centers can pass resistancefree through each other, this is not true in general, and it is never possible if particles are models of small bodies.) To illustrate the latter, the admission of infinite forces having bounded impulse is a widely used technique for modeling impacts, but such forces are not admissible under the axiomatic of McKinsey et al. (3). As Destouches (1)has observed, in “physic0-logical problems”, both logical conditions and physical interpretations arise. The latter remain at the level of intuitive semantics, and this presents difficulties in creating and testing an axiomatization. To date, no axiomatization of Newtonian particle mechanics exists whose elements comprise primitives, defined terms, definitions, axioms, theorems and proofs, in which the primitives and axioms are examined for independence, consistency and completeness, and which is isomorphic to present-day practice in the solution of dynamics problems within the compass of Newtonian particle mechanics. This has led Bunge to observe (5) that “no generally accepted axiomatization of elementary [particle] mechanics seems to exist. . . the best of them (McKinsey et al.) has unjustifiably failed to attract the attention of physicists”. This paper does not have the ambitious aim of constructing an axiomatic of Newtonian particle mechanics which satisfies the criteria listed above. Instead, it is our purpose to suggest an organization of classical particle mechanics which legitimizes and clarifies the methods and concepts utilized in present-day applications of solving dynamics problems whether they are or are not admissible under “strictly Newtonian” viewpoints. II. The Models
The practitioner constructs a “mathematical model”, and the axiomaticist deals with a so-called “intended model”. In discussing these, our language remains of necessity at the colloquial level of intuitive semantics. An observable will be called a prototype. Then, we shall say: The mathematical model of a prototype classical particle mechanics.
The process of mapping the prototype
150
is its image
in the domain
into its mathematical
of
model is called
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An Organization of Classical Particle Mechanics
an idealization. It involves ignoring properties of the prototype which are thought by the practitioner to have either no influence on the phenomenon under study (for instance, the color or temperature of a massive object), or only a negligible influence (for instance, quantities which are “very small” compared to others). The mathematical model comprises components which have (probably) no counterpart in the world of observables. Thus, particles are objects having zero volume but positive mass and hence, infinite density; forces are taken to act at points thus producing infinite bearing stresses; infinite forces of bounded impulse are admitted, as are massless connections which may or may not be infinitely rigid; matter is replaced by mass; and mass and time are taken to be infinitely divisible. The construction of a successful mathematical model requires insight, experience, intuition and good luck; it is non-axiomatic in character and is more nearly an art than a science. An axiomatization of a fairly complete body of knowledge may be regarded as a form of description (2), and what is being described is the “intended model”. We shall say: The intended model of our organization of classical particle mechanics is the collection of concepts and methods used presently to solve problems which deal with the motion of massive bodies acted on by forces.
If our organization of classical particle mechanics is isomorphic to the intended model, there exists strong evidence to believe that it is consistent (i.e. no statement can be demonstrated to be both true and false) because proper application of Newtonian particle mechanics has a long, varied, and unblemished history of consistency on the domain of its applicability. However, that statement should not be confused with a proof of consistency. We shall say: Our organization of classical particle mechanics is adequate if the catalog of its concepts and methods is isomorphic to present-day practice in the application of this discipline.
In organizing classical particle mechanics, the elements of logic, arithmetic, analysis including vector algebra and calculus, and the elements of topology and set theory are tacitly adopted. Thus, the primitive notions of these disciplines are incorporated here without special mention. III. The Spaces
Consider the Euclidean reference frames
3-space
X3 and the class
93 = {bk = (6:. G;, 6:);
&=1,2,...}
where the Sr, (i = 1,2,3) are three linearly independent X3, and
I
X3 = X: X = f: Xicf Vol. 313, No. 3, pp. 149-164, Printed in Great Britain
i=l
March
1981
’
(Xl, X2*X3);
of global bases
xiE@‘,
or (1)
unit vectors spanning
(i=1,2,3)
I
(2)
151
R. M. Rosenberg where (5;’is the metrized line of reals. It is well known that X3 is homogeneous and isotropic. We may regard X3 as the mathematical model of physical 3-space, and a reference frame bk as the mathematical model of an observer. Next, we introduce two one-dimensional metric spaces. One is E = {t: t E @?}
(3)
called the “time space”. The other is the half space SJ2={m: mECE’; O
(4)
called the “mass space”. The names assigned to these spaces suggest their interpretation. It is implied that mass and time are bounded scalar continua which possess the additivity property. Moreover, non-positive masses do not exist. Mass may be regarded as the mathematical model of matter. IV. Particles In colloquial language, a particle is an object having zero volume and finite constant mass, i.e. a “point-mass” which has at every instant of time a unique position relative to an observer. Since “position relative to an observer” and “mass” are essential properties of every particle we introduce them jointly by means of the 4-space which is the Cartesian product
L24=x3xm={(x,m):
xEX3; mExR},
(5)
where x is the position relative to a b” E ‘23. We can now define a particle. D.l. A point P, = (Y, m,) is a particle having position xr E X3 and mass m, E 1Dzif and only if (i) there exists a mapping SE+, U4, namely x’(t), m,(t), such thatt (a) x’(t) E X3 is a single-valued vector function for all t ES, and every xi(t), (i = 1,2,3) is continuous on 5X,piecewise of class C’ on any compact subset of &, and piecewise of class C2 on any subset of 5E on which it is of class C’; (b) m,(t) = m,(t,,) b m,Vt E SC; to E &; (ii) For two distinct particles P, and Pr, x’(t) = x’(t) if there exists a to E & such that x’(tJ = x’(t,J; (iii) for two distinct particles P, and Pf, x’(t) # x’(t) Vr E a if there exists a to E % such that x’(tO) # x’(t,,). The primitives are m, and x’. The verbal interpretation of D.l is as follows: (i) ensures (a) that a particle can and does have one and only one position relative to an observer at any tBy a common
152
abuse of notation
we denote a function
as well as its value by the same symbol.
Iournal of The
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An Organization of Classical Particle Mechanics
instant of time, but it can have isolated velocity and acceleration discontinuities, and (b) that the mass of every particle is positive and constant for all time; (ii) and (iii) constitute the axiom of impenetrability; (ii) says that if two particles have the same position at some instant of time they have the same position for all time and are thus equivalent to a single particle of mass equal to the summed masses and (iii) says that particles having different positions at some instant have different positions for all time. Thus, one and only one particle can occupy a given position at any instant of time. It is part of D. 1 that the particle velocity i’(t) k dx’(t)/dt may have isolated discontinuities. The general mapping from E into 114 is illustrated in Fig. 1, and the particular mapping m,(t) = m,(t,J of D.l (i) (b) is illustrated in Fig. 2. Of course, these illustrations suffer from the defect that they are at best projections onto a 2-dimensional space (the page) of a 4-dimensional space, but it is hoped that they will nevertheless clarify the procedure. The impenetrability principle is illustrated in Fig. 3 where (a) refers to D.l (ii) and (b) to D.l (iii). We denote by SE, = {tj: xr(tj)
not defined;
tj E E;
j = 1,2,.
. .)
the set of instants (empty, finite, or denumerably infinite) at which the motion of P, has velocity discontinuities. The particle P, = (mr, x,(t)) EU4. If we project that point “down the maxis” onto X3 we obtain the “particle projection” x’(t) E X3; it is permanently associated with the r’th particle of mass m, The set of all particle projections onto X3 at t E 5X is P.l.
q’(t)
=(x’(t):
r = 1,2,. . ., s (00)
C X3.
This proposition asserts that particles exist and the number of particles in the universe is finite. The new primitive is s.
FIG. 1. The mapping from E into U4. Vol. 313. No. 3. pp. 149-164, March Printed in Great Britain
1981
153
R. M. Rosenberg Particle T
trajectory
in II4
d
Plone
m&t
1~ m&t,1
I I I Trajectory
FIG. 2. The mapping
V. Massless
m,(t) = m,(Q
in x3
from & into !?JL
Connections
In colloquial language, the term “massless connection” denotes an element having negligible mass to which one or more than one body (which can be modeled as a particle) is attached permanently; the length and shape of such connections may be fixed (as in rigid systems of particles for instance), or variable (as in “massless” coil or leaf springs, for instance). The concept of the massless connection arises naturally in prototypes in which some components have “very small” mass compared to that of other components. Then, in the construction of the mathematical model, it is natural to delete the mass of those components whose mass is regarded as negligible. The massless connection is particularly troublesome from the standpoint of axiomatics because in Newtonian particle mechanics forces can only act on massive elements. Some of these difficulties were pointed out by Stadler (6) who resolved them in a way different from that suggested here by postulating a
m
m
I
9 mr’
pr’
Xr’ =xr’/
(0)
FIG. 3. Visualization
154
(b) of the impenetrability
principle
of D.l (ii) and (iii).
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An Organization of Classical Particle Mechanics
separate axiom of angular momentum not derivable from the linear momentum axiom. We shall use the term “line segment” to mean a set in X3 which is homeomorphic to a straight line segment, and we denote such a set by Ea. If there are particle projections which coincide at some instant to E E with the line segment Ga, we denote the set of these coincidences by D.2.
L(G)
to EE.
i Ea f-l @3(tO);
(6)
If that set is not empty, we call Ga a structure Ga, i.e., EU = Ea(to) iff 3,(&J f 4.
D.3.
(7)
Then we have the proposition P.2.
G,(t)
= E&(&l) vt
E E.
(8)
The verbalization of P.2 is that if particle projections lie in a line segment at some instant of time they remain in that line segment for all time. Finally, we define the massless component G,2: of a structure G, by stating D.4.
6: = G,\S,:
x” = (x,O1,xzn, x37
x2 = fz(P),
x3 = f3(p);
E
G,;
Xl
=
fl(P),
p a parameter.
Thus G,*, is the arc of a curve in the 3-space X3. The manner in which structures are generated is illustrated in Fig. 4 in which some particle projections coincide with the line segment while others do not. VI. Systems of Particles
When the term “system of particles” is not clearly defined, difficulties are likely to arise in the application of Newton’s third law and in the treatment of those variable-mass dynamics problems which are within the compass of classical particle mechanics. For instance, McKinsey et al. (3) suggest that variable-mass dynamics problems such as rocket flight (where the rocket mass decreases as fuel is expended) require the introduction of a time-dependent particle mass. However, the concept of variable particle mass is totally foreign to Newtonian dynamics, as stated in D.l (i) (b). Thus if timedependent particle mass were admitted one would not have
I
2
I
m,
‘0
dx
dr
r
dr = m,.? + const.
These difficulties disappear with the introduction Vol. 313. No. 3. pp. 149-164, Printed in Great Britain
March
1981
of a boundary 2X. 155
R. M. Rosenberg
FIG. 4.
D.5.
The boundary
A structure 6,: C, = 2.
% is a surface homeomorphic
G = x E 3E3: i
to the spherical surface
x2= R*;
(9)
i=l
We can now define the configuration space by stating D.6. The configuration solid sphere
space B is the set of points homeomorphic
XEX~: $x:
to the
(10)
. I
Verbalized, D.6 states that the configuration space is the set of points within and on the boundary. Then, the particles of the system are defined as follows: D.7.
The particles of the system are the members of the set ‘$3(t) = {x’(t):
(11)
x’(t) E B; r = 1,2,. . . , n(t); t E 5X).
This definition states that the particles of the system at the time t are those whose projections at the time t are within or on the boundary of the configurations space (5. We can now state the axiom P.3.
156
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An Organization of Classical Particle Mechanics
whose verbal interpretation is that the configuration space always contains at least one particle. The new primitives are !?I and n(t). We note the following properties: (i) The total membership n(t) of the “population” of the configuration space is a diophantine time function, i.e., particle projections may pass in either direction through the boundary and may thus be acquired or lost by the system; (ii) the total mass of the particles of the system at the time t is n 1)
M(t)=
m, h r=l
(13)
VII. The Mass Center
One of the very useful concepts of Newtonian particle dynamics is the of the system. We state D.8. The mass center of the particles of the system at the time t E 5Xis the point g(t) E X3 such that mass center of the particles
E(t)
t$m, = g
m$(t).
(14)
Under certain circumstances, and in some respects to be discussed later, mass centers behave like projections of particles of mass M(t). This has led McKinsey et al. (3) to observe that “. . . physicists are not quite agreed . . . as to what is meant by a particle. . ., should small bodies serve. . . or should we use centers of mass?” To clarify this issue we state: Mass centers are not particles.
To prove this theorem we note that
because of D.l (i) (b) which requires that the particle mass be constant, and because of D.l (ii) and (iii) which require that one and only one particle can occupy a given position. As pointed out by McKinsey et al. (3) the latter is violated, for instance, when a bullet is shot through a doughnut where two mass centers may coincide at one instant without coinciding for all time. Finally, we define the system as the set
m
(15)
In words, the system is the set of particle projections ‘$3C B together with the configuration space B. The notion of the system is important if we wish to study several systems and their interaction. Vol. 313, No. 3, pp. 14%162. March Printed in Great Britain
1981
157
R. M. Rosenberg VIII. Remarks
on Newton’s
Second Law
The dynamical axioms describe the interaction of particles and structures with forces. They define, in fact, the mapping X ++ U4 of D.l (i). Undoubtedly, the most far-reaching of all axioms of particle mechanics is Newton’s second law. It is usually given in a form equivalent to the following statement: There exists a reference frame and a time (called by Newton “absolute space” and “absolute time”) such that with respect to that reference frame and that time, the acceleration a(t) of a particle is permanently proportional to the force F which acts on the particle, or mu(t) = F ‘it. For this equation to have meaning, Now, under present-day practice, are admitted, i.e. forces such that
F must have finite magnitude for all t. infinitely large forces of bounded impulse
I
f
I = lim F(T) dT# 0 r+rj $ is bounded. Moreover, as already stated, it is present-day practice to admit forces which “act on” massless components of structures. To legitimize the first of these, Newton’s second law will be given in its integrated form called the “axiom of bounded momentum” and to legitimize the latter, a new axiom will be required. Moreover, the phrase “the force acts on (or at) a point” needs clarification. IX
Force
The interaction between forces on the one hand and particles and/or structures on the other is conventionally described in physical 3-space. However, the incorporation of this interaction into X3 meets with the difficulty that the set X3 is not a physical space, i.e. it contains neither particles nor forces; instead, its members are only points occupied by particle projections, points occupied by massless components of structures, and points not occupied by either. Thus, for forces to “act on” particle projections and on massless components of structures, a dynamically meaningful mapping of the force into X3 must be devised. We introduce a 3-dimensional 0space with origins 0 (Q = 0) called the “force space” and spanned by the base b” E 23. Consider now the directed line segment dQ = (Q,, QZ, Q3) E 0 and a point x = (x,, x2, x3) E X3 relative to the same base b”. Then we map Q = 0 +x and Q-R, x, R E X3 such that
F = (F!, F2, F3) = R -x where
158
Fi = Qj, (i = 1,2,3).
Then
E X3
we say: the force
(16)
F acts on (or at) the point Journal
of The Franklin Institute Pergamon Press Ltd.
An Organization of Classical Particle Mechanics x. In general, Q = Q(t), and we require that Jk Q(7) d7 exist on any interval rto, t1 E SE. The process described here provides for the treatment of force-particle structure interaction in X3 and it defines precisely the meaning of the phrase “a force acts on (or at) a point”. This mapping is illus’trated in Fig. 5. The new primitive is Q.
X. The Dynamical Axioms We introduce the integrated form of Newton’s axiom of bounded momentum (7).
second law called here the
P.4. There exists a set of reference frames @l= {g} C 58 and a time space E* C i3, and for every r = 1,2,. . . , n(t) a vector-valued integrable function F’ =
F’(xl, x2,. . .
)
xs; i’, i*, . . . ,i’“;
t)
such that for every t E X* and for every g E @, (i) the vectors T(t) are bounded for all t E E*, (ii) the vectors i’(t) satisfy for all t E SE* and for every r = 1,2,. . . , n(t) the equation i’(t)
=
$
I
et,
F’(xl, x2,. . . , xs; i’, si.‘,. . . , 1”; 7) d7 + i’(to),
(17)
fll
where x’(t) is the particle projection relative to g, and the ir(to) are arbitrary constant vectors; (iii) the vectors i-‘(t) are single-valued and continuous for all t E E:r =
FIG. 5. The mapping of a force into X3-space. Vol. 313, No. 3. pp. 149-164, March Printed in Great Britain
1981
159
R. M. Rosenberg &*\E, where F’dTf
0, 00; j = 1,2,. . . .
(18)
I
The new primitives are the set of “Galilean” frames @, and the “absolute” time space E*. The function F’ is called the resultant force acting on x’. The quantity m,_?(t) is called the linear momentum of the particle P,, and the integral Ji F’ d7 is called the impulse of F’ during the time interval [to, t] E ST’*.
Verbalized, P.3 states: With respect to a Galilean frame and absolute time, the linear momentum of every particle is permanently bounded, and the change of linear momentum during any time interval is equal to the impulse of the resultant force acting on that particle over that same time interval. Let us introduce the notation n(t) Ej
=
U
n(t) SE,;
iy
=
r=l
u Q’,.
(19)
r=l
Then
is the set of instants continuities. Therefore,
09:
m,f’=F’(x’,x*
when no particle of the system we have the theorems
has velocity
,...,
r=1,2
x”; i’,i2 ,...,
3;
t)VtEE:;
,...,
dis-
n(t) (21)
and (SN):
If and only if Ej = #J
m,.Y = F’(x’, x2,. . . , x’; i’, 1*, . . . ,1”; t) Vt E X*;
r= 1,2 ,...,
n(t). (22)
When 5Zj# 4 we call the system “Newtonian” (N), and when 5Ej= 4 we call it “strictly Newtonian” (SN). Theorem (N) states that the motion of every particle of the system satisfies Newton’s second law almost always; (SN) states that in the absence of unbounded forces (of bounded momentum) the motion of every particle of the system satisfies Newton’s second law for all time. The proof of these theorems follows from differentiating (16) with respect to t E iE*. To deal with forces F’$ (y = 1,2,. . . , C,) at points x”, in the massless components of a structure we consider the set of these forces $Ja = {F;: 160
y = 1,2,. . . , Cm}. Journal
of The Franklin Institute Pergamon Press Ltd.
An Organization Then the massless connection
of Classical Particle Mechanics
axiom is
P.5.
(23)
It states that every massless component of every structure is permanently in static equilibrium. In the absence of such an axiom, massless components of structures acted on by nonvanishing forces would, by Newton’s second law, have infinite acceleration because FZ 0, m = 0 implies ial = 00 in ma = F. Stadler (6) points out that his independent axiom of angular momentum is equivalent to the statement that the constraint forces are in static equilibrium, and that observation is similar to the massless connection axiom; in fact, when the only forces acting on massless components of structures are constraint forces the two axioms are completely equivalent. In Fig. 6, we illustrate the meaning of P.5 on an example in which not all forces on massless connections are constraint forces. One of the most useful theorems of Newtonian mechanics deals with the motion of the mass center of a system, and one of the most useful techniques in solving dynamics problems utilizes the so-called free-body diagram. The former requires Newton’s third law in the strong form,? the latter requires a similar law in the weak form. Thus we introduce Newton’s third law in the strong form by means of P.6.
The resultant force F’ acting on x’ is of the form
FIG. 6. The spring-restrained
pendulum.
A and B are fixed points
tThe strong form of Nkwton’s third law requires not only that internal (the weak form) but that the forces of each pair be collinear.
Vol. 313. No. 3. pp. 149-164, Printed in Great Britain
March
1981
forces
in X3. vanish
in pairs
161
R. M. Rosenberg where 0; _=O;(xr,
&
xk) = ?\Ir;((x’ - xk()
Vk, r;
k# r
is called the resultant of the internal forces, and *\I(Jx’- xkl) is a scalar function of the distance Ixr - xkl. The new primitive is 9’;. The quantity 0’ is called the resultant of the external forces acting on x’. The free-body diagram is a technique by means of which geometrical constraints are replaced by forces which have the same effect as the constraints, i.e., they enforce the constraints. These forces may be interaction forces between particles on the one hand and massless components on the other, or they may be interaction forces between portions of a massless component. Newton’s third law does not include these forces because it only speaks of interaction forces acting on particles of the system. Thus, a new axiom is required to legitimize the technique of the free-body diagram. That axiom bears a close resemblance to Hamel’s “Befreiungsprinzip” (4) (liberation principle) in which rigid connections are “liberated” from the rigidity requirement (i.e. they become deformable), and then, forces are introduced which restore the undeformed configuration. However, within our organization, the Befreiungsprinzip must be broadened to include non-rigid massless connections, and it must conform to the free-body diagram technique which provides for separating parts of massless components of structures either from each other or from particles which are on the structure. Thus we introduce the free-body axiom. P.7. A structure G,, of a system % may be replaced by two connected subsets GJ+) and Gap(-) with x;(+)EG~J+) and x;(--)EG,,,(-_) if and only if a force F;(+) acts at xE(+) and a force Fz(-) at xz(-) such that F”d(-) = -F;(+) and G,,,(+) U Gap(-) = Ga when x”,(+) = x%(-) = x; E G,X. It is noted that the forces F;(+) and F%(-) satisfy the weak form for internal forces, i.e. they vanish in pairs. This axiom permits subdividing the structure at any point x; so long as D.l (i) (a) is not violated, i.e. the structure cannot be subdivided so that a particle projection belongs to both Q,,(+) and Gap(-). The example of Fig. 6 is used in Fig. 7 to illustrate the technique of the free-body diagram.
XI
Mass Center Motion
We can now examine in what measure mass centers behave like particles. The well known theorem of the motion of the mass center may be stated in the form: If and only if n(t) = n(QVt E SE’*;to E SE*, the motion of the mass center F
162
Journal of The
Franklin Institute Pergamon Press Ltd.
An Organization of Classical Particle Mechanics
FIG. I. A free-body
of a system
diagram
% of particles
of the spring-restrained mass-axis is now shown.)
pendulum
of Fig. 6. (The
satisfies (24)
where d(to) is an arbitrary vector. The proof of this theorem is found in nearly all elementary texts on dynamics and will not be repeated here; it makes appeal to Newton’s third law in the strong form. The theorem states that, if the number of particles of a system is fixed during some time interval [to, t] E SE*, the motion of the mass center is the same during that time interval as that of a particle of mass M(to), n(tij) located at the mass center g, and acted on by the resultant Cp= X @’ of the r=l external forces acting on the system. Let us now consider several constantmass systems !Jln,(t>={@B,(t),Bs: with mass centers ~~ and system overlapping” in the sense that
6=1,2
masses
!Xzs,n ‘& = C#IV’S’, 8” = 1,2,. Then,
there
exists
a boundary
,...,
M,(t,).
. . , A;
3, and a time interval
A} These
systems
are “non-
6’ # 6”. [to, tA] E E* such that
where %?Ibis the boundary of CIA. Expressed in words, if we consider several non-overlapping systems during a time interval in which the number of particles of each remains constant, we can find a boundary ‘Sz, large enough so that all boundaries of the non-overlapping systems are in the interior of, or on the new boundary %?l~.In that case, each mass center g,, (6 = 1,2,. . . , A) Vol. 313, No. 3, PP. 149-164. Printed in Great Britain
March
1981
163
R. M. Rosenberg satisfies (23). However, (ii) and (iii).
two distinct
mass centers
g;i and 8: need not satisfy
D.l
XIZ. Conclusion The preceding definitions and axioms constitute our organization of classical particle mechanics. This organization admits the analysis of rigid or non-rigid constant or variable-mass systems of particles under forces whose amplitude may be finite, or infinite with bounded impulse, and the forces may be acting on particles or on massless components of structures. The proof of its isomorphism with present-day practice in analyzing dynamical systems would require a complete catalog of every concept and method used to solve problems of the motion of massive bodies and systems of bodies under forces and a complete catalog of the concepts and methods admitted under our organization, and finally the demonstration of a one-to-one correspondence between them. This demonstration has not been made partly because it is
tedious and difficult, and partly because it is considered unnecessary. Under (15) and (16), all systems not comprising structures, as defined in (7), are admitted whether the forces are bounded, or unbounded with bounded impulse, and under (22) massless connections and forces acting on them are also admitted. Finally, the free-body diagram technique is admitted under P.7. Thus, while there is no formal proof of the adequacy of our organization of classical particle mechanics, all methods, techniques and concepts known to us to be in present-day use for solving problems within the domain of Newtonian particle mechanics are legitimate under that organization. Acknowledgements Many fruitful discussions with Professor George Leitmann significant influence on this paper; it is by his choice that co-author.
have had a he is not a
References (1) J. L. Destouches, “The Axiomatic Method” (Edited by L. Henkin, P. Suppes and A. Tarski), pp. 390-402, North Holland, Amsterdam, 1959. (2) Quoted from R. von Mises, “Kleines Lehrbuch des Positivsimus”, p. 114. W. P. van Stockum & Zoon, The Hague, 1939. (Quote translated by R.M.R.) (3) J. C. C. McKinsey, A. C. Sugar and P. Suppes, J. Rat. Mech. Anal., Vol. 2, p. 253, 1953. (4) G. Hamel, “Handbuch der Physik”, pp. l-42, J. Springer, Berlin, 1927. See also G. Hamel, “Theoretische Mechanik”, pp. 508-525, Springer-Verlag, Berlin, 1949. (5) M. Bunge, “Foundations of Physics”, p. 130. Springer-Verlag, New York, 1967. (6) W. Stadler, unpublished paper (personal communication, 1980). (7) R. M. Rosenberg, “Analytical Dynamics of Discrete Systems”, p. 10, Plenum Press, New York, 1980.
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Journal
of The Franklin Institute Pergamon Press Ltd.