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Unformatted text preview: Inadequacy of the usual Newtonian formulation for certain problems in
particle mechanics
W. Stadler Division of Engineering, San Francisco State University, San Francisco, California 94132
(Received 2 January 1980; accepted for publication 15 September 1981) The problem of the rigid double pendulum is used to show that the Newtonian axioms are
inadequate to obtain a solution. It is further shown that an additional independent axiom, such as
the law of moment of momentum, must be introduced to make the solution possible. I. INTRODUCTION Although solved correctly some 300 years ago, the prob
lem of the rigid double pendulum is an example which suf
ﬁces to convince the reader of the need for an additional
postulate when dealing with constrained particle motion.
It will be shown that the problem of the rigid double pendu
lum has no solution when Newton’s third law is taken to
include the centrality (or collinearity) of the internal forces.
When centrality is not assumed the “principle of angular
momentum” no longer follows as a theorem and must be
postulated as an independent law. The application of the
law of linear momentum and that of moment of momen
tum then leads to a noncentral internal force system. There are a number of other problems which exhibit the
same difﬁculty; e.g., a mass point constrained to slide on
the massless rod of a simple pendulum. The analysis here is
conﬁned to the rigid double pendulum, for simplicity. II. NEWTONIAN FORMULATION By way of background, the main ingredients of the me
chanics of a system of particles are now summarized. Consider a set (or system) .ﬂ of m particles
P,, i = 1,...,m, with masses m,, respectively. The position
vector of P, relative to an arbitrary ﬁxed origin in an iner tial reference frame is r, =x} il +x§i2 +x§i3, where (x,',x?,x§) are rectangular Cartesian coordinates and (il ,i2,i3) is a ﬁxed orthonormal basis. The velocity and ac
celeration of P, at the instant t of time are denoted by r..(t)
and m t ), respectively. For a system of particles in which all forces are assumed
to arise from the interaction of the particles with one an
other, the forces acting on a subsystem JV Q J , of n<m
particles, may be subdivided into external and internal
forces. The external forces acting on JV are those exerted
by particles in % \JV on particles in JV, while the internal
forces are those exerted by particles in JV on each other.
The external force acting on P, is denoted by F), while the
internal force exerted on P, by P]. is denoted by FU. Either of
these forces may generally be a function of the current posi
tions and velocities of all n particles as well as time. The usual formulation of Newtonian particle mechan—
icsI consists of the following postulatesz: (i) Newton’s second law in the form Fm + (S: Far) = mm) (I) 1:1 for each particle P, in the system JV. Of course, (1) includes
Newton’s ﬁrst law for a single particle. 595 Am. J. Phys. 50(7), July 1982 (ii) Newton’s third law in the form
Fri“) = —— FIJI(t) and F11“): 0 (2) fort: l,...,n; j= 1,...,n.
(iii) The internal forces F” (t) are central 3; that is, [l'.(t)l'j(t)]><F.y(t)=0 (3) for i = 1,...,n; j = 1,...,n._ Furthermore, the masses of the particles are always as
sumed to be nonzero. For any given particle P, in the system the problem to be
solved in particle mechanics belongs to one of the following
categories: (a) The postion r, (t) of a particle P, is completely speci
ﬁed as a function of the time t. (b) The particle P, is constrained to move on a speciﬁed
surface or curve. (c) All of the forces acting on P, are prescribed as func
tions of the kinematic variables or some of the forces are
prescribed together with classes of permissible velocities or
accelerations. A problem is considered solved when the motion r, (t)
and the external and internal forces have been obtained as
functions of time in such a way as to satisfy Newton’s laws
(i) and (ii). On the basis of assumptions (i)—(iii) it is easy to establish
the following theorem. Theorem I: Moment of momentum. Assume that the
forces, masses, and accelerations of 'a system of particles
satisfy the postulates (i)—(iii). Then , H0 = M, where H0 is
the moment of momentum of the subsystem JV gel! with
respect to a point 0 ﬁxed in an inertial reference frame and
where M0 is the total moment about 0 of the external forces
acting on JV. (See Fig. 1.) Proof. For convenience, let 0 be the origin. For each par
ticle, the application of Newton’s second law (i) implies VF..(z)+ 2 Fat) =m.i.(t) (4) i=1 Fig. 1. Central force assumption. © 1982 American Association of Physics Teachers 595 for i = 1, .. .,n. Cross multiplication of each equation by r, (t )
and a summation over 1' yield n 2 r.(t)><F.(t)+ i )"3 r.(t)><F.,(t) 1= 1 l = 1 j = 1
= 2 miriltleiltl (5)
i = l
Newton’s third law (ii), together with the assumption that
the internal forces are central (iii), then implies that Hom= i m.r.(r)><i.~(t) l=l = 2 r.~(t)><F.(t) = Mo(t)~ (6)
i = I
In applications, Theorem 1 is habitually used as if it were
an independent axiom; that is, no check is made as to
whether the hypotheses of the theorem are satisﬁed or not.
However, the use of the theorem obviously is justiﬁed only
if all of the hypotheses are met. As will be shown, the hy
potheses are, in fact, not satisﬁed in the case of the rigid
double pendulum. Indeed, the postulates (i)—(iii) are insufﬁ
cient to solve this problem.
Preliminary to the discussion of the rigid double pendu
lum, it is instructive to ﬁrst analyze the simple pendulum in
the light of the above remarks. III. SIMPLE PENDULUM In the standard treatment of the simple pendulum, the
freebody diagram of the particle P is drawn as in Fig. 2,
with the tensile force T = — Te, and weight
W = mg(cos 0 e, — sin 6 es) of the particle. An applica
tion of Newton’s second law yields [9 +(g/a)sin0=0 , (7)
as the eocomponent equation, and
T=mgcos0 +mal92 (8) as the e,component equation. Equation (7) may be used
(subject to apprOpriate initial conditions) to obtain 0 (t ), and
(8) then gives the tension required to maintain this motion.
It should be emphasized that the tension T was not speci
ﬁed as regards its functional dependence on 49 (t ) but rather
that this dependence was derived through the application
of postulates and geometrical constraints. Note that, in ad
dition to Newton’s second law, the solution was based on
the assumed centrality of the internal force T. The above problem also serves as an illustration of the
use of Theorem 1. With 0 as the ﬁxed point and with W as
the external force, one again obtains (7). Next, Newton’s
second law, together with the central force assumption, is used to calculate T. Thus, in the case of the simple Pendu
lum, the use of Theorem 1 is justiﬁed since a central force 596 Am. J. Phys, Vol. 50, No. 7, July 1982 Fig. 2. Standard simple pendulum. \ Fig. 3.
pendulum. Variant of the simple does, in fact, suﬁice to sustain the motion. Suppose now that one attempted to solve the problem
without making use of the central force assumption. To this
end, let the internal force on P be represented as
F = F,e, + F9 eg. The use of Newton’s second law yields F, = mg sine +maé, (9)
F, = —mgcos€—ma€2. (10) One thus has two equations in the three unknowns 0, F9,
and F,, so that an additional condition must be introduced
before a solution is possible. (See Fig. 3.)
As one such condition, the central force assumption, F9
= 0, would again yield (7) and (8). Alternatively, one may
assume Ho = M0 as an independent axiom. With F consi
dered as an internal force, and with W as the external force,
one obtains man's): —mga sin6. (11) By substituting (1 1) into (9), one deduces that F 9 = 0, and
F, may be determined as before. It follows that in this problem, the central force assump
tion and an independent law of moment of momentum are
equivalent in the sense that one may be deduced from the
other once Newton’s second law is assumed. This equiv
alence does not hold in general, however, as the next exam
ple demonstrates. IV. RIGID DOUBLE PENDULUM In this section, a simple problem in the constrained mo
tion of particles is presented which cannot be solved with
the central force assumption but for which an independent
ly postulated axiom of moment of momentum does lead to
a solution. It thus appears that the latter postulate allows
solutions for a class of problems for which the former pos
tulate fails to provide one. Problem. Consider a system of three particles P0,P,,P2
having masses m0,m,,m2, respectively, and connected by
means of a single massless rod. The particle P0 is ﬁxed4 and
the rod is hinged at 0 in such a way that the system may
move as a pendulum in the xy plane, as illustrated in Fig. 4.
The only external forces acting on the system are the
weights of the particles and the reaction force at 0. The rod Fig. 4. Rigid double pendulum. W. Stadler 596 is initially at rest in the horizontal (0y) position. Obtain the
angular displacement 0 as a function of the time. Strictly speaking, the simple pendulum and the problem
as posed above do not conform to the Newtonian formula
tion in Sec. II, because of the inclusion of such concepts as
strings and rods. However, these idealizations are artiﬁces
which serve to make the pendulum models physically plau
sible; they are not essential to the problem statement. In
order to cast the pendulum problems within the framework
of particle mechanics, the rods and strings must be re
placed by suitable statements about the intemal forces Fij
and by geometrical constraints on the motion. Consider then the rigid double pendulum as posed with
in the framework of Sec. II. It is idealized as a system of
three particles P0, P,,P2 aligned along a straight line for all
time and moving on concentric circles. The particles are
acted upon by the external forces comprised of the weights
of the particles, W0,W,,W2, acting vertically downward,
and the reaction F0 at 0. In addition, the action of the con
necting members upon the particles is idealized as a set of
internal forces {F,0,F0,,F20,F02,F12,F2, }. The unknown re
action, the intemal forces and the motion are to be
determined. Ideally, any axioms postulated for the solution of dyna
mical problems provide necessary and sufﬁcient conditions
for that solution; that is, for the determination of the mo
tion and the forces (or possibly, their resultants). For a giv
en set of unknowns it is generally accepted that there must
be at least as many conditions available for their determina
tion as there are unknowns. With reference to the rigid
double pendulum, the process of the introduction of un
knowns and the provision of an equal number of conditions
for their determination will now be illustrated. The solution begins with the postulation of Newton’s
second law (i) for each particle as the ﬁrst relationship
between the unknowns. Based on the freebodies of Fig. 5,
one has the equations of motion given by wo+Fo+F01+F02=mofm
W1+F10+F12=mifn (12)
W2 + F2: + F20 = ”12.13 The set of vector unknowns in this system of equations is
iFo’FonFio,F02:F20:F12F21:1'0sl'1’r2i, so that there are 20 unknown component functions of time
for whose determination one has 6 component equations in
(12). The geometric constraints are introduced next. For the
particle 1"o one has r0 = 0; the presumed circular motion of
the particle Pl, namely, r1 = a,e,, yields Fig. 5. Rigid double pendulum as a con
strained particle motion. 597 Am. J. Phys., Vol. 50, No. 7, July 1982' i‘l = a,( — 49 2e, + deg) as the expression for the accelera
tion; ﬁnally, since the particle P2 remains aligned with P1,
one has r2 = (oz/am]. The use of these 5 conditions leaves
the unknowns iFO’FODFIOF’02’F201F12,F2110 } , when suitable initial conditions have been prescribed. Thus
there now are 15 unknowns and 6 equations. An application of Newton’s third law (ii) leaves equa
tions of motion of the form wo+Fo—F10_on=0’ wl+F10+F12=mlf1’ (13) W2 — F12 + F20 = m2(aZ/al)i:l 2
with the set of unknowns reduced to {F0,F10,F20,F,2,0},
leaving 9 unknowns and 6 equations. Evidently, in the Newtonian analysis, the effect of the
internal forces on a given particle consists of a resultant
internal force due to all of the other particles. Conversely,
this resultant is all that one may expect to determine when
nothing concerning the functional form of the internal
forces has been speciﬁed. With this in mind, Eq. (13) may be
written as W0+Fo—Rl—R2=0, W1+ R1: mlil’ R1: F10 + 1:"12 , (14) W2 + R2 = mziaz/aiifi: R2 = — F12 + F20  The new set of unknowns is {F0,R,,R2,0 }. With the
weights written as W, =m,.g(cos0e, —sint9e9) i=0,l,2, (15) the system (14) of 6 equations in 7 unknowns may now be
expressed in the form
F0 = " [(mlal + "12%)?2 + (”'0 + m1 + m2)gcos 01%
+ “"1101 + "1202)0 + (me + "11+ m2)gsin 0199 ,
. (16)
R, = — m,.(a,62 +gcos 6)e,
+ m,(a,é +g sin 0)e,9 1': 1,2. An additional condition is needed to solve this system of
equations. Usually, this condition is taken to be the as
sumed centrality of the internal forces, condition (iii)
above. When used here, this condition comprises two sepa
rate conditions requiring that the «2,9 components of RI and
R2 vanish. It then follows from (16) that a = — (g/a,)sin 6 and a = — (g/az)sin a. (17) Since (12 is not equal to a, in general, the es components of
R1 and R2 cannot vanish simultaneously and one is led to
the conclusion that the internal force system cannot be cen
tral. It follows that this problem has no solution within the
traditional framework of Newtonian dynamics. _ Suppose now that instead of assuming centrality of the
internal forces, one postulates an independent law of mo
ment of momentum. Thus applying (6), one obtains é: _ mlal +m2a2
mlai + m2“; and hence, by integration, 9 2 = mm
mm? + mzaﬁ where the inital conditions 9 (0) = 0 and 6 (0) = 1r/2 have gsin6 (18) gcos 0, (19) W. Stadler 597 been used.
The corresponding forces are then given by cos 6
F0: — g2 2 [(mo‘l‘ml‘l'mzlimlal +m20§l
"2101 + mzaz
+ 2(m101 + mZazlzler + gZSIn 0 2
mlal + mzaz
X [(7710 + m1 + mzllmﬂi + mzagl
—‘ (mlal + mzazlzlee (20)
and
m,g cos 0 2 2
R: = — 2 [mlal +mza2 2
ma, +m2a2 m, sin 6
‘l‘ 2ailm1al + mzazl]er + 2g 2
mlal + "1202 >< [mla] +m2a§ —a,(m,al +m2a2)]ea 1’: 1,2. (21) This internal force system clearly is not central. V. CONCLUSIONS The preceding example shows that there are simple
problems in constrained particle motion for which the tra
ditional Newtonian axioms are insufﬁcient to obtain a solu
tion. It has also been shown that by invoking the law of
moment of momentum (as a separate postulate) the prob
lem of the rigid double pendulum became soluble. It is of interest to note that for the present problem, the
postulation of an independent law of moment of momen—
tum is equivalent to the following postulates in the sense
that Eq. (18) is the consequence of their application: (a) The law of conservation of kinetic and potential ener
gy. A choice of zero potential energy at a vertical distance
02 from 0 yields V(t) = m,g(az ——a1 cos 0) + ngaz(l — cos 0) +m0gaz ,
T(t)= ma? + magma for the potential and the kinetic energy, respectively. The
conservation of the total energy is expressed by
V(t ) + T(t ) = const. A differentiation with respect to t re~
sults in Eq. (18). (b) The system of internal forces is in static equilibrium.
This, of course, may be expressed in terms of either the
virtual work of the forces or in terms of a summation of forces and moments.
The total virtual work of the system of internal forces vanishes. With admissible virtual displacements given by
6n, = 0, 5r, = a,60e,,, and 6r2 = a260e9, this statement
takes on the form Rl6l'l + R2'6r2 = [(mlaﬁ + mza§)é +(m,a1+ m2a2)g sin 9 ]60 = 0. 598 Am. J. Phys., Vol. 50, No. 7, July 1982 Since this must be true for any 66, the expression in brack
ets must vanish. The summation of internal forces and the summation of
their moments with respect to any ﬁxed point are zero.
With R0 = F0l + F02 as the internal force acting on the
particle P0 one has R0+R, +R2=0
and
1x710=r,xR,+r2xR,=o. The latter equation again results in Eq. (18). In closing, it may be remarked that within the frame
work of the dynamics of deformable media one does, in
fact, postulate Euler’s second law of motion, the law of
moment of momentum, as an independent axiom. It is also
interesting to note that from a historical viewpoint, the in
ception of the law of moment of momentum preceded the
Newtonian postulates. The historical development is
traced by Truesdell.2 ACKNOWLEDGMENTS The author is grateful to J. Casey for many discussions
and suggestions related to the subject matter of this paper.
In addition, he wishes to thank G. Leitmann and R. Rosen
berg for their comments and encouragement. 11. Newton, Philosophiae Naturalis Principia Mathematica. The printing
thereof was completed by S. Pepys on 5 July 1686; the book was pub
lished in London in 1687. The most commonly cited English translation
is that given in 1729 by Andrew Motte as revised by Florian Cajori: Sir
Isaac Newton’s Mathematical Principles of Natural Philosophy and his
The System of the World (University of California, Berkeley, CA, 1960). 2Reference 1, Vol. 1, p. 13. Newton’s laws have been stated here in the
usual modern form. They cannot be found precisely in this form any
wherein the Princzpia, although all the basic ideas are there. If Newton’s
second law is postulated about a system of particles, that is, the total
external force is set equal to the total rate of change of the linear momen
tum of the system, and if it is assumed that the law also applies to every
subsystem, then Newton’s third law in the form (ii) follows as a theorem.
If Newton’s second law is taken as a statement about the motion of an individual particle, as is done here, then the third law does not follow as a
theorem but must be independently postulated. For further critiques and
historical comments concerning Newton’s laws, the works of I. B. Cohen
[Texas Quart. X (3), 127—157 (1967)], C. A. Truesdell [Essays in the His
tory of Mechanics (SpringerVerlag, New York, 1968)] , and I. Szabé
[Geschichte der Mechanischen Prinzipien (Birkhiiuser, Stuttgart, 1977)]
are recommended. 3Sometimes (ii) and (iii) together are put forward as a strong form of New
ton‘s third law; but it seems to be more in keeping with Newton’s own
statement and usage to regard (ii) alone as representing the third law. [See
Ref. 1, The System of the World, p. 570.] 4The particle PO has been included only to conform to the Newtonian idea
that forces act on masses. W. Stadler 598 ...
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