Editing Newton's laws of motion (section)

==Relation to other physical theories==
===Thermodynamics and statistical physics===
[[File:Brownian motion large.gif|thumb|A simulation of a larger, but still microscopic, particle (in yellow) surrounded by a gas of smaller particles, illustrating [[Brownian motion]]]]
In [[statistical physics]], the [[kinetic theory of gases]] applies Newton's laws of motion to large numbers (typically on the order of the [[Avogadro constant|Avogadro number]]) of particles. Kinetic theory can explain, for example, the [[pressure]] that a gas exerts upon the container holding it as the aggregate of many impacts of atoms, each imparting a tiny amount of momentum.<ref name=":5" />{{Rp|page=62}}

The [[Langevin equation]] is a special case of Newton's second law, adapted for the case of describing a small object bombarded stochastically by even smaller ones.<ref name=":4" />{{Rp|page=235}} It can be written<math display="block">m \mathbf{a} = -\gamma \mathbf{v} + \mathbf{\xi} \, </math>where <math>\gamma</math> is a [[drag coefficient]] and <math>\mathbf{\xi}</math> is a force that varies randomly from instant to instant, representing the net effect of collisions with the surrounding particles. This is used to model [[Brownian motion]].<ref>{{Cite journal |last=Mermin |first=N. David |author-link=N. David Mermin |date=August 1961 |title=Two Models of Brownian Motion |url=http://aapt.scitation.org/doi/10.1119/1.1937823 |journal=[[American Journal of Physics]] |language=en |volume=29 |issue=8 |pages=510–517 |doi=10.1119/1.1937823 |bibcode=1961AmJPh..29..510M |issn=0002-9505}}</ref>

===Electromagnetism===
Newton's three laws can be applied to phenomena involving [[electricity]] and [[magnetism]], though subtleties and caveats exist.

[[Coulomb's law]] for the electric force between two stationary, [[electric charge|electrically charged]] bodies has much the same mathematical form as Newton's law of universal gravitation: the force is proportional to the product of the charges, inversely proportional to the square of the distance between them, and directed along the straight line between them. The Coulomb force that a charge <math>q_1</math> exerts upon a charge <math>q_2</math> is equal in magnitude to the force that <math>q_2</math> exerts upon <math>q_1</math>, and it points in the exact opposite direction. Coulomb's law is thus consistent with Newton's third law.<ref>{{Cite journal|last=Kneubil|first=Fabiana B.|date=2016-11-01|title=Breaking Newton's third law: electromagnetic instances|url=https://iopscience.iop.org/article/10.1088/0143-0807/37/6/065201|journal=[[European Journal of Physics]] |volume=37|issue=6|pages=065201|doi=10.1088/0143-0807/37/6/065201|bibcode=2016EJPh...37f5201K |s2cid=126380404 |issn=0143-0807}}</ref>

Electromagnetism treats forces as produced by ''fields'' acting upon charges. The [[Lorentz force law]] provides an expression for the force upon a charged body that can be plugged into Newton's second law in order to calculate its acceleration.<ref>{{Cite book|last=Tonnelat|first=Marie-Antoinette|url=https://www.worldcat.org/oclc/844001|title=The principles of electromagnetic theory and of relativity.|date=1966|publisher=D. Reidel|isbn=90-277-0107-5|location=Dordrecht|oclc=844001|author-link=Marie-Antoinette Tonnelat}}</ref>{{Rp|page=85}} According to the Lorentz force law, a charged body in an electric field experiences a force in the direction of that field, a force proportional to its charge <math>q</math> and to the strength of the electric field. In addition, a ''moving'' charged body in a magnetic field experiences a force that is also proportional to its charge, in a direction perpendicular to both the field and the body's direction of motion. Using the vector [[cross product]],<math display="block">\mathbf{F} = q \mathbf{E} + q \mathbf{v} \times \mathbf{B}.</math>
[[Image:Cyclotron_motion.jpg|right|thumb|The Lorentz force law in effect: electrons are bent into a circular trajectory by a magnetic field.]]If the electric field vanishes (<math>\mathbf{E} = 0</math>), then the force will be perpendicular to the charge's motion, just as in the case of uniform circular motion studied above, and the charge will circle (or more generally move in a [[helix]]) around the magnetic field lines at the [[Cyclotron motion|cyclotron frequency]] <math>\omega = qB/m</math>.<ref name=":4">{{Cite book|last=Reichl|first=Linda E.|url=https://www.worldcat.org/oclc/966177746|title=A Modern Course in Statistical Physics|date=2016|publisher=Wiley-VCH|isbn=978-3-527-69048-0|edition=4th|location=Weinheim, Germany|oclc=966177746|author-link=Linda Reichl}}</ref>{{Rp|page=222}} [[Mass spectrometry]] works by applying electric and/or magnetic fields to moving charges and measuring the resulting acceleration, which by the Lorentz force law yields the [[mass-to-charge ratio]].<ref>{{Cite book |last1=Chu |first1=Caroline S. |chapter=Introduction to Modern Techniques in Mass Spectrometry |date=2010 |chapter-url=https://doi.org/10.1007/978-1-60327-233-9_6 |title=Biomedical Applications of Biophysics |pages=137–154 |editor-last=Jue |editor-first=Thomas |place=Totowa, NJ |publisher=Humana Press |language=en |doi=10.1007/978-1-60327-233-9_6 |isbn=978-1-60327-233-9 |access-date=2022-03-24 |last2=Lebrilla |first2=Carlito B.}}</ref>

Collections of charged bodies do not always obey Newton's third law: there can be a change of one body's momentum without a compensatory change in the momentum of another. The discrepancy is accounted for by momentum carried by the electromagnetic field itself. The momentum per unit volume of the electromagnetic field is proportional to the [[Poynting vector]].<ref name="Panofsky1962"/>{{Rp|184}}<ref>{{Cite journal |last1=Bonga |first1=Béatrice |last2=Poisson |first2=Eric |last3=Yang |first3=Huan |date=November 2018 |title=Self-torque and angular momentum balance for a spinning charged sphere |url=http://aapt.scitation.org/doi/10.1119/1.5054590 |journal=[[American Journal of Physics]] |language=en |volume=86 |issue=11 |pages=839–848 |doi=10.1119/1.5054590 |arxiv=1805.01372 |bibcode=2018AmJPh..86..839B |s2cid=53625857 |issn=0002-9505}}</ref>

There is subtle conceptual conflict between electromagnetism and Newton's first law: [[Maxwell's equations|Maxwell's theory of electromagnetism]] predicts that electromagnetic waves will travel through empty space at a constant, definite speed. Thus, some inertial observers seemingly have a privileged status over the others, namely those who measure the [[speed of light]] and find it to be the value predicted by the Maxwell equations. In other words, light provides an absolute standard for speed, yet the principle of inertia holds that there should be no such standard. This tension is resolved in the theory of special relativity, which revises the notions of ''space'' and ''time'' in such a way that all inertial observers will agree upon the speed of light in vacuum.{{refn|group=note|Discussions can be found in, for example, Frautschi et al.,<ref name=":0" />{{Rp|page=215}} Panofsky and Phillips,<ref name="Panofsky1962">{{Cite book|last1=Panofsky|first1=Wolfgang K. H.|url=https://www.worldcat.org/oclc/56526974|title=Classical Electricity and Magnetism|last2=Phillips|first2=Melba|date=2005|publisher=Dover Publications|isbn=0-486-43924-0|edition=2nd|location=Mineola, N.Y.|oclc=56526974|author-link=Wolfgang Panofsky|author-link2=Melba Phillips|orig-date=1962}}</ref>{{Rp|page=272}} Goldstein, Poole and Safko,<ref name=":6">{{Cite book |last1=Goldstein |first1=Herbert |author-link=Herbert Goldstein |title-link=Classical Mechanics (Goldstein) |title=Classical Mechanics |last2=Poole |first2=Charles P. |last3=Safko |first3=John L. |date=2002 |publisher=Addison Wesley |isbn=0-201-31611-0 |edition=3rd |location=San Francisco |oclc=47056311}}</ref>{{Rp|page=277}} and Werner.<ref>{{Cite journal |last=Werner |first=Reinhard F. |date=2014-10-09 |title=Comment on "What Bell did" |journal=[[Journal of Physics A: Mathematical and Theoretical]] |volume=47 |issue=42 |pages=424011 |bibcode=2014JPhA...47P4011W |doi=10.1088/1751-8113/47/42/424011 |s2cid=122180759 |issn=1751-8113}}</ref>}}

===Special relativity===
{{further|Relativistic mechanics|Acceleration (special relativity)}}
In special relativity, the rule that Wilczek called "Newton's Zeroth Law" breaks down: the mass of a composite object is not merely the sum of the masses of the individual pieces.<ref name=":7">{{Cite book |last=Choquet-Bruhat |first=Yvonne |url=https://www.worldcat.org/oclc/317496332 |title=General Relativity and the Einstein Equations |date=2009 |publisher=Oxford University Press |isbn=978-0-19-155226-7 |location=Oxford |oclc=317496332 |author-link=Yvonne Choquet-Bruhat}}</ref>{{Rp|page=33}} Newton's first law, inertial motion, remains true. A form of Newton's second law, that force is the rate of change of momentum, also holds, as does the conservation of momentum. However, the definition of momentum is modified. Among the consequences of this is the fact that the more quickly a body moves, the harder it is to accelerate, and so, no matter how much force is applied, a body cannot be accelerated to the speed of light. Depending on the problem at hand, momentum in special relativity can be represented as a three-dimensional vector, <math>\mathbf{p} = m\gamma \mathbf{v}</math>, where <math>m</math> is the body's [[rest mass]] and <math>\gamma</math> is the [[Lorentz factor]], which depends upon the body's speed. Alternatively, momentum and force can be represented as [[four-vector]]s.<ref>{{Cite book|last1=Ellis|first1=George F. R.|url=https://www.worldcat.org/oclc/44694623|title=Flat and Curved Space-times|last2=Williams|first2=Ruth M.|date=2000|publisher=Oxford University Press|isbn=0-19-850657-0|edition=2nd|location=Oxford|oclc=44694623|author-link=George F. R. Ellis|author-link2=Ruth Margaret Williams}}</ref>{{Rp|page=107}}

Newton's third law must be modified in special relativity. The third law refers to the forces between two bodies at the same moment in time, and a key feature of special relativity is that simultaneity is relative. Events that happen at the same time relative to one observer can happen at different times relative to another. So, in a given observer's frame of reference, action and reaction may not be exactly opposite, and the total momentum of interacting bodies may not be conserved. The conservation of momentum is restored by including the momentum stored in the field that describes the bodies' interaction.<ref>{{cite book|last=French |first=A. P. |author-link=Anthony French |title=Special Relativity |year=1968 |isbn=0-393-09804-4 |publisher=W. W. Norton and Company |page=224}}</ref><ref>{{Cite journal |last=Havas |first=Peter |date=1964-10-01 |title=Four-Dimensional Formulations of Newtonian Mechanics and Their Relation to the Special and the General Theory of Relativity |url=https://link.aps.org/doi/10.1103/RevModPhys.36.938 |journal=[[Reviews of Modern Physics]] |language=en |volume=36 |issue=4 |pages=938–965 |doi=10.1103/RevModPhys.36.938 |bibcode=1964RvMP...36..938H |issn=0034-6861 |quote=...the usual assumption of Newtonian mechanics is that the forces are determined by the simultaneous positions (and possibly their derivatives) of the particles, and that they are related by Newton's third law. No such assumption is possible in special relativity since simultaneity is not an invariant concept in that theory.}}</ref>

Newtonian mechanics is a good approximation to special relativity when the speeds involved are small compared to that of light.<ref>{{Cite book |last=Stavrov |first=Iva |title=Curvature of Space and Time, with an Introduction to Geometric Analysis |title-link=Curvature of Space and Time, with an Introduction to Geometric Analysis |date=2020 |publisher=American Mathematical Society |isbn=978-1-4704-6313-7 |location=Providence, Rhode Island |oclc=1202475208}}</ref>{{Rp|page=131}}

===General relativity===
[[General relativity]] is a theory of gravity that advances beyond that of Newton. In general relativity, the gravitational force of Newtonian mechanics is reimagined as curvature of [[spacetime]]. A curved path like an orbit, attributed to a gravitational force in Newtonian mechanics, is not the result of a force deflecting a body from an ideal straight-line path, but rather the body's attempt to fall freely through a background that is itself curved by the presence of other masses. A remark by [[John Archibald Wheeler]] that has become proverbial among physicists summarizes the theory: "Spacetime tells matter how to move; matter tells spacetime how to curve."<ref name="Wheeler">{{Cite book|last=Wheeler|first=John Archibald|url=https://books.google.com/books?id=zGFkK2tTXPsC&pg=PA235|title=Geons, Black Holes, and Quantum Foam: A Life in Physics|date=2010-06-18|publisher=W. W. Norton & Company|isbn=978-0-393-07948-7|language=en|author-link=John Archibald Wheeler}}</ref><ref>{{Cite journal|last=Kersting|first=Magdalena|date=May 2019|title=Free fall in curved spacetime—how to visualise gravity in general relativity|journal=[[Physics Education]] |volume=54|issue=3|pages=035008|doi=10.1088/1361-6552/ab08f5|bibcode=2019PhyEd..54c5008K |s2cid=127471222 |issn=0031-9120|doi-access=free|hdl=10852/74677|hdl-access=free}}</ref> Wheeler himself thought of this reciprocal relationship as a modern, generalized form of Newton's third law.<ref name="Wheeler" /> The relation between matter distribution and spacetime curvature is given by the [[Einstein field equations]], which require [[tensor calculus]] to express.<ref name=":7" />{{Rp|page=43}}<ref>{{Cite book|last=Prescod-Weinstein|first=Chanda|url=https://www.worldcat.org/oclc/1164503847|title=The Disordered Cosmos: A Journey into Dark Matter, Spacetime, and Dreams Deferred|date=2021|publisher=Bold Type Books|isbn=978-1-5417-2470-9|location=New York, NY|oclc=1164503847|author-link=Chanda Prescod-Weinstein}}</ref>

The Newtonian theory of gravity is a good approximation to the predictions of general relativity when gravitational effects are weak and objects are moving slowly compared to the speed of light.<ref name=":6" />{{Rp|page=327}}<ref>{{Cite book|last=Goodstein|first=Judith R.|url=https://www.worldcat.org/oclc/1020305599|title=Einstein's Italian Mathematicians: Ricci, Levi-Civita, and the Birth of General Relativity|date=2018|publisher=American Mathematical Society|isbn=978-1-4704-2846-4|location=Providence, Rhode Island|pages=143|oclc=1020305599|author-link=Judith R. Goodstein}}</ref>

===Quantum mechanics===
[[Quantum mechanics]] is a theory of physics originally developed in order to understand microscopic phenomena: behavior at the scale of molecules, atoms or subatomic particles. Generally and loosely speaking, the smaller a system is, the more an adequate mathematical model will require understanding quantum effects. The conceptual underpinning of quantum physics is [[Bell's theorem|very different from that of classical physics]]. Instead of thinking about quantities like position, momentum, and energy as properties that an object ''has'', one considers what result might ''appear'' when a [[Measurement in quantum mechanics|measurement]] of a chosen type is performed. Quantum mechanics allows the physicist to calculate the probability that a chosen measurement will elicit a particular result.<ref>{{cite journal|last=Mermin|first=N. David|author-link=N. David Mermin|year=1993|title=Hidden variables and the two theorems of John Bell|journal=[[Reviews of Modern Physics]]|volume=65|issue=3|pages=803&ndash;815|arxiv=1802.10119|bibcode=1993RvMP...65..803M|doi=10.1103/RevModPhys.65.803|s2cid=119546199 |quote=It is a fundamental quantum doctrine that a measurement does not, in general, reveal a pre-existing value of the measured property.}}</ref><ref>{{Cite journal|last1=Schaffer|first1=Kathryn|last2=Barreto Lemos|first2=Gabriela|date=24 May 2019|title=Obliterating Thingness: An Introduction to the "What" and the "So What" of Quantum Physics|journal=[[Foundations of Science]] |volume=26 |pages=7–26 |language=en|arxiv=1908.07936|doi=10.1007/s10699-019-09608-5|issn=1233-1821|s2cid=182656563}}</ref> The [[Expectation value (quantum mechanics)|expectation value]] for a measurement is the average of the possible results it might yield, weighted by their probabilities of occurrence.<ref>{{Cite journal|last1=Marshman|first1=Emily|last2=Singh|first2=Chandralekha|author-link2=Chandralekha Singh|date=2017-03-01|title=Investigating and improving student understanding of the probability distributions for measuring physical observables in quantum mechanics|journal=[[European Journal of Physics]]|volume=38|issue=2|pages=025705|doi=10.1088/1361-6404/aa57d1|bibcode=2017EJPh...38b5705M |s2cid=126311599 |issn=0143-0807|doi-access=free}}</ref>

The [[Ehrenfest theorem]] provides a connection between quantum expectation values and Newton's second law, a connection that is necessarily inexact, as quantum physics is fundamentally different from classical. In quantum physics, position and momentum are represented by mathematical entities known as [[Hermitian operator]]s, and the [[Born rule]] is used to calculate the expectation values of a position measurement or a momentum measurement. These expectation values will generally change over time; that is, depending on the time at which (for example) a position measurement is performed, the probabilities for its different possible outcomes will vary. The Ehrenfest theorem says, roughly speaking, that the equations describing how these expectation values change over time have a form reminiscent of Newton's second law. However, the more pronounced quantum effects are in a given situation, the more difficult it is to derive meaningful conclusions from this resemblance.{{refn|group=note|Details can be found in the textbooks by, e.g., Cohen-Tannoudji et al.<ref name="Cohen-Tannoudji">{{cite book|last1=Cohen-Tannoudji |first1=Claude |last2=Diu |first2=Bernard |last3=Laloë |first3=Franck |title=Quantum Mechanics |author-link1=Claude Cohen-Tannoudji |publisher=John Wiley & Sons |year=2005 |isbn=0-471-16433-X |translator-first1=Susan Reid |translator-last1=Hemley |translator-first2=Nicole |translator-last2=Ostrowsky |translator-first3=Dan |translator-last3=Ostrowsky}}</ref>{{Rp|242}} and Peres.<ref>{{cite book|last=Peres|first=Asher|author-link=Asher Peres |title=Quantum Theory: Concepts and Methods |title-link=Quantum Theory: Concepts and Methods|publisher=[[Kluwer]]|year=1993|isbn=0-7923-2549-4|oclc=28854083}}</ref>{{Rp|302}}}}