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Lockheed Martin FB-22. The Lockheed Martin/Boeing F-22 Raptor is an American twin-engine all-weather stealth fighter aircraft developed and produced for the United States Air Force (USAF). As a product of the USAF's Advanced Tactical Fighter (ATF) program the aircraft was designed as an air superiority fighter, but also incorporates ground ...
- Mathematical Description
- Solutions
- History
- Other Problems Involving Three Bodies
- N-Body Problem
- External Links
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 Euler found three families of periodic solutions in which the three masses are collinear at each instant. See Euler's three-body problem. 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 fam...
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.The 828-metre (2,717 ft) tall Burj Khalifa in Dubai has been the tallest building since 2010. The Burj Khalifa has been classified as megatall. A diagram showing the tallest buildings as of 2024 This is a list of the tallest buildings.Tall buildings, such as skyscrapers, are intended here as enclosed structures with continuously occupiable floors and a height of at least 340 metres (1,120 ft).
The Maldives is the smallest country in Asia. Including the sea, the territory spans roughly 90,000 square kilometres (35,000 sq mi), and a land area of 298 square kilometres (115 sq mi). The Maldives is one of the world's most geographically dispersed sovereign states, and the smallest Muslim-majority country by land area.
The Burj Khalifa [a] (known as the Burj Dubai prior to its inauguration) is a skyscraper in Dubai, United Arab Emirates. It is the world's tallest structure. With a total height of 829.8 m (2,722 ft, or just over half a mile) and a roof height (excluding antenna, but including a 242.6 m spire) [2] of 828 m (2,717 ft), the Burj Khalifa has been ...
Stoicism is a school of Hellenistic philosophy that flourished in Ancient Greece and Ancient Rome. [1] The Stoics believed that the practice of virtue is enough to achieve eudaimonia: a well-lived life. The Stoics identified the path to achieving it with a life spent practicing the four virtues in everyday life: wisdom, courage, temperance or ...
