The relation between momentum conservation and Newton's third law revisited

0.1 MB·English

by  Rodolfo A. Diaz

The relation between momentum conservation and Newton’s third
6
law revisited
0
0
2
Rodolfo A. Diaz∗, William J. Herrera†
n
a Universidad Nacional de Colombia, Departamento de F´ısica, Bogot´a-Colombia.
J

6
1 Abstract
A very important part of the foundations of clas-

] sical mechanics lies on one hand in Newton’s third
h
p Under certain conditions usually fulfilled in classical law or, on the other hand, on the principle of conser-
- mechanics, the principle of conservation of linear mo- vation of linear momentum. Thus, the link between
s
mentum and Newton’s third law are equivalent. How- bothapproachesis ofgreatestinterestinconstructing
s
a ever, the demonstration of this fact is usually incom- the formalism of classical Physics. Commom texts of
l
c plete in textbooks. We shall show here that to demon- mechanics [1], usually state that Newton’s third law
. stratetheequivalence, werequiretheexplicit useofthe automaticallyleads to the principle of conservationof
s
c principle of superposition contained in Newton’s sec- linear momentum. However, the converse is also true
i
s ond law. On the other hand, under some additional under certain conditions; the proof in reverse order is
y conditions the combined laws of conservation of lin- either absent or restricted to systems of two particles
h
ear and angular momentum, are equivalent to New- in most textbooks. We start from the statement of
p
ton’s third law with central forces. The conditions for the principle of linear momentum conservation for a
[
such equivalence apply in many scenarios of classical closed and isolated system of particles‡, with respect

3
mechanics; once again the principle of superposition to a certain inertial frame
v
contained in Newton’s second law is the clue.
9
9 PACS: {01.30.Pp,01.55.+b,45.20.Dd} P1+P2+...+Pn =constant. (1)
0 Keywords: emphNewton’s laws, conservation of lin-
Deriving with respect to time, gives
5
ear momentum, conservation of angular momentum.
0
4 Resumen dP1 + dP2 +...+ dPn =0, (2)
0 dt dt dt
/
cs Bajo ciertas condiciones que se cumplen con frecuen- but according to Newton’s second law, ddPti refers to
i cia en la meca´nica cla´sica, los principios de conser- the total force applied to the i−th particle. Further,
s
y vacio´n del momento lineal y la tercera ley de Newton since there are no external forces Eq. (2) becomes
h sonequivalentes. Noobstante, lademostracio´n deeste
n n n
:p hechoesta´usualmenteincompletaenlostextossobreel XF1j +XF2j +...+XFnj = 0
v tema. En este art´ıculo se demuestra dicha equivalen- j6=1 j6=2 j6=n
i cia, para lo cual requerimos del uso expl´ıcito del prin-
X n n
cipio de superposicio´n contenido en la segunda ley de ⇒ F = 0. (3)
r XX ij
a Newton. Por otro lado, bajo algunas condiciones adi- i=1 j6=i
cionales las leyes combinadas de conservacio´n del mo-
mento angular y del momento lineal, son equivalentes Where Fij stands for the force on the i−th particle
a la tercera ley de Newton con fuerzas centrales. Las due to the j−th particle. In the case of two particles,
condiciones para esta u´ltima equivalencia tambi´en son Eq. (3) leads to Newton’s third law automatically.
va´lidas en muchos escenarios de la meca´nica cla´sica; However, in the case of an arbitrary number of parti-
una vez ma´s el principio de superposicio´n contenido cles, Newton’s third law is only a sufficient condition
en la segunda ley de Newton es la clave para la de- in this step. The proof of necessity is the one that is
mostracio´n. usually absent in textbooks.
Inordertoprovethenecessity,weshallusetheprin-
Palabras clave: Leyes de Newton, conservacio´n del
ciple of superposition stated in Newton’s second law.
momento lineal, conservacio´n del momento angular.
‡For a closed system, we mean a system in which particles
∗[email protected] are the same at all times, i.e. no particle interchange occurs
†[email protected] withthesurroundings.
1
2 Rodolfo A. Diaz, William J. Herrera
Considering a system of n particles, let us take a cou- ofmomentumconservationisstillheldwhileNewton’s
pleofparticleskandl. Theyundergotheforceofeach third law is not valid any more, from which follows
otherF andF respectively,plustheinternalforces the advantage of formulating the empirical principles
kl lk
due to the other particlesof the system. However,ac- of classical mechanics in terms of the concept of mo-
cording to the second law, the forces F and F are mentum. Evenwhenthe assumptionsgivenaboveare
kl lk
notalteredbythepresenceoftherestoftheforces(i.e. satisfied,theformulationintermsofmomentumisad-
theotherforcesdonotinterferewiththem). Therefore vantageous [2]. Nevertheless, we emphasize that un-
ifwewithdrewtheotherparticlesofthesystemleaving der the conditions cited above, Newton’s third law is
the particles k and l in the same position, the forces equivalent to the principle of conservation of linear
F andF wouldbe the same as those when allpar- momentum, but the complete proofof that statement
kl lk
ticles were interacting. Now, after the withdrawal of requires the principle of superposition of forces estab-
theotherparticles,oursystemconsistsoftwoisolated lished by Newton’s second law.
particles for which the third law is evident. Hence On the other hand, by a similar argument we can
Fkl = −Flk. We proceed in the same way for all the show the equivalence of combined conservationof lin-
pairs of particles and obtain that Fij = −Fji for all ear and angular momentum with Newton’s third law
i,j in the system. Observe that the proof of necessity with central forces. Starting from the conservation of
requires the use of the principle of superposition con- angular momentum for a closed isolated system with
tainedin Newton’s secondlaw. Since wehavedemon- respect to an inertial frame
strated the necessity and sufficiency, we have proved
the equivalence. Notwithstanding, this equivalence is L=L1+L2+...+Ln =constant (4)
based on many implicit assumptions
and deriving this equation we find
1. Newton’ssecondlaw is valid: Asiswellknown,in
scenariossuchasquantummechanicstheconcept dL1+dL2+...+dLn =0.
of force is not meaningful any more. dt dt dt
2. The time runs in the same way for all inertial FromthedefinitionofL andtakingintoaccountthat
i
observers: We have used this condition since in the system is isolated, the derivative of the total an-
the time derivative of Eq. (2) we do not mention gular momentum reads
whatinertialsystemwehaveusedtomeasurethe
time. Besides, this condition is necessary to as- n n
dL  
sumethattheforceisequalinallinertialsystems. = r × F =0.
dt X i X ij
i=1 j6=i 
3. All the momentum of the system is carried by
the particles: In this approach we are ignoring
Under the assumptionF =−F (obtainedfromthe
ij ji
the possible storage or transmission of momen-
conservation of linear momentum) we can show the
tum from the fields generatedby the interactions
following identity by induction
(see discussion in Ref. [3]).
4. The signals transmitting the interactions travel n n n−1 n
 
instantaneously: In Eq. (1), each momentum Pi X ri×XFij = XX[(ri−rj)×Fij].
is supposed to be measured at the same time. If i=1 j6=i  i=1 j>i
any particle of the system changes its momen-
Clearly, Newton’s third law with central forces (i.e.
tum at the time t; then to preserve the law of
F = −F and (r −r ) parallel to F ) is a suf-
conservation of momentum (at the time t), it is ij ji i j ij
ficient condition for the conservation of angular and
necessary for the rest of the particles to change
linear momentum (we shall refer to the third law
theirmomentaat the same time,insuchaway
with central forces as the strong version of Newton’s
thattheycancelthechangeofmomentumcaused
third law henceforth). To prove necessity we resort
by the i−th particle. This fact is in turn related
again to the argument of isolating one pair of parti-
withtheconditionthatallthemomentumbecar-
cles k,l without changing their positions. Since this
riedby the particles(mechanicalmomentum). In
two-particle system is now isolated, its total angular
other words,the other particlesmust learnofthe
momentum must be constant, and remembering that
changeinmomentumofthei−thparticleinstan-
the forces F = −F have not changed either, we
taneously. kl lk
have
Ashasbeenemphasizedintheliterature,eveninthe dL(two particles)
case in which all these assumptions fail, the principle =(rk−rl)×Fkl =0.
dt
Equivalence between momentum and Newton’s third law 3
Now, since both particles have different positions and Fundamental University Physics, Vol I, Mechan-
we are assuming that they are interacting (F 6=0)§, ics (Addison-Wesley Publishing Co., Massachus-
kl
we obtain that (r −r) must be parallel to F . We sets,1967);B.R.Lindsay,PhysicalMechanics,3rd
k l kl
can proceed in a similar way for all possible pairs of Ed. (D. Van Nostrand Company, Princeton, New
particles in the system. In this case we have used the Jersey,1961);K.R.Symon, Mechanics. (Addison-
combined laws of conservation of linear and angular WesleyPublishingCo.,Massachusetts,1960),Sec-
momentum because Newton’s third law in its weak ond Ed.
version was assumed since the beginning. Of course,
[2] Edward A. Desloge, “The empirical foundation
the conditions for this equivalence to hold are those
of classical dynamics”, Am. J. Phys. 57, 704-706
cited above, but with analogous assumptions for an-
(1989).
gular momentum as well.
As before, when the conditions for this equivalence
[3] E.Gerjuoy,“OnNewton’sThirdLawandtheCon-
fail, the laws of conservation of linear and angular
servation of Momentum”, Am. J. Phys. 17, 477-
momentum are still valid, while the strong version of
482 (1949).
Newton’s third law no longer holds.
An important issue arises when we consider non- [4] For a revision, see for example: L. Eisenbud, “On
isolated systems, since we have assumed that the sys- the Classical Laws of Motion”, Am. J. Phys. 26,
tem is isolated throughout this treatment. If we add 144-159 (1958); G. H. Duffey Theoretical Physics,
external forces, once again the principle of superpo- page 36 (Houghton Mifflin Company, Boston,
sition states that the internal forces do not interfere 1973); Edward A. Desloge, Classical Mechanics
with them, and so Newton’s third law is maintained. Vol. I (John Wiley & Sons, Inc., 1982); R. B.
Asimilarargumentholdsforpossibleexternaltorques LindsayandH.Margenau,Foundations of Physics
and the Newton’s third law in its strong version. (Dover Publications, Inc., New York, 1957).
In conclusion, we have proved that under certain
conditions the principle of linear momentum conser-
vation is equivalent to the weak version of the New-
ton’sthirdlaw. Analogously,undersimilarconditions,
the combined laws of conservation of linear and an-
gular momentum are equivalent to the strong version
of Newton’s third law. We emphasize that for both
demonstrations we should invoke the principle of su-
perposition contained in Newton’s second law¶. Fi-
nally, it is worth mentioning that the suppositions
neccesary to obtain these equivalences are implicit
in the original formalism of classical mechanics [4].
Therefore, such equivalences deserve more attention,
at least until reaching relativity, quantum mechanics
or classical (quantum) field theories.
The authors wish to thank Dr. H´ector Mu´nera for
useful discussions and Dr. Victor Tapia for his sug-
gestions concerning the bibliography.
References
[1] D. Kleppner and R. Kolenkow,An introduction to
mechanics (McGRAW-HILLKOGAKUSHALTD,
1973);R.ResnickandD.Halliday,Physics (Wiley,
NewYork,1977),3rdEd.;M.AlonsoandE.Finn,
§ThecaseofFkl=0couldbeconsideredasatrivialrealiza-
tionofthestrongversionofNewton’sthirdlaw.
¶Of course, Newton’s first law is also in the background of
allthistreatment, byassumingtheexistence ofinertialframes
inwhichtheotherlawsofNewtonandtheconservationofmo-
mentaarevalid.