Editing Newton's laws of motion (section)

==Rigid-body motion and rotation==
{{main|Rigid-body motion}}

A rigid body is an object whose size is too large to neglect and which maintains the same shape over time. In Newtonian mechanics, the motion of a rigid body is often understood by separating it into movement of the body's [[center of mass]] and movement around the center of mass.

===Center of mass===
{{main|Center of mass}}
[[File:Masocentro1.jpg|alt=Fork-cork-toothpick object balanced on a pen on the toothpick part|thumb|The total center of mass of the [[Fork|forks]], [[Stopper (plug)|cork]], and [[toothpick]] is on top of the pen's tip.]]
Significant aspects of the motion of an extended body can be understood by imagining the mass of that body concentrated to a single point, known as the center of mass. The location of a body's center of mass depends upon how that body's material is distributed. For a collection of pointlike objects with masses <math>m_1, \ldots, m_N</math> at positions <math>\mathbf{r}_1, \ldots, \mathbf{r}_N</math>, the center of mass is located at <math display="block">\mathbf{R} = \sum_{i=1}^N \frac{m_i \mathbf{r}_i}{M},</math> where <math>M</math> is the total mass of the collection. In the absence of a net external force, the center of mass moves at a constant speed in a straight line. This applies, for example, to a collision between two bodies.<ref>{{Cite journal |last=Lyublinskaya |first=Irina E. |date=January 1998 |title=Central collisions—The general case |url=http://aapt.scitation.org/doi/10.1119/1.879949 |journal=[[The Physics Teacher]] |language=en |volume=36 |issue=1 |pages=18–19 |doi=10.1119/1.879949 |bibcode=1998PhTea..36...18L |issn=0031-921X}}</ref> If the total external force is not zero, then the center of mass changes velocity as though it were a point body of mass <math>M</math>. This follows from the fact that the internal forces within the collection, the forces that the objects exert upon each other, occur in balanced pairs by Newton's third law. In a system of two bodies with one much more massive than the other, the center of mass will approximately coincide with the location of the more massive body.<ref name=":2" />{{Rp|pages=22-24}}

===Rotational analogues of Newton's laws===
When Newton's laws are applied to rotating extended bodies, they lead to new quantities that are analogous to those invoked in the original laws. The analogue of mass is the [[moment of inertia]], the counterpart of momentum is [[angular momentum]], and the counterpart of force is [[torque]].

Angular momentum is calculated with respect to a reference point.<ref>{{Cite journal |last1=Close |first1=Hunter G. |last2=Heron |first2=Paula R. L. |date=October 2011 |title=Student understanding of the angular momentum of classical particles |url=http://aapt.scitation.org/doi/10.1119/1.3579141 |journal=[[American Journal of Physics]] |language=en |volume=79 |issue=10 |pages=1068–1078 |doi=10.1119/1.3579141 |bibcode=2011AmJPh..79.1068C |issn=0002-9505}}</ref> If the displacement vector from a reference point to a body is <math>\mathbf{r}</math> and the body has momentum <math>\mathbf{p}</math>, then the body's angular momentum with respect to that point is, using the vector [[cross product]], <math display="block">\mathbf{L} = \mathbf{r} \times \mathbf{p}.</math> Taking the time derivative of the angular momentum gives <math display="block"> \frac{d\mathbf{L}}{dt} = \left(\frac{d\mathbf{r}}{dt}\right) \times \mathbf{p} + \mathbf{r} \times \frac{d\mathbf{p}}{dt} = \mathbf{v} \times m\mathbf{v} +  \mathbf{r} \times \mathbf{F}.</math> The first term vanishes because <math>\mathbf{v}</math> and <math>m\mathbf{v}</math> point in the same direction. The remaining term is the torque, <math display="block">\mathbf{\tau} = \mathbf{r} \times \mathbf{F}.</math> When the torque is zero, the angular momentum is constant, just as when the force is zero, the momentum is constant.<ref name=":2" />{{Rp|pages=14-15}} The torque can vanish even when the force is non-zero, if the body is located at the reference point (<math>\mathbf{r} = 0</math>) or if the force <math>\mathbf{F}</math> and the displacement vector <math>\mathbf{r}</math> are directed along the same line.

The angular momentum of a collection of point masses, and thus of an extended body, is found by adding the contributions from each of the points. This provides a means to characterize a body's rotation about an axis, by adding up the angular momenta of its individual pieces. The result depends on the chosen axis, the shape of the body, and the rate of rotation.<ref name=":2" />{{Rp|page=28}}

===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>