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Introduction. When a body is free to rotate around an axis, torque must be applied to change its angular momentum. The amount of torque needed to cause any given angular acceleration (the rate of change in angular velocity) is proportional to the moment of inertia of the body.
The three laws of motion were first stated by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica ( Mathematical Principles of Natural Philosophy ), originally published in 1687. [3] Newton used them to investigate and explain the motion of many physical objects and systems.
- Mathematical Description
- Solutions
- History
- Other Problems Involving Three Bodies
- N-Body Problem
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The mathematical statement of the three-body problem can be given in terms of the Newtonian equations of motion for vector positions r i = ( x i , y i , z i ) {\displaystyle \mathbf {r_{i}} =(x_{i},y_{i},z_{i})} of three gravitationally interacting bodies with masses m i {\displaystyle m_{i}} : where H {\displaystyle {\mathcal {H}}} is the Hamilton...
General solution
There is no general closed-form solution to the three-body problem.In other words, it does not have a general solution that can be expressed in terms of a finite number of standard mathematical operations. Moreover, the motion of three bodies is generally non-repeating, except in special cases. However, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists an analytic solution to the three-body problem in the form of a Puiseux series, specifically a power series in t...
Special-case solutions
In 1767, Leonhard Eulerfound three families of periodic solutions in which the three masses are collinear at each instant. In 1772, Lagrange found a family of solutions in which the three masses form an equilateral triangle at each instant. Together with Euler's collinear solutions, these solutions form the central configurations for the three-body problem. These solutions are valid for any mass ratios, and the masses move on Keplerian ellipses. These four families are the only known solution...
Numerical approaches
Using a computer, the problem may be solved to arbitrarily high precision using numerical integration although high precision requires a large amount of CPU time. There have been attempts of creating computer programs that numerically solve the three-body problem (and by extension, the n-body problem) involving both electromagnetic and gravitational interactions, and incorporating modern theories of physics such as special relativity. In addition, using the theory of random walks, an approxim...
The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Philosophiæ Naturalis Principia Mathematica, when Newton was trying to figure out if any long term stability is possible, especially the system of our Earth, the Moon, and the Sun. He was guided under the major Renaissanc...
The term "three-body problem" is sometimes used in the more general sense to refer to any physical problem involving the interaction of three bodies. A quantum-mechanical analogue of the gravitational three-body problem in classical mechanics is the helium atom,in which a helium nucleus and two electrons interact according to the inverse-square Cou...
The three-body problem is a special case of the n-body problem, which describes how n objects move under one of the physical forces, such as gravity. These problems have a global analytical solution in the form of a convergent power series, as was proven by Karl F. Sundman for n = 3 and by Qiudong Wang for n > 3 (see n-body problem for details). Ho...
Chenciner, Alain (2007). "Three body problem". Scholarpedia. 2 (10): 2111. Bibcode:2007SchpJ...2.2111C. doi:10.4249/scholarpedia.2111.Amyotrophic lateral sclerosis ( ALS ), also known as motor neurone disease ( MND) or Lou Gehrig's disease in the United States, is a rare, terminal neurodegenerative disorder that results in the progressive loss of both upper and lower motor neurons that normally control voluntary muscle contraction. [3] .
5S methodology. 5S resource corner at Scanfil Poland factory in Sieradz. 5S is a workplace organization method that uses a list of five Japanese words: seiri (整理), seiton (整頓), seisō (清掃), seiketsu (清潔), and shitsuke (躾). These have been translated as 'sort', 'set in order', 'shine', 'standardize', and 'sustain'. [1]
Today's featured picture. Trithemis annulata, commonly known as the violet dropwing, is a species of dragonfly in the family Libellulidae. It is found in most of Africa, the Middle East, and southern Europe. Males of this species are violet-red with red veins in the wings, while females are yellow and brown.
Theodor Herzl was the founder of the modern Zionist movement. In his 1896 pamphlet Der Judenstaat, he envisioned the founding of a future independent Jewish state during the 20th century. Zionism (/ ˈ z aɪ. ə n ɪ z əm / ZY-ə-niz-əm; Hebrew: צ י ו נו ת, romanized: Ṣīyyonūt, IPA:; derived from Zion) is an ethnic or ethno-cultural nationalist movement that emerged in Europe in ...
