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- Eulers_three-body_problem abstract "In physics and astronomy, Euler's three-body problem is to solve for the motion of a particle that is acted upon by the gravitational field of two other point masses that are fixed in space. This problem is exactly solvable, and yields an approximate solution for particles moving in the gravitational fields of prolate and oblate spheroids. This problem is named after Leonhard Euler, who discussed it in memoirs published in 1760. Important extensions and analyses were contributed subsequently by Lagrange, Liouville, Laplace, Jacobi, Darboux, Le Verrier, Velde, Hamilton, Poincaré, Birkhoff and E. T. Whittaker, among others.Euler's problem also covers the case when the particle is acted upon by other inverse-square central forces, such as the electrostatic interaction described by Coulomb's law. The classical solutions of the Euler problem have been used to study chemical bonding, using a semiclassical approximation of the energy levels of a single electron moving in the field of two atomic nuclei, such as the diatomic ion HeH2+. This was first done by Wolfgang Pauli in his doctoral dissertation under Arnold Sommerfeld, a study of the first ion of molecular hydrogen, namely the Hydrogen molecule-ion H2+. These energy levels can be calculated with reasonable accuracy using the Einstein–Brillouin–Keller method, which is also the basis of the Bohr model of atomic hydrogen. More recently, as explained further in the quantum-mechanical version, analytical solutions to the eigenenergies have been obtained: these are a generalization of the Lambert W function. By treating Euler's problem as a Liouville dynamical system, the exact solution can be expressed in terms of elliptic integrals. For convenience, the problem may also be solved by numerical methods, such as Runge–Kutta integration of the equations of motion. The total energy of the moving particle is conserved, but its linear and angular momentum are not, since the two fixed centers can apply a net force and torque. Nevertheless, the particle has a second conserved quantity that corresponds to the angular momentum or to the Laplace–Runge–Lenz vector as limiting cases.The Euler three-body problem is known by a variety of names, such as the problem of two fixed centers, the Euler–Jacobi problem, and the two-center Kepler problem. Various generalizations of Euler's problem are known; these generalizations add linear and inverse cubic forces and up to five centers of force. Special cases of these generalized problems include Darboux's problem and Velde's problem.".
- Eulers_three-body_problem wikiPageExternalLink eulerarchive.maa.org.
- Eulers_three-body_problem wikiPageID "1504593".
- Eulers_three-body_problem wikiPageLength "19769".
- Eulers_three-body_problem wikiPageOutDegree "80".
- Eulers_three-body_problem wikiPageRevisionID "665373076".
- Eulers_three-body_problem wikiPageWikiLink Angular_momentum.
- Eulers_three-body_problem wikiPageWikiLink Arnold_Sommerfeld.
- Eulers_three-body_problem wikiPageWikiLink Astronomy.
- Eulers_three-body_problem wikiPageWikiLink Atomic_nucleus.
- Eulers_three-body_problem wikiPageWikiLink Bohr_model.
- Eulers_three-body_problem wikiPageWikiLink Bonnets_theorem.
- Eulers_three-body_problem wikiPageWikiLink Carl_Gustav_Jacob_Jacobi.
- Eulers_three-body_problem wikiPageWikiLink Category:Orbits.
- Eulers_three-body_problem wikiPageWikiLink Central_force.
- Eulers_three-body_problem wikiPageWikiLink Charles_Coulson.
- Eulers_three-body_problem wikiPageWikiLink Computer_algebra_system.
- Eulers_three-body_problem wikiPageWikiLink Constant_of_motion.
- Eulers_three-body_problem wikiPageWikiLink Coulombs_law.
- Eulers_three-body_problem wikiPageWikiLink Dihydrogen_cation.
- Eulers_three-body_problem wikiPageWikiLink E._T._Whittaker.
- Eulers_three-body_problem wikiPageWikiLink Einstein–Brillouin–Keller_method.
- Eulers_three-body_problem wikiPageWikiLink Electric_field.
- Eulers_three-body_problem wikiPageWikiLink Electron.
- Eulers_three-body_problem wikiPageWikiLink Electrostatics.
- Eulers_three-body_problem wikiPageWikiLink Elliptic_coordinate_system.
- Eulers_three-body_problem wikiPageWikiLink Elliptic_integral.
- Eulers_three-body_problem wikiPageWikiLink Energy.
- Eulers_three-body_problem wikiPageWikiLink Experimental_mathematics.
- Eulers_three-body_problem wikiPageWikiLink George_David_Birkhoff.
- Eulers_three-body_problem wikiPageWikiLink Gravitational_field.
- Eulers_three-body_problem wikiPageWikiLink Gravity.
- Eulers_three-body_problem wikiPageWikiLink Henri_Poincaré.
- Eulers_three-body_problem wikiPageWikiLink Hydrogen.
- Eulers_three-body_problem wikiPageWikiLink Imaginary_number.
- Eulers_three-body_problem wikiPageWikiLink Implicit_function.
- Eulers_three-body_problem wikiPageWikiLink Inverse-square_law.
- Eulers_three-body_problem wikiPageWikiLink Ion.
- Eulers_three-body_problem wikiPageWikiLink Jacobi_integral.
- Eulers_three-body_problem wikiPageWikiLink Jean_Gaston_Darboux.
- Eulers_three-body_problem wikiPageWikiLink Joseph-Louis_Lagrange.
- Eulers_three-body_problem wikiPageWikiLink Joseph_Liouville.
- Eulers_three-body_problem wikiPageWikiLink Kepler_problem.
- Eulers_three-body_problem wikiPageWikiLink Kinetic_energy.
- Eulers_three-body_problem wikiPageWikiLink Lagrangian_point.
- Eulers_three-body_problem wikiPageWikiLink Lambert_W_function.
- Eulers_three-body_problem wikiPageWikiLink Laplace–Runge–Lenz_vector.
- Eulers_three-body_problem wikiPageWikiLink Leonhard_Euler.
- Eulers_three-body_problem wikiPageWikiLink Limiting_case_(mathematics).
- Eulers_three-body_problem wikiPageWikiLink Liouville_dynamical_system.
- Eulers_three-body_problem wikiPageWikiLink Momentum.
- Eulers_three-body_problem wikiPageWikiLink Parametric_equation.
- Eulers_three-body_problem wikiPageWikiLink Physics.
- Eulers_three-body_problem wikiPageWikiLink Pierre-Simon_Laplace.
- Eulers_three-body_problem wikiPageWikiLink Planet.
- Eulers_three-body_problem wikiPageWikiLink Potential_energy.
- Eulers_three-body_problem wikiPageWikiLink Prolate_spheroidal_coordinates.
- Eulers_three-body_problem wikiPageWikiLink Runge–Kutta_methods.
- Eulers_three-body_problem wikiPageWikiLink Star.
- Eulers_three-body_problem wikiPageWikiLink Three-body_problem.
- Eulers_three-body_problem wikiPageWikiLink Urbain_Le_Verrier.
- Eulers_three-body_problem wikiPageWikiLink William_Rowan_Hamilton.
- Eulers_three-body_problem wikiPageWikiLink Wolfgang_Pauli.
- Eulers_three-body_problem wikiPageWikiLinkText "Euler's three-body problem".
- Eulers_three-body_problem wikiPageWikiLinkText "circular restricted three-body problem".
- Eulers_three-body_problem wikiPageWikiLinkText "quantum mechanical three body problem".
- Eulers_three-body_problem wikiPageWikiLinkText "restricted three-body problem".
- Eulers_three-body_problem wikiPageWikiLinkText "special cases".
- Eulers_three-body_problem wikiPageWikiLinkText "three-body problem".
- Eulers_three-body_problem wikiPageUsesTemplate Template:Cite_book.
- Eulers_three-body_problem wikiPageUsesTemplate Template:Cite_journal.
- Eulers_three-body_problem wikiPageUsesTemplate Template:Reflist.
- Eulers_three-body_problem subject Category:Orbits.
- Eulers_three-body_problem type Astrodynamic.
- Eulers_three-body_problem type Page.
- Eulers_three-body_problem comment "In physics and astronomy, Euler's three-body problem is to solve for the motion of a particle that is acted upon by the gravitational field of two other point masses that are fixed in space. This problem is exactly solvable, and yields an approximate solution for particles moving in the gravitational fields of prolate and oblate spheroids. This problem is named after Leonhard Euler, who discussed it in memoirs published in 1760.".
- Eulers_three-body_problem label "Euler's three-body problem".
- Eulers_three-body_problem sameAs Q4177149.
- Eulers_three-body_problem sameAs Problema_de_Euler_dos_três_corpos.
- Eulers_three-body_problem sameAs m.056hxg.
- Eulers_three-body_problem sameAs Reštringovaný_problém.
- Eulers_three-body_problem sameAs Q4177149.
- Eulers_three-body_problem wasDerivedFrom Eulers_three-body_problem?oldid=665373076.
- Eulers_three-body_problem isPrimaryTopicOf Eulers_three-body_problem.