Editing Newton's laws of motion (section)

===Multi-body gravitational system===
{{main article|Two-body problem|Three-body problem}}
[[File:Three-body Problem Animation.gif|thumb|Animation of three points or bodies attracting to each other]]
Newton's law of universal gravitation states that any body attracts any other body along the straight line connecting them. The size of the attracting force is proportional to the product of their masses, and inversely proportional to the square of the distance between them. Finding the shape of the orbits that an inverse-square force law will produce is known as the [[Kepler problem]]. The Kepler problem can be solved in multiple ways, including by demonstrating that the [[Laplace–Runge–Lenz vector]] is constant,<ref>{{Cite journal |last=Mungan |first=Carl E. |date=2005-03-01 |title=Another comment on "Eccentricity as a vector" |url=https://iopscience.iop.org/article/10.1088/0143-0807/26/2/L01 |journal=[[European Journal of Physics]] |volume=26 |issue=2 |pages=L7–L9 |doi=10.1088/0143-0807/26/2/L01 |s2cid=121740340 |issn=0143-0807}}</ref> or by applying a duality transformation to a 2-dimensional harmonic oscillator.<ref>{{Cite journal |last=Saggio |first=Maria Luisa |date=2013-01-01 |title=Bohlin transformation: the hidden symmetry that connects Hooke to Newton |url=https://iopscience.iop.org/article/10.1088/0143-0807/34/1/129 |journal=[[European Journal of Physics]] |volume=34 |issue=1 |pages=129–137 |doi=10.1088/0143-0807/34/1/129 |bibcode=2013EJPh...34..129S |s2cid=119949261 |issn=0143-0807}}</ref> However it is solved, the result is that orbits will be [[conic section]]s, that is, [[ellipse]]s (including circles), [[parabola]]s, or [[hyperbola]]s. The [[Orbital eccentricity|eccentricity]] of the orbit, and thus the type of conic section, is determined by the energy and the angular momentum of the orbiting body. Planets do not have sufficient energy to escape the Sun, and so their orbits are ellipses, to a good approximation; because the planets pull on one another, actual orbits are not exactly conic sections.

If a third mass is added, the Kepler problem becomes the three-body problem, which in general has no exact solution in [[Closed-form expression|closed form]]. That is, there is no way to start from the differential equations implied by Newton's laws and, after a finite sequence of standard mathematical operations, obtain equations that express the three bodies' motions over time.<ref name="Barrow-Green1997">{{cite book |last=Barrow-Green |first=June |author-link=June Barrow-Green |title=Poincaré and the Three Body Problem |title-link=Poincaré and the Three-Body Problem |publisher=American Mathematical Society |year=1997 |isbn=978-0-8218-0367-7 |pages=8–12 |bibcode=1997ptbp.book.....B}}</ref><ref name="Barrow-Green2008">{{cite book |last=Barrow-Green |first=June |title=The Princeton Companion to Mathematics |title-link=The Princeton Companion to Mathematics |publisher=Princeton University Press |year=2008 |isbn=978-0-691-11880-2 |editor-last1=Gowers |editor-first1=Timothy |editor-link1=Timothy Gowers |pages=726–728 |chapter=The Three-Body Problem |oclc=682200048 |author-link=June Barrow-Green |editor-last2=Barrow-Green |editor-first2=June |editor-link2=June Barrow-Green |editor-last3=Leader |editor-first3=Imre |editor-link3=Imre Leader}}</ref> [[Numerical methods for ordinary differential equations|Numerical methods]] can be applied to obtain useful, albeit approximate, results for the three-body problem.<ref>{{Cite journal |last1=Breen |first1=Barbara J. |last2=Weidert |first2=Christine E. |last3=Lindner |first3=John F. |last4=Walker |first4=Lisa May |last5=Kelly |first5=Kasey |last6=Heidtmann |first6=Evan |date=April 2008 |title=Invitation to embarrassingly parallel computing |url=http://aapt.scitation.org/doi/10.1119/1.2834738 |journal=[[American Journal of Physics]] |language=en |volume=76 |issue=4 |pages=347–352 |doi=10.1119/1.2834738 |bibcode=2008AmJPh..76..347B |issn=0002-9505}}</ref> The positions and velocities of the bodies can be stored in [[Variable (computer science)|variables]] within a computer's memory; Newton's laws are used to calculate how the velocities will change over a short interval of time, and knowing the velocities, the changes of position over that time interval can be computed. This process is [[Loop (computing)|looped]] to calculate, approximately, the bodies' trajectories. Generally speaking, the shorter the time interval, the more accurate the approximation.<ref>{{Cite journal |last=McCandlish |first=David |date=July 1973 |editor-last=Shirer |editor-first=Donald L. |title=Solutions to the Three-Body Problem by Computer |url=http://aapt.scitation.org/doi/10.1119/1.1987423 |journal=[[American Journal of Physics]] |language=en |volume=41 |issue=7 |pages=928–929 |doi=10.1119/1.1987423 |issn=0002-9505}}</ref>