Download 1 Contributions of Newton and Euler towards F =ma Ajay Sharma

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Contributions of Newton and Euler towards F =ma
Ajay Sharma
Fundamental Physics Society. His Mercy Enclave Post Box 107 GPO Shimla 171001 HP
India
Email [email protected]
Abstract
Newton’s second law of motion …….‘The alteration of motion is ever proportional to the motive force
impress'd; and is made in the direction of the right line in which that force is impress'd.’ remained
unchanged in all three editions ( 1687,1713 and 1726) of the Principia. Newton did not give any
equation based upon second law either in algebraic or differential forms. In 1750 Swiss mathematician
Leonhard Euler independently i.e. using extrinsic references frames (a system of three orthogonal
Cartesian axes), gave mathematical equation F =md 2s/dt2 =ma. Euler’s equation is incorrectly regarded
as equation for the Principia’s second law of motion. Now an attempt has been made to derive equation
of motion for the Principia’s second law of motion, using various relevant terms ( absolute motion,
relative motion, celerity as motion , definition of force etc. ) defined by Newton is the Principia.
Obviously mathematical equation on the Principia’s second law of motion, turns out to be F =k(v-u), k
has dimensions MT-1. The dimensional analysis was initiated by Fourier in 1822 , unit of force dyne was
defined in 1861 and newton in 1948. Thus there are two laws of motion i.e. the Principia’s second law
of motion F = k(v-u) , and Euler’s law of motion F =ma. Thus Newton’s second law which is taught in
standard literature, in fact is statement for Euler’s equation F=ma. Newton’s statement as given in the
Principia page 19 is neglected completely. Any equation not given by Newton should not be called
Newton’s second law of motion.
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1.0
Newton’s Second Law of Motion
Isaac Newton stated the second law of motion [1,2] in The Mathematical Principles of Natural
Philosophy at page 19 in 1687. Initially the law was stated in Latin, then translated to English. The law
remained same in all three editions of the Principia published in 1687, 1713 and 1726.
Law II ( Latin) Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam
rectam qua vis illa imprimitur.
Law II (English) ‘The alteration of motion is ever proportional to the motive force impress'd; and is
made in the direction of the right line in which that force is impress'd.’
Newton explained the definition of second law of motion in the following way:
‘If any force generates a motion, a double force will generate double the motion, a
triple force triple the motion, whether that force be impressed altogether and at once, or gradually and
successively. And this motion (being always directed the same way with the generating force), if the
body moved before, is added to or subducted from the former motion, according as they directly
conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to
produce a new motion compounded from the determination of both.”
Here Newton did not give any mathematical equation how to calculate magnitude of
force, so just like definition the explanation is also qualitative. There is nothing in the Principia’s second
law about acceleration and nothing about a rate of change as confirmed by Pourciau, article titled Is
Newton’s second law really Newton’s ? [3,4]. The extensively used equation F=ma nowhere occurs in
any edition of the Principia [ 6]. Truesdell [ 5 ], it has been clearly stated that F=ma appears nowhere in
Newton's Principia, but was clearly stated for the first time by Euler in 1750. It is strange that Newton’s
original second law of motion as in the Principia is completely neglected and not mentioned in the
standard textbooks. The definition based upon Euler’s equation F=ma is taught as Newton’s second law
of motion in existing literature. But Newton has not given it in any way. Euler’s law must be taught as
Euler’s law independently, there is no objection , but should not be taught as Newton’s law. The
implications of the effect are being discussed especially assessing in the correct equation for the
Principia’s second law.
2.0
Useful terms in the Principia to understand Second Law of Motion.
(i) Definition of Rest:
When the position of a body with respect to its surroundings does not change
with time, it is said to be at rest. In this case, velocity v = 0 as distance moved is zero.
Definition of motion: Motion is a change in position of an object with respect to time and its reference
point. Thus in this case body moves v>0 as distance moved is non-zero. So both the states of rest and
motion are described in terms velocity.
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(ii) Absolute Motion (motus absolutus)
Newton [1-2] has defined motion as absolute motion in Scholium of DEF.VIII, page 10.
It is translation of body from one absolute space to other. In Physics translation is defined as
movement of body from one point to other i.e. body possesses velocity.
Relative motion. It is the translation from one relative place from other.
Further in same section understanding of motion of ship with respect to earth Newton has expressed it
in terms of velocity.
Absolute motion = motion ≡ velocity or speed
Thus, alteration in motion = (v-u)
(1)
(iii) Motion as Celerity (velocity) .
While explaining Definition VIII at page 8, Newton stated, “Wherefore the Accelerative force will stand
in the same relation to Motive, as celerity does motion.”
Thus motion is directly related to celerity (velocity or speed).
(vi) Quantis motus or Quantity of motion
Newton defined ‘ Quality of motion’ is defined in DEF. II at page 2. Absolute motion is defined in
Scholium at page 10. Both must be clearly understood before interpretation.
‘The Quantity of Motion is the measure of the same, arising from the velocity and quantity of matter
conjunctively.’
As both the terms are different, hence explained at different places. If the meaning of both is same, then
would have been explained in the same line or sentence. Thus definition of Quantity of motion and
absolute motion are entirely different hence defined under different headings , so ‘motion’ cannot
replace ‘Quantity of motion’ and vice versa. If both are regarded as having same meaning, then it is
misinterpretation and contradictory to concepts laid down by Newton in the Principia.
(v) Alteration and ‘rate of change’. In original definition of Newton’s second law of motion as given
in all three editions of the Principia (1687, 1712, 1726) , the word alteration is used. It means change in
two stages ( Q2-Q1). However usually the ‘rate of change of ’ is used, instead of alteration , it is not
justified [3-4]. Thus finally it is concluded that there is no resemblance between
d
and ( Q2 –Q1).
dt
Also the terms ‘motion’ and ‘Quantity of motion’ are entirely different. These terms cannot be regarded
as replacements of each other.
(vi) Definition (IV) of force. “An impressed force is an action exerted upon a body, in order to change
its state , either of rest or of moving uniformly forward in right line.” Now ,
state of moving uniformly forward in right line
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It is state of ………..uniform velocity forward in right line.
Thus impressed force is related with velocity. This definition of impressed force in other words is First
Law of Motion [1-2]. Also impressed force is associated with Second Law of Motion. Thus in Newton’s
Principia, impressed force is associated with velocity.
(vii) First Law of Motion. “Every body perseveres in its state of rest or of moving uniformly forward
in a right line, unless in so far as it is compelled to change that state by forces impressed thereon.”
‘state of moving uniformly forward in straight line’: is changed by impressed force.
‘state of uniform velocity in forward in straight line’ : is changed by impressed force.
velocity : is changed by impressed force.
Thus Newton’s first law of motion relates impressed force with velocity.
Thus according to definition of Absolute motion, definition of impressed force,
definition of first law of motion , comparison of Accelerative and Motive forces and current
definitions of motion and rest ; the impressed force is related with velocity or speed. The force depends
upon velocity. In the statement of second law of motion and subsequent explanation Newton meant
motion as velocity or speed.
3.0 In the Principia F=ma was not derived
C. Truesdell [ 5 ] has clearly stated that F=ma is not derived by Newton in the Principia in any edition.
This equation was derived for first time by Leonhard Euler [ 7,8 ] in 1750. According to Stanford
Encyclopedia of Philosophy [6] — ‘The modern F=ma form of Newton's second law nowhere occurs
in any edition of the Principia . Further the obvious question with the second law is what Newton means
by “a change in motion.” If he had meant a change in what we call momentum — that is, if he had
meant, in modern notation, Δmv — the proper phrasing would have been “a change in the quantity of
motion.”
But in the definition of Newton’s second law of motion, as given in the Principia, the
phrasing is ‘change in motion’. Newton persisted with this phrase in all three editions ( 1687, 1713 and
1726) . Newton wrote ‘change in motion’ means ‘change in velocity i.e.(v-u)’. Thus the law means, F
α (v-u) or F = k(v-u). In the interpretation of such a significant law then all the terms must be logically
and scientifically considered, not by speculations and arbitrary permutations and combinations.
Further according to The Stanford Encyclopedia of Philosophy [6] – “If this
(geometrical) way of interpreting the second law seems perverse, keep in mind that the geometric
mathematics Newton used in the Principia — and others were using before him — had no way of
representing acceleration as a quantity in its own right. Newton, of course, could have conceptualized
acceleration as the second derivative of distance with respect to time within the framework of the
symbolic calculus. This indeed is the form in which Jacob Hermann presented the second law in
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his Phoronomia of 1716 (and Euler in the 1740s). But the geometric mathematics used in
the Principia offered no way of representing second derivatives.”
Newton started the beginning of the laying of foundations of Physics by defining
the basic concepts . The basic terms such as mass, inertia , force , rest, motion , gravity, centripetal force
etc. were defined in the Principia. However the, physical quantities, dimensions, units etc. were defined
afterwards. These terms were gradually developed by others later on, and the process is still on. In this
sense Newton was much ahead of time, as he perceived origin of physics defining basic terms.
During that period the physical phenomena were largely interpreted or understood the basis of
definitions, axioms, concepts and laws etc , as it was just beginning of establishing the laws. It will not
be incorrect to conclude that at that time quantitative measurements may not be stringent requirement of
the interpretation, as it was just beginning of the developments of the concepts. It is just possible the
formation of equations and interpretations, was next phase of development of concepts. It is justified by
the facts that physical quantities, units and dimensions were defined much after and then significantly
quantitative measurements started. The mathematical equation based on second law was not given by
Newton; later on others interpreted the laws as suited to them. However scientific logic must always
prevail.
Also at that time there was no mode of communication that newly formulated law, is
globally circulated for interactions as we have journals, internet now. In this way laws are quickly
critically analyzed and further improved. Newton and Leibniz shared perceptions on the calculus by
sending communication by boats/ships. It may be regarded as one reason that Newton did not derive
mathematical equation, as we do in now days.
The dimensional analysis was started by Fourier in 1822 [9,10 ], and
dimensions of force are based upon F =ma. However equation for the Principia’s second law of motion
is F =k(v-u). Also the unit of force dyne [11] was initially defined in 1861 about 184 years after
publication of the first edition of the Principia. But this definition was unacceptable to Committee of the
British Association for the advancement of Science [12 ]. The 9th Conférence Générale des Poids et
Mesures held in 1948, then adopted the name "newton" for unit of force in resolution 7[13]. The
concept of force was initially defined in 1687, so it took considerable time for developments of law and
process is still on. In case concepts of units, dimensions and related mathematical methods were
available at Newton’s time, he would have given equation for the Principia’s second law of motion.
Also Principia does not express anything about acceleration and ‘rate of change’ [3,6]
4.0 F =k(v-u) is equation for second law as given in the Principia
The second law of motion is clear statement, hence mathematical equation should be directly
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based upon it, thus no possibility of speculations and extrapolations.
From the definition Newton’s second law of motion we have,
alteration in motion α
F α (v-u)
impressed force
or F = k (v-u)
[2]
where k is coefficient of proportionality which has dimensions MT -1. Like numerous other coefficients of
proportionality, the value of k also depends upon experimental conditions.
d 2s
5.0 Euler derived F=ma = m
for first time in 1750.
dt 2
(i) In 14th century French philosopher Jean Buridan (1300 – 1358) propagated Impetus Theory [14] and
defined equation similar to momentum.
Impetus = weight (mass) × velocity
(3)
(ii) In 1716, Jacob Hermann published a Latin text called “Phoronomia”, meaning the science of motion
[15]. He stated a equation dc =pdt, where c stands for “celeritas” meaning speed, and p stands for
“potentia”, meaning force.
or
p=
dc
= or Force
dt
(5).
Herman called it force (potentia), it is believed that this equation is stands for F =ma , but is not so.
However in today’s notation it is acceleration. Newton, even in the Principia used “celeritas” as speed.
The current definition of force was given by Leonhard Euler mathematically as F =ma . It relates force
with mass and acceleration.
(iii) Further momentum was defined by J Jenning's in Miscellanea [16] in Latin in 1721 as
M = Q× V
or p =mv
(6)
where Q is quantity of matter and V is velocity, and M is momentum . In the next definition the
velocity or speed is defined by dividing distance with time i.e. S/t. These developments took place after
second edition (1713) and before third edition (1726) of the Principia. But Newton preferred not to
mention them at all and kept original Second Laws Of Motion in third edition , as in first edition (1687)
of the Principia. The modern F=ma form of Newton's second law is not stated in any edition of
the Principia [6], and even second derivatives were not represented.
(iv) Swiss mathematician Leonhard Euler
Leonhard Euler used extrinsic references frames (a system of three orthogonal Cartesian axes)
and formulated equation for motion as below [7-8]:
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2Mddx=Pdt2, 2Mddy=Qdt2, 2Mddz=Rdt2,
or P = 2 M
d 2x
d2y
d 2z
,
Q
=
,
R
=
2
M
2
M
dt 2
dt 2
dt 2
where M is the mass and P, Q, and R the components of the force on the
axis (the coefficient 2 depended on the unit of measure). If the unit of measure is such that
coefficient 2 is one then result is
F=m
d 2s
=
dt 2
ma
(7)
The eq.(7 ) is called equation for second law of motion, however it never corresponds to the
Principia’s definition of the second law of motion. This relates force with acceleration, regarded as
complete as relates mass with force. So it was readily accepted by scientific community, but it is not
equation of force for second law of motion given by Newton in the Principia. Further it is expressed in
differential forms so easily to solve usual calculus methods.
6.0 Euler’s and Newton’s equations and definitions.
Euler’s equation (F=ma) and definition
Euler derived equation F=ma mathematically in independent derivation [17 ]. It corresponds to definition
‘The rate of change of momentum is proportional to the motive force impressed and takes place along
straight line in which force is impressed.’
In Principia’s form of second law of motion, arbitrarily the word ‘motion’ was replaced by momentum
(units and dimensions are different), and word ‘alteration’ was changed by ‘rate of change’. But these
are unscientific speculations, extrapolations and replacements. Thus definition of the Principia’s second
law of motion changes. But this altered definition also leads to Euler’s mathematical equation F=ma.
Further the modification is only done if the law has some limitations, and after modification it solves the
purpose. Here it is difficult to understand why definition of second law of motion is altered , and what
was achieved in altering the definition of the Principia’s law of motion. Thus none of the questions
have answer.
However Newton did not change the definition of the second law of motion as
given in the Principia. Apparently this arbitrary alteration occurred after 1750 when Euler gave F=ma. It
is not consistent in anyway. Thus second law of motion as we study now is contributed by Euler, not by
Newton. Newton’s second law of motion is in the Principia at page 19.
Newton’s definition and equation, F =k(v-u)
Newton’s second law of motion as given in the Principia is
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‘The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction
of the right line in which that force is impress'd.’
The second law of motion is clear statement, hence mathematical equation should be directly based upon
it, there is no possibility of speculations and extrapolations.
Thus mathematical equation for this is
F α (v-u)
or F = k (v-u)
(2)
where k is coefficient of proportionality which has dimensions MT -1. Like numerous other coefficients of
proportionality, the value of k also depends upon experimental conditions. Further eq.(2) is expressed as
F =k
ds
dt
(8)
The eq.(1) is useful in explaining that second law reduces to first law if no external force acts on body
( F= 0, v=u ). Any law must not be associated with Newton (or any scientist) which is not given by
Newton (or any scientist). Thus there are two laws of motion i.e. the Principia’s second law of motion
F = k(v-u) , and Euler’s law of motion or Newton’s arbitrarily altered law of motion i.e. F =ma. The
scientific logic must prevail everywhere in understanding the phenomena and doctrines be transparently
discussed for development of science.
Acknowledgements
Author is highly indebted to Dr R Stanley , Dr Steve Crothers and Anjana Sharma for critical
discussions and encouragements at various stages of the work.
References
[1] Newton , Isaac Mathematical Principles of Natural Philosophy pp.19-20 , London, 1727 , translated
by Andrew Motte from the Latin.
[2] http://books.google.co.in/books?
id=Tm0FAAAAQAAJ&pg=PA1&redir_esc=y#v=onepage&q&f=false
[3] B Pourciau , ‘ Is Newton’s second law really Newton’s ? American Journal of Physics , 79 1015,
2011
[4] B. Pourciau, “Newton’s interpretation of Newton’s second law of motion.
law,” Arch. Hist. Exact Sci. 60, 157–207 (2006).
[5] C. Truesdell, Arch. History. Exact. Sci 1, 3 (1960),
[6]
http://plato.stanford.edu/entries/newton-principia/#ThrEdiPri
[7] Euler, Leonhard Recherches sur le mouvement des corps célestes en général, Mémoires de
l’académie des sciences de Berlin 3, 93-143 or Opera, ser. 2, vol. 25, 1 – 44 (1747).
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[8] Euler, Leonhard (1750) “Découverte d’un nouveau principe de Mécanique”, Mém., 6, 185217 or Opera, ser. 2, vol. 5, 81 – 108, (1750).
[9] J. Fourier. The analytical theory of heat. Cambridge University Press, 1822.
[10] R. Martins, The origin of dimensional analysis. J. Franklin Institute, 331, 331 (1981)
[11] Rossiter, William (1879). Dictionary of Scientific Terms. London and Glasgow: William Collins,
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[12] Everett, ed. "First Report of the Committee for the Selection and Nomenclature of Dynamical and
Electrical Units". Forty-third Meeting of the British Association for the Advancement of Science.
Bradford: Johna Murray. p. 223. Retrieved 8 April 2012
[13] International Bureau of Weights and Measures (1977), The international system of units (330–331)
(3rd ed.), U.S. Dept. of Commerce, National Bureau of Standards, p. 17
[14 ] Pedersen, Olaf (1993-03-26). Early physics and astronomy: a historical introduction. CUP Archive.
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[15] Herman Jacob , Phoronomia, sive De viribus et motibus corporum solidorum et fluidorum libri duo
, Amstelædami, apud R. & G. Wetstenios, h. ff., 1716
[16] Jennings, John (1721). Miscellanea in Usum Juventutis Academicae. Northampton: R. Aikes & G.
Dicey. p. 67.
[17] Rao, Anil Vithala Dynamics of particles and rigid bodies Cambridge University Press. p. 355
( 2006)