Editing Newton's laws of motion (section)

=== After the ''Principia'' ===
[[File:Page_157_from_Mechanism_of_the_Heaven,_Mary_Somerville_1831.png|thumb|right|Page 157 from ''Mechanism of the Heavens'' (1831), [[Mary Somerville]]'s expanded version of the first two volumes of Laplace's ''Traité de mécanique céleste.''<ref>{{Cite journal |last=Patterson |first=Elizabeth C. |date=December 1969 |title=Mary Somerville |url=https://www.cambridge.org/core/product/identifier/S0007087400010232/type/journal_article |journal=[[The British Journal for the History of Science]] |language=en |volume=4 |issue=4 |pages=311–339 |doi=10.1017/S0007087400010232 |s2cid=246612625 |issn=0007-0874 |quote=In no sense was it a mere translation of Laplace's work. Instead it endeavoured to explain his method ". . . by which these results were deduced from one general equation of the motion of matter" and to bring the reader's mathematical skill to the point where the exposition of Laplace's mathematics and ideas would be meaningful—then to give a digest in English of his great work. Diagrams were added when necessary to the original text and proofs of various problems in physical mechanics and astronomy included. ... [F]or almost a hundred years after its appearance the book continued to serve as a textbook for higher mathematics and astronomy in English schools.}}</ref> Here, Somerville deduces the inverse-square law of gravity from [[Kepler's laws of planetary motion]].]]
Newton expressed his second law by saying that the force on a body is proportional to its change of motion, or momentum. By the time he wrote the ''Principia,'' he had already developed calculus (which he called "[[Fluxion|the science of fluxions]]"), but in the ''Principia'' he made no explicit use of it, perhaps because he believed geometrical arguments in the tradition of [[Euclid]] to be more rigorous.<ref>{{Cite book |last=Baron |first=Margaret E. |url=https://www.worldcat.org/oclc/892067655 |title=The Origins of Infinitesimal Calculus |publisher=Pergamon Press |date=1969 |isbn=978-1-483-28092-9 |edition=1st |location=Oxford |oclc=892067655 |author-link=Margaret Baron}}</ref>{{Rp|page=15}}<ref>{{Cite journal |last=Dunlop |first=Katherine |date=May 2012 |title=The mathematical form of measurement and the argument for Proposition I in Newton's Principia |url=http://link.springer.com/10.1007/s11229-011-9983-8 |journal=[[Synthese]] |language=en |volume=186 |issue=1 |pages=191–229 |doi=10.1007/s11229-011-9983-8 |s2cid=11794836 |issn=0039-7857}}</ref> Consequently, the ''Principia'' does not express acceleration as the second derivative of position, and so it does not give the second law as <math>F = ma</math>. This form of the second law was written (for the special case of constant force) at least as early as 1716, by [[Jakob Hermann]]; [[Leonhard Euler]] would employ it as a basic premise in the 1740s.<ref>{{cite web|url=https://plato.stanford.edu/entries/newton-principia/ |title=Newton's ''Philosophiae Naturalis Principia Mathematica'' |website=[[Stanford Encyclopedia of Philosophy]] |date=2007-12-20 |accessdate=2022-03-06 |first=George |last=Smith}}</ref> Euler pioneered the study of rigid bodies<ref>{{Cite journal |last1=Marquina |first1=J. E. |last2=Marquina |first2=M. L. |last3=Marquina |first3=V. |last4=Hernández-Gómez |first4=J. J. |date=2017-01-01 |title=Leonhard Euler and the mechanics of rigid bodies |url=https://iopscience.iop.org/article/10.1088/0143-0807/38/1/015001 |journal=[[European Journal of Physics]] |volume=38 |issue=1 |pages=015001 |doi=10.1088/0143-0807/38/1/015001 |bibcode=2017EJPh...38a5001M |s2cid=125948408 |issn=0143-0807}}</ref> and established the basic theory of fluid dynamics.<ref>{{Cite book |last=Hesse |first=Mary B. |url=https://www.worldcat.org/oclc/57579169 |title=Forces and Fields: The Concept of Action at a Distance in the History of Physics |date=2005 |publisher=Dover Publications |isbn=978-0-486-44240-2 |edition=Dover reprint |location=Mineola, N.Y. |pages=189 |oclc=57579169 |author-link=Mary Hesse |orig-date=1961}}</ref> [[Pierre-Simon Laplace]]'s five-volume ''[[Traité de mécanique céleste]]'' (1798–1825) forsook geometry and developed mechanics purely through algebraic expressions, while resolving questions that the ''Principia'' had left open, like a full theory of the [[Tide|tides]].<ref>{{cite web|url=https://plato.stanford.edu/entries/newton/ |title=Isaac Newton |website=[[Stanford Encyclopedia of Philosophy]] |first=George |last=Smith |date=2007-12-19 |access-date=2022-03-06 |quote=These advances in our understanding of planetary motion led Laplace to produce the four principal volumes of his ''Traité de mécanique céleste'' from 1799 to 1805, a work collecting in one place all the theoretical and empirical results of the research predicated on Newton's ''Principia''. From that time forward, Newtonian science sprang from Laplace's work, not Newton's.}}</ref>

The concept of energy became a key part of Newtonian mechanics in the post-Newton period. Huygens' solution of the collision of hard spheres showed that in that case, not only is momentum conserved, but kinetic energy is as well (or, rather, a quantity that in retrospect we can identify as one-half the total kinetic energy). The question of what is conserved during all other processes, like inelastic collisions and motion slowed by friction, was not resolved until the 19th century. Debates on this topic overlapped with philosophical disputes between the metaphysical views of Newton and Leibniz, and variants of the term "force" were sometimes used to denote what we would call types of energy. For example, in 1742, [[Émilie du Châtelet]] wrote, "Dead force consists of a simple tendency to motion: such is that of a spring ready to relax; [[Vis viva|living force]] is that which a body has when it is in actual motion." In modern terminology, "dead force" and "living force" correspond to potential energy and kinetic energy respectively.<ref>{{Cite journal |last=Reichenberger |first=Andrea |date=June 2018 |title=Émilie Du Châtelet's interpretation of the laws of motion in the light of 18th century mechanics |url=https://linkinghub.elsevier.com/retrieve/pii/S0039368118300177 |journal=[[Studies in History and Philosophy of Science Part A]] |language=en |volume=69 |pages=1–11 |doi=10.1016/j.shpsa.2018.01.006|pmid=29857796 |bibcode=2018SHPSA..69....1R |s2cid=46923474 }}</ref> Conservation of energy was not established as a universal principle until it was understood that the energy of mechanical work can be dissipated into heat.<ref>{{Cite journal |last=Frontali |first=Clara |date=September 2014 |title=History of physical terms: "energy" |url=https://iopscience.iop.org/article/10.1088/0031-9120/49/5/564 |journal=[[Physics Education]] |volume=49 |issue=5 |pages=564–573 |doi=10.1088/0031-9120/49/5/564 |bibcode=2014PhyEd..49..564F |s2cid=122097990 |issn=0031-9120}}</ref><ref>{{Cite web |last=Gbur |first=Greg |author-link=Greg Gbur |date=2018-12-10 |title=History of the Conservation of Energy: Booms, Blood, and Beer (Part 1) |url=https://skullsinthestars.com/2018/12/10/history-of-the-conservation-of-energy-booms-blood-and-beer-part-1/ |access-date=2022-03-07 |website=Skulls in the Stars |language=en}} {{Cite web |date=2018-12-29 |title=History of the Conservation of Energy: Booms, Blood, and Beer (Part 2) |url=https://skullsinthestars.com/2018/12/28/history-of-the-conservation-of-energy-booms-blood-and-beer-part-2/ |access-date=2022-03-07  |language=en}} {{Cite web |date=2019-08-25 |title=History of the Conservation of Energy: Booms, Blood, and Beer (Part 3) |url=https://skullsinthestars.com/2019/08/24/history-of-the-conservation-of-energy-booms-blood-and-beer-part-3/ |access-date=2022-03-07 |language=en}}</ref> With the concept of energy given a solid grounding, Newton's laws could then be derived within formulations of classical mechanics that put energy first, as in the Lagrangian and Hamiltonian formulations described above.

Modern presentations of Newton's laws use the mathematics of vectors, a topic that was not developed until the late 19th and early 20th centuries. Vector algebra, pioneered by [[Josiah Willard Gibbs]] and [[Oliver Heaviside]], stemmed from and largely supplanted the earlier system of [[Quaternion|quaternions]] invented by [[William Rowan Hamilton]].<ref>{{Cite journal |last1=Silva |first1=Cibelle Celestino |last2=de Andrade Martins |first2=Roberto |date=September 2002 |title=Polar and axial vectors versus quaternions |url=http://aapt.scitation.org/doi/10.1119/1.1475326 |journal=[[American Journal of Physics]] |language=en |volume=70 |issue=9 |pages=958–963 |doi=10.1119/1.1475326 |bibcode=2002AmJPh..70..958S |issn=0002-9505}}</ref><ref>{{cite book|first=Karin |last=Reich |author-link=Karin Reich |chapter=The Emergence of Vector Calculus in Physics: The Early Decades |pages=197–210 |title=Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar |editor-first=Gert |editor-last=Schubring |series=Boston Studies in the Philosophy of Science |volume=187 |publisher=Kluwer |isbn=978-9-048-14758-8 |oclc=799299609 |year=1996}}</ref>