Editing Newton's laws of motion (section)

=== Nonlinear dynamics ===
{{main|Chaos theory}}
[[File:Demonstrating Chaos with a Double Pendulum.gif|thumb|Three double pendulums, initialized with almost exactly the same initial conditions, diverge over time.]]
Newton's laws of motion allow the possibility of [[Chaos theory|chaos]].<ref name=":3">{{Cite journal|last1=Masoliver|first1=Jaume|last2=Ros|first2=Ana|date=2011-03-01|title=Integrability and chaos: the classical uncertainty|url=https://iopscience.iop.org/article/10.1088/0143-0807/32/2/016|journal=[[European Journal of Physics]] |volume=32|issue=2|pages=431–458|doi=10.1088/0143-0807/32/2/016|arxiv=1012.4384 |bibcode=2011EJPh...32..431M |s2cid=58892714 |issn=0143-0807}}</ref><ref>{{Cite journal|last=Laws|first=Priscilla W.|author-link=Priscilla Laws|date=April 2004|title=A unit on oscillations, determinism and chaos for introductory physics students|url=http://aapt.scitation.org/doi/10.1119/1.1649964|journal=[[American Journal of Physics]]|language=en|volume=72|issue=4|pages=446–452|doi=10.1119/1.1649964|bibcode=2004AmJPh..72..446L |issn=0002-9505}}</ref> That is, qualitatively speaking, physical systems obeying Newton's laws can exhibit sensitive dependence upon their initial conditions: a slight change of the position or velocity of one part of a system can lead to the whole system behaving in a radically different way within a short time. Noteworthy examples include the three-body problem, the [[double pendulum]], [[dynamical billiards]], and the [[Fermi–Pasta–Ulam–Tsingou problem]].

Newton's laws can be applied to [[fluid]]s by considering a fluid as composed of infinitesimal pieces, each exerting forces upon neighboring pieces. The [[Euler equations (fluid dynamics)|Euler momentum equation]] is an expression of Newton's second law adapted to fluid dynamics.<ref name="Zee2020">{{cite book|last=Zee |first=Anthony |author-link=Anthony Zee |title=Fly by Night Physics |publisher=Princeton University Press |year=2020 |pages=363–364 |isbn=978-0-691-18254-4 |oclc=1288147292}}</ref><ref>{{Cite journal |last1=Han-Kwan |first1=Daniel |last2=Iacobelli |first2=Mikaela |date=2021-04-07 |title=From Newton's second law to Euler's equations of perfect fluids |url=https://www.ams.org/proc/2021-149-07/S0002-9939-2021-15349-5/ |journal=[[Proceedings of the American Mathematical Society]] |language=en |volume=149 |issue=7 |pages=3045–3061 |doi=10.1090/proc/15349 |s2cid=220127889 |issn=0002-9939|doi-access=free |arxiv=2006.14924 }}</ref> A fluid is described by a velocity field, i.e., a function <math>\mathbf{v}(\mathbf{x},t)</math> that assigns a velocity vector to each point in space and time. A small object being carried along by the fluid flow can change velocity for two reasons: first, because the velocity field at its position is changing over time, and second, because it moves to a new location where the velocity field has a different value. Consequently, when Newton's second law is applied to an infinitesimal portion of fluid, the acceleration <math>\mathbf{a}</math> has two terms, a combination known as a [[material derivative|''total'' or ''material'' derivative]]. The mass of an infinitesimal portion depends upon the fluid [[density]], and there is a net force upon it if the fluid pressure varies from one side of it to another. Accordingly, <math>\mathbf{a} = \mathbf{F}/m</math> becomes
<math display="block"> \frac{\partial v}{\partial t} + (\mathbf{\nabla} \cdot \mathbf{v}) \mathbf{v} = -\frac{1}{\rho} \mathbf{\nabla}P + \mathbf{f} ,</math>
where <math>\rho</math> is the density, <math>P</math> is the pressure, and <math>\mathbf{f}</math> stands for an external influence like a gravitational pull. Incorporating the effect of [[viscosity]] turns the Euler equation into a [[Navier–Stokes equation]]:
<math display="block"> \frac{\partial v}{\partial t} + (\mathbf{\nabla} \cdot \mathbf{v}) \mathbf{v} = -\frac{1}{\rho} \mathbf{\nabla}P + \nu \nabla^2 \mathbf{v} + \mathbf{f} ,</math>
where <math>\nu</math> is the [[Viscosity#Kinematic viscosity|kinematic viscosity]].<ref name="Zee2020"/>