View source for History of centrifugal and centripetal forces

In [[physics]], the '''history of centrifugal and centripetal forces''' illustrates a long and complex evolution of thought about the nature of [[force]]s, [[Principle of relativity|relativity]], and the nature of [[physical law]]s.

==Huygens, Leibniz, Newton, and Hooke==

Early scientific ideas about [[centrifugal force]] were based upon [[intuition (knowledge)|intuitive perception]], and [[circular motion]] was considered somehow more "natural" than [[straight-line motion]]. According to Domenico Bertoloni-Meli:
:For [[Christiaan Huygens|Huygens]] and [[Isaac Newton|Newton]] centrifugal force was the result of a curvilinear motion of a body; hence it was located in nature, in the object of investigation. According to a more recent formulation of classical mechanics, centrifugal force depends on the choice of how phenomena can be conveniently represented. Hence it is not located in nature, but is the result of a choice by the observer. In the first case a mathematical formulation mirrors centrifugal force; in the second it creates it.<ref>
{{cite journal
 | title = The Relativization of Centrifugal Force
 | author = Domenico Bertoloni Meli
 | journal = Isis
 | volume = 81
 | issue = 1
 | date = March 1990
 | pages = 23–43
 | publisher = The University of Chicago Press on behalf of The History of Science Society
 | ref =Reference-idMeli1990
 | jstor = 234081
 | doi = 10.1086/355247
 | s2cid = 144526407
 }}</ref>

Christiaan Huygens coined the term "centrifugal force" in his 1659 ''De Vi Centrifuga''<ref>
{{cite web
 |url=http://www.bun.kyoto-u.ac.jp/~suchii/inertia.html
 |title=Inertia
 |author=Soshichi Uchii
 |date=October 9, 2001
 |accessdate=2008-05-25
 }}</ref> and wrote of it in his 1673 ''Horologium Oscillatorium'' on [[pendulum]]s. In 1676–77, Isaac Newton combined [[Kepler's laws of planetary motion]] with Huygens' ideas and found<blockquote>the proposition that by a centrifugal force reciprocally as the square of the distance a planet must revolve in an ellipsis about the center of the force placed in the lower umbilicus of the ellipsis, and with a radius drawn to that center, describe areas proportional to the times.<ref>{{Cite web|title=Anni Mirabiles|url=https://www.laphamsquarterly.org/revolutions/anni-mirabiles|access-date=2020-08-27|website=Lapham's Quarterly|language=en}}</ref></blockquote>Newton coined the term "[[centripetal force]]" (''vis centripeta'') in his discussions of [[gravity]] in his ''[[De motu corporum in gyrum]]'', a 1684 manuscript which he sent to [[Edmond Halley]].<ref>
{{cite book
 | url = https://books.google.com/books?id=lIZ0v23iqRgC&q=centrifugal+centripetal+newton+huygens+de-motu-corporum+horologium-oscillatorium&pg=PA5 
 | title = The Mathematical Papers of Isaac Newton
 | volume = VI
 | publisher = Cambridge: University Press
 | isbn = 978-0-521-04585-8
 | year = 2008
 }}</ref>

[[Gottfried Leibniz]] as part of his "[[solar vortex theory]]" conceived of centrifugal force as a real outward force which is induced by the circulation of the body upon which the force acts.  An inverse cube law centrifugal force appears in an equation representing planetary [[orbit]]s, including non-circular ones, as Leibniz described in his 1689 ''Tentamen de motuum coelestium causis''.<ref>
{{cite book
 | title = Revolutions in Mathematics
 | author = Donald Gillies
 | publisher = Oxford: University Press
 | year = 1995
 | isbn = 978-0-19-851486-2
 | page = 130
 | url = https://books.google.com/books?id=HANUEkUdULcC&q=Leibniz+centrifugal+circular+%22tentamen+de+motuum+coelestium+causis%22&pg=PA130  
 }}</ref> Leibniz's equation is still used today to solve planetary orbital problems, although his solar vortex theory is no longer used as its basis.<ref name=goldstein>
{{cite book
 | title = Classical mechanics
 | edition = 2nd
 | author = Herbert Goldstein
 | publisher = Addison-Wesley
 | year = 1980
 | isbn = 978-0-201-02918-5
 | page = 74
 | url = https://books.google.com/books?id=9M8QAQAAIAAJ&q=intitle:classical+intitle:mechanics+inauthor:goldstein+3-12
 }}</ref>

Leibniz produced an equation for planetary orbits in which the centrifugal force appeared as an outward inverse cube law force in the radial direction:<ref name="Linton">
{{cite book
 | title = From Eudoxus to Einstein: a history of mathematical astronomy
 | author = Christopher M. Linton
 | publisher = Cambridge University Press
 | year = 2004
 | isbn = 978-0-521-82750-8
 | pages = 264–285
 | url = https://books.google.com/books?id=B4br4XJFj0MC&q=Leibnitz+on+centrifugal+force&pg=PA285
 }}</ref>

:<math> \ddot r = -k/r^{2} + l^{2}/r^{3}</math>.

Newton himself appears to have previously supported an approach similar to that of Leibniz.<ref name=Swetz/> Later, Newton in his ''Principia'' crucially limited the description of the dynamics of planetary motion to a frame of reference in which the point of attraction is fixed. In this description, Leibniz's centrifugal force was not needed and was replaced by only continually inward forces toward the fixed point.<ref name=Linton/> Newton objected to Leibniz's equation on the grounds that it allowed for the centrifugal force to have a different value from the centripetal force, arguing on the basis of his [[Newton's third law|third law of motion]], that the centrifugal force and the centripetal force must constitute an equal and opposite action-reaction pair. In this however, Newton was mistaken, as the [[reactive centrifugal force]] which is required by the third law of motion is a completely separate concept from the centrifugal force of Leibniz's equation.<ref name=Swetz>
{{cite book
 | url = https://books.google.com/books?id=gqGLoh-WYrEC&q=reaction+fictitious+rotating+frame+%22centrifugal+force%22&pg=PA269
 | title = Learn from the masters!
 | author = Frank Swetz
 | publisher = MAA
 | year = 1997
 | pages = 268–269
 | isbn = 978-0-88385-703-8
 }}</ref><ref name="xnewton">
{{cite web
 |url=http://www.phy.hr/~dpaar/fizicari/xnewton.html
 |title=Newton, Sir Isaac
 |accessdate=2008-05-25
 }}</ref>

Huygens, who was, along with Leibniz, a neo-Cartesian and critic of Newton, concluded after a long correspondence that Leibniz's writings on celestial mechanics made no sense, and that his invocation of a harmonic vortex was logically redundant, because Leibniz's radial equation of motion follows trivially from Newton's laws.  Even the most ardent modern defenders of the cogency of Leibniz's ideas acknowledge that his harmonic vortex as the basis of centrifugal force was dynamically superfluous.<ref>A. R. Hall, Philosophers at War, 2002, pp 150-151</ref>

It has been suggested that the idea of circular motion as caused by a single force was introduced to Newton by [[Robert Hooke]].<ref name="xnewton"/>

Newton described the role of centrifugal force upon the height of the oceans near the equator in the ''[https://archive.org/stream/newtonspmathema00newtrich/newtonspmathema00newtrich_djvu.txt Principia]'':
{{Blockquote
 |Since the centrifugal force of the parts of the earth, arising from the earth's diurnal motion, which is to the force of gravity as 1 to 289, raises the waters under the equator to a height exceeding that under the poles by 85472 Paris feet, as above, in Prop. XIX., the force of the sun, which we have now shewed to be to the force of gravity as 1 to 12868200, and therefore is to that centrifugal force as 289 to 12868200, or as 1 to 44527, will be able to raise the waters in the places directly under and directly opposed to the sun to a height exceeding that in the places which are 90 degrees removed from the sun only by one Paris foot and 113 V inches; for this measure is to the measure of 85472 feet as 1 to 44527.
 |Newton: ''Principia'' Corollary to Book II, Proposition XXXVI. Problem XVII
 }}

The effect of centrifugal force in countering gravity, as in this behavior of the tides, has led centrifugal force sometimes to be called "false gravity" or "imitation gravity" or "quasi-gravity".<ref name=Volovik>
{{cite book
 |url=https://books.google.com/books?id=-tyXuduShHUC&q=centrifugal+quasi-gravity&pg=PA200
 |page=200
 |title=Artificial black holes
 |author=M. Novello, Matt Visser & G. E. Volovik
 |isbn=981-02-4807-5
 |publisher=World Scientific
 |year=2002
 }}</ref>

==Eighteenth century==

It wasn't until the latter half of the 18th century that the modern "[[fictitious force]]" understanding of the centrifugal force as a pseudo-force artifact of rotating reference frames took shape.<ref>
{{cite journal
 | title = Newton's Orbit Problem: A Historian's Response
 | journal = The College Mathematics Journal
 | volume = 25
 | issue = 3
 | pages = 193–200
 | issn = 0746-8342
 | doi = 10.2307/2687647
 | year = 1994
 | author = Wilson
 | publisher = Mathematical Association of America
 | jstor = 2687647
 }}</ref> In a 1746 [[mémoire|memoir]] by [[Daniel Bernoulli]], "the idea that the centrifugal force is fictitious emerges unmistakably."<ref name=Meli>[[#Reference-idMeli1990|Meli 1990]], "The Relativization of Centrifugal Force".</ref> Bernoulli, in seeking to describe the motion of an object relative to an arbitrary point, showed that the magnitude of the centrifugal force depended on which arbitrary point was chosen to measure circular motion about. Later in the 18th century [[Joseph Louis Lagrange]] in his ''Mécanique Analytique'' explicitly stated that the centrifugal force depends on the rotation of a system of [[perpendicular]] [[axis of rotation|axes]].<ref name=Meli/> In 1835, [[Gaspard-Gustave Coriolis]] analyzed arbitrary motion in rotating systems, specifically in relation to waterwheels. He coined the phrase "compound centrifugal force" for a term which bore a similar mathematical expression to that of centrifugal force, albeit that it was multiplied by a factor of two.<ref>
{{cite book
 | title = A History of Mechanics
 | author = René Dugas and J. R. Maddox 
 | publisher = Courier Dover Publications
 | year = 1988
 | isbn = 0-486-65632-2
 | page = 387
 | url = https://books.google.com/books?id=vPT-JubW-7QC&pg=PA374
 }}</ref> The force in question was perpendicular to both the [[velocity]] of an object relative to a rotating frame of reference and the [[axis of rotation]] of the frame. Compound centrifugal force eventually came to be known as the [[Coriolis Force]].<ref>Persson, Anders (July 1998). [http://www.science.unitn.it/~fisica1/fisica1/appunti/mecc/appunti/cinematica/Coriolis_persson.pdf "How Do We Understand the Coriolis Force?"]. ''Bulletin of the American Meteorological Society'' '''79''' (7): pp. 1373–1385. {{ISSN|0003-0007}}.</ref><ref>
{{cite book
 | title = The Fundamental Equations of Dynamics and its Main Coordinate Systems Vectorially Treated and Illustrated from Rigid Dynamics
 | author = Frederick Slate
 | publisher = Berkeley, CA: University of California Press
 | year = 1918
 | page = [https://archive.org/details/fundamentalequat00slat/page/137 137]
 | url = https://archive.org/details/fundamentalequat00slat
 | quote = compound centrifugal force coriolis.
 }}</ref>

==Absolute versus relative rotation==

The idea of centrifugal force is closely related to the notion of [[absolute rotation]]. In 1707 the Irish bishop [[George Berkeley]] took issue with the notion of [[absolute space]], declaring that "motion cannot be understood except in relation to our or some other body". In considering a solitary globe, all forms of motion, uniform and accelerated, are unobservable in an otherwise empty universe.<ref name=Harrison>
{{cite book
 |title=Cosmology |author= Edward Robert Harrison
 |url=https://books.google.com/books?id=-8PJbcA2lLoC&pg=PA237
 |page=237
 |isbn=0-521-66148-X
 |year=2000
 |publisher=Cambridge University Press
 |edition =2nd
 }}</ref> This notion was followed up in modern times by [[Ernst Mach]]. For a single body in an empty universe, motion of any kind is inconceivable. Because rotation does not exist, centrifugal force does not exist. Of course, addition of a speck of matter just to establish a reference frame cannot cause the sudden appearance of centrifugal force, so it must be due to rotation relative to the entire mass of the universe.<ref name=Mach>
{{cite book
 |title=The science of mechanics
 |author=Ernst Mach
 |page=[https://archive.org/details/sciencemechanic01jourgoog/page/n54 33]
 |quote=Try to fix Newton's bucket and rotate the heaven of fixed stars and then prove the absence of centrifugal forces 
 |url= https://archive.org/details/sciencemechanic01jourgoog
 |publisher=The Open Court Publishing Co.
 |year=1915
 |isbn=0-87548-202-3
 }}</ref> The modern view is that centrifugal force is indeed an indicator of rotation, but relative to those frames of reference that exhibit the simplest laws of physics.<ref name=Carlo>
{{cite book
 |title=Einstein and Aquinas 
 |page=27
 |author= J. F. Kiley, W. E. Carlo
 |chapter-url=https://books.google.com/books?id=Icyclh_-suUC&pg=PA27
 |chapter=The epistemology of Albert Einstein
 |isbn=90-247-0081-7
 |year=1970
 |publisher=Springer
 }}</ref> Thus, for example, if we wonder how rapidly our galaxy is rotating, we can make a model of the galaxy in which its rotation plays a part. The rate of rotation in this model that makes the observations of (for example) the flatness of the galaxy agree best with physical laws as we know them is the best estimate of the rate of rotation<ref name=Genz>{{cite book
 |title=Nothingness
 |author=Henning Genz
 |page=275
 |url=https://books.google.com/books?id=Cn_Q9wbDOM0C&q=%22rotation+of+the+universe%22&pg=PA274
 |isbn=0-7382-0610-5
 |year=2001
 |publisher=Da Capo Press
 }}{{Dead link|date=October 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> (assuming other observations are in agreement with this assessment, such as isotropy of the [[cosmic background radiation|background radiation of the universe]]).<ref name="Thompson">
{{cite book|author=J. Garcio-Bellido|chapter-url=https://books.google.com/books?id=3TrsMTmbr-sC&q=CMB+%22rotation+of+the+universe%22&pg=PA32|title=Advances in Astronomy|publisher=Imperial College Press|year=2005|isbn=1-86094-577-5|editor=J. M. T. Thompson|page=32, §9|chapter=The Paradigm of Inflation}}</ref>

==Role in developing the idea of inertial frames and relativity==

{{Main article|Inertial frame of reference}}

In the [[Bucket argument|rotating bucket]] experiment, Newton observed the shape of the surface of water in a bucket as the bucket was spun on a rope. At first the water is flat, then, as it acquires the same rotation as the bucket, it becomes parabolic.  Newton took this change as evidence that one could detect rotation relative to "absolute space" experimentally, in this instance by looking at the shape of the surface of the water.

Later scientists pointed out (as did Newton) that the laws of mechanics were the same for all observers that differed only by uniform translation; that is, all observers that differed in motion only by a constant velocity. Hence, "absolute space" was not preferred, but only one of a set of frames related by [[Galilean transformation]]s.<ref name=Pippard>
{{cite book  |url= https://books.google.com/books?id=wOQ6je9knG8C&q=intitle:Twentieth+intitle:Century+intitle:Physics 
 |title=Twentieth Century Physics
 |author=Laurie M. Brown, Abraham Pais & A. B. Pippard
 |isbn=0-7503-0310-7 |publisher=CRC Press
 |year=1995
 |pages=256–257}}</ref>
{{Blockquote
 |By the end of the nineteenth century, some physicists had concluded that the concept of absolute space is not really needed...they used the law of inertia to define the entire class of inertial frames. Purged of the concept of absolute space, Newton's laws do single out the class of inertial frames of reference, but assert their complete equality for the description of all mechanical phenomena.
 | Laurie M. Brown, Abraham Pais, A. B. Pippard: ''Twentieth Century Physics'', pp. 256-257
 }}

Ultimately this notion of the transformation properties of physical laws between frames played a more and more central role.<ref name=symmetries>
The idea of transformation properties of physical laws under various transformations is a central topic in modern physics, related to such basic concepts as [[Conservation law (physics)|conservation law]]s like conservation of energy and momentum through [[Noether's theorem]]. See, for example, 
{{cite book
 |author=Harvey R. Brown
 |title=Physical Relativity
 |page= 180
 |url= https://books.google.com/books?id=T6IVyWiPQksC&q=symmetry++conservation+Noether&pg=PA180
 |isbn=0-19-927583-1
 |year=2005
 |publisher=Oxford University Press 
 }}, and 
{{cite book
 |title=Einstein Studies in Russia
 | author = Gennady Gorelik
 | authorlink = Gennady Gorelik
 |editor=Yuri Balashov |editor2=Vladimir Pavlovich Vizgin
 | page=''The problem of conservation laws and the Poincare quasigroup in general relativity''; pp. 17 ''ff'' 
 |url= https://books.google.com/books?id=9vs4rrgLYw8C&q=symmetry++conservation+Noether&pg=PA26 
 |publisher=Birkhäuser
 |year=2002
 |isbn=0-8176-4263-3
 }} and 
{{cite book
 |title=Laws of Nature
 |author=Peter Mittelstaedt & Paul Weingartner 
 |url= https://books.google.com/books?id=w0ttLgyKqBsC&q=spacetime+symmetries+%22three+conservation+principles%22&pg=PA80 
 |page=80
 |isbn=3-540-24079-9
 |year=2005
 |publisher=Springer
 }}</ref>  
It was noted that accelerating frames exhibited "fictitious forces" like the centrifugal force. These forces did not behave under transformation like other forces, providing a means of distinguishing them. This peculiarity of these forces led to the names ''inertial forces'', ''pseudo-forces'' or ''fictitious forces''. In particular, fictitious forces ''did not appear at all'' in some frames: those frames differing from that of the fixed stars by only a constant velocity. In short, a frame tied to the "fixed stars" is merely a member of the class of "inertial frames", and absolute space is an unnecessary and logically untenable concept. The preferred, or "inertial frames", were identifiable by the ''absence of fictitious forces''.<ref name=Rothman2>
{{cite book
 |title=Discovering the Natural Laws: The Experimental Basis of Physics
 |author= Milton A. Rothman
 |page=[https://archive.org/details/discoveringnatur0000roth/page/23 23]
 |url= https://archive.org/details/discoveringnatur0000roth
 |url-access=registration
 |quote=reference laws of physics.
 |isbn=0-486-26178-6
 |publisher=Courier Dover Publications
 |year=1989 
 }}</ref><ref name=Borowitz>
{{cite book 
 |title=A Contemporary View of Elementary Physics
 |page=138
 |publisher=McGraw-Hill
 |year=1968
 |url= https://books.google.com/books?as_q=&num=10&btnG=Google+Search&as_epq=The+effect+of+his+being+in+the+noninertial+frame+is+to+require+the+observer+to&as_oq=&as_eq=&as_brr=0&lr=&as_vt=&as_auth=&as_pub=&as_sub=&as_drrb=c&as_miny=&as_maxy=&as_isbn= 
 |asin= B000GQB02A
 |author=Sidney Borowitz & Lawrence A. Bornstein
 }}</ref><ref name=Arnold2>
{{cite book
 |title=Mathematical Methods of Classical Mechanics
 |page=129
 |author=V. I. Arnol'd
 |isbn=978-0-387-96890-2
 |year=1989
 |url= https://books.google.com/books?as_q=&num=10&btnG=Google+Search&as_epq=additional+terms+called+inertial+forces.+This+allows+us+to+detect+experimentally&as_oq=&as_eq=&as_brr=0&lr=&as_vt=&as_auth=&as_pub=&as_sub=&as_drrb=c&as_miny=&as_maxy=&as_isbn=
 |publisher=Springer}}
</ref>
{{Blockquote
 |The effect of his being in the noninertial frame is to require the observer to introduce a fictitious force into his calculations….|Sidney Borowitz and Lawrence A Bornstein in ''A Contemporary View of Elementary Physics'', p. 138}}  
{{Blockquote|The equations of motion in a non-inertial system differ from the equations in an inertial system by additional terms called inertial forces. This allows us to detect experimentally the non-inertial nature of a system.
 |V. I. Arnol'd: ''Mathematical Methods of Classical Mechanics'' Second Edition, p. 129}}

The idea of an inertial frame was extended further in the [[special theory of relativity]]. This theory posited that ''all'' physical laws should appear of the same form in inertial frames, not just the laws of mechanics. In particular, [[Maxwell's equations]] should apply in all frames. Because Maxwell's equations implied the same speed of light in the vacuum of [[free space]] for all inertial frames, inertial frames now were found to be related not by Galilean transformations, but by [[Poincaré transformation]]s, of which a subset is the [[Lorentz transformation]]s. That posit led to many ramifications, including [[Lorentz contraction]]s and [[relativity of simultaneity]]. Einstein succeeded, through many clever [[thought experiments]], in showing that these apparently odd ramifications in fact had very natural explanation upon looking at just how measurements and clocks actually were used. That is, these ideas flowed from [[operational definition]]s of measurement coupled with the experimental confirmation of the constancy of the [[speed of light]].

Later the general theory of relativity further generalized the idea of frame independence of the laws of physics, and abolished the special position of inertial frames, at the cost of introducing [[curved space-time]]. Following an analogy with centrifugal force (sometimes called "artificial gravity" or "false gravity"), gravity itself became a fictitious force,<ref name=VonBaeyer>
{{cite book
 |title=The Fermi Solution: Essays on science
 |isbn=0-486-41707-7
 |year=2001
 |publisher=Courier Dover Publications
 |edition=Reprint of 1993 
 |url= https://books.google.com/books?id=VhJr9Qx8ohsC&q=gravity+like+%22centrifugal+force%22&pg=PA78 
 |page=78
 |author=Hans Christian Von Baeyer
 }}</ref> as enunciated in the [[equivalence principle]].<ref name=Giancoli>
{{cite book
 |title=Physics for Scientists and Engineers with Modern Physics
 |author=Douglas C. Giancoli 
 |url= https://books.google.com/books?id=xz-UEdtRmzkC&q=%22principle+of+equivalence%22&pg=PA155
 |page=155
 |year=2007
 |publisher=Pearson Prentice Hall
 |isbn=978-0-13-149508-1 }}</ref>
{{Blockquote
 |''The principle of equivalence: ''There is no experiment observers can perform to distinguish whether an acceleration arises because of a gravitational force or because their reference frame is accelerating
 |Douglas C. Giancoli ''Physics for Scientists and Engineers with Modern Physics'', p. 155
 }}

In short, centrifugal force played a key early role in establishing the set of inertial frames of reference and the significance of fictitious forces, even aiding in the development of general relativity.

==The modern conception==

The modern interpretation is that [[Centrifugal force (rotating reference frame)|centrifugal force in a rotating reference frame]] is a pseudo-force that appears in equations of motion in [[rotating reference frame|rotating frames of reference]], to explain effects of [[inertia]] as seen in such frames.<ref>
{{cite book
 | title = Four Lectures on Relativity and Space
 | author = Charles Proteus Steinmetz
 | publisher = Kessinger Publishing
 | year = 2005
 | isbn = 1-4179-2530-2
 | page = 49
 | url = https://books.google.com/books?id=69v4uH5xBEMC&q=centrifugal+force+inertia&pg=PA49
 }}</ref>

Leibniz's centrifugal force may be understood as an application of this conception, as a result of his viewing the motion of a planet along the radius vector, that is, from the standpoint of a special reference frame rotating with the planet.<ref name=Linton/><ref name=Swetz/><ref>
{{cite journal
 | title = The celestial mechanics of Leibniz in the light of Newtonian criticism
 | journal = Annals of Science
 | author = E. J. Aiton
 | publisher = Taylor & Francis
 | volume = 18
 | issue = 1
 | date = 1 March 1962
 | pages = 31–41
 | url = 
 | doi = 10.1080/00033796200202682
 }}</ref> Leibniz introduced the notions of ''vis viva'' (kinetic energy)<ref name=Russell>
{{cite book
 |title=A Critical Exposition of the Philosophy of Leibniz
 |author= Bertrand Russell
 |url=https://books.google.com/books?id=R7GauFXXedwC&pg=PA96
 |page=96
 |isbn=0-415-08296-X
 |year=1992
 |edition=Reprint of 1937 2nd
 |publisher=Routledge
 }}</ref> and ''action'',<ref name="Lefèvre">
{{cite book
 |title=Between Leibniz, Newton, and Kant
 |author=Wolfgang Lefèvre
 |url=https://books.google.com/books?id=BbIV21eN4jEC&pg=PA39
 |page=39
 |isbn=0-7923-7198-4
 |year=2001
 |publisher=Springer
 }}</ref> which eventually found full expression in the [[Lagrangian mechanics|Lagrangian formulation of mechanics]]. In deriving Leibniz's radial equation from the Lagrangian standpoint, a rotating reference frame is not used explicitly, but the result is equivalent to that found using Newtonian vector mechanics in a co-rotating reference frame.<ref name=goldstein2002>
{{cite book
 | title =  Classical Mechanics 
 | author = Herbert Goldstein
 | publisher = San Francisco : Addison Wesley
 | year = 2002
 | isbn = 0-201-31611-0
 | pages = 74–77, 176
 }}</ref><ref>
{{cite book
 | title = Classical Mechanics
 | author = John Taylor
 | publisher = University Science Books
 | year = 2005
 | isbn = 1-891389-22-X
 | pages = 358–359
 | url = https://books.google.com/books?id=P1kCtNr-pJsC&q=centrifugal-force%20co-rotating-frame%20inauthor%3ATaylor&pg=PA358
 }}</ref><ref>
{{cite journal
| last        = Whiting
| first       = J.S.S. 
|date=November 1983
| title       = Motion in a central-force field
| journal     = Physics Education
| volume      = 18
| issue       = 6
| pages       = 256&ndash;257
| issn        = 0031-9120
| url         = http://www.iop.org/EJ/article/0031-9120/18/6/102/pev18i6p256.pdf
| accessdate  = May 7, 2009
| doi        = 10.1088/0031-9120/18/6/102
|bibcode = 1983PhyEd..18..256W | s2cid = 250750602 
}}</ref>

==References==
{{Reflist|3}}

{{DEFAULTSORT:History Of The Concepts Of Centrifugal And Centripetal Forces}}
[[Category:Rotation]]
[[Category:Force]]