Math 2.0

Mechanics of particles and systems Particle systems and celestial mechanics Problem

Euler's three-body problem

Builds on Newton's law of universal gravitationHamiltonian systemFirst integralIntegrable systemThree-body problem

Problem

Let G>0G > 0 be the gravitational constant, let m1,m2>0m_1, m_2 > 0 and let c>0c > 0. Two point masses m1m_1 and m2m_2 are fixed at the points (0,0,−c)(0, 0, -c) and (0,0,c)(0, 0, c) of ℝ3\R^3. A particle at x(t)∈ℝ3x(t) \in \R^3 moves under their Newtonian attraction: x′′=−Gm1(x−(0,0,−c))r13−Gm2(x−(0,0,c))r23,x'' = - \frac{G m_1 (x - (0, 0, -c))}{r_1^3} - \frac{G m_2 (x - (0, 0, c))}{r_2^3}, where r1r_1 and r2r_2 are the distances from xx to the two fixed points. Given the initial position and velocity of the particle, determine its motion.

Remarks

The problem is also called the problem of two fixed centres. It is solved: Euler showed in 1760 that it can be integrated exactly. It is a Hamiltonian system with three degrees of freedom and three independent first integrals in involution, so it is an integrable system. They are the energy, the component of the angular momentum along the axis through the two centres, and a third integral found by separating variables. In prolate spheroidal coordinates ξ=(r1+r2)/(2c)\xi = (r_1 + r_2)/(2c), η=(r1−r2)/(2c)\eta = (r_1 - r_2)/(2c) and the angle φ\varphi around the axis, the Hamilton–Jacobi equation separates, and the solution is given by elliptic integrals.

The problem differs from the three-body problem because the two centres do not move. It is used as an approximation for a satellite near an oblate planet and, in quantum mechanics, for the hydrogen molecule ion.

Reading path

To understand this page from the ground up, read these 49 pages in order. The foundations come first.

  1. Derivative concept
  2. Differential equation concept
  3. Euclidean space object
  4. Euclidean vector concept
  5. First integral concept
  6. Function concept
  7. Hamiltonian system concept
  8. Integrable system concept
  9. Interval concept
  10. Ordinary differential equation concept
  11. Velocity concept
  12. Fréchet derivative definition
  13. Cross product definition
  14. Ordinary differential equation definition
  15. Function of several variables concept
  16. Vector-valued function concept
  17. Lipschitz condition definition
  18. Classical mechanics concept
  19. n-body problem problem
  20. Limit of a function of several variables concept
  21. Momentum concept
  22. Partial derivative concept
  23. Solution of an ordinary differential equation definition
  24. Inertial frame of reference concept
  25. Three-body problem problem
  26. Continuity in R^n definition
  27. Differentiable function of several variables definition
  28. Partial derivative definition
  29. Initial value problem definition
  30. Newton's laws of motion statement
  31. Differentiable on an open set definition
  32. Gradient definition
  33. Higher partial derivatives definition
  34. Picard–Lindelöf hypotheses definition
  35. Iterated derivative definition
  36. Picard operator definition
  37. C^k function characterised by continuous derivatives statement
  38. Integral equation form of an initial value problem statement
  39. An iterate of the Picard operator is a contraction statement
  40. Vector field definition
  41. Picard–Lindelöf theorem statement
  42. Conservative vector field definition
  43. Classical mechanics definition
  44. Momentum definition
  45. Potential energy definition
  46. Angular momentum definition
  47. Central force definition
  48. Newton's law of universal gravitation statement
  49. Euler's three-body problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Euler%27s_three-body_problem
About this page
TagFXPHA1T3NF
URIm2://FXPHA1T3NF
Kindproblem
WritingW0 stub
MSC70F07, 70H06, 70F15
StatusThis copy of the site has no build data. Run m2 status FXPHA1T3NF.
AboutThree-body problem
UsesNewton's law of universal gravitation, Hamiltonian system, First integral, Integrable system, Three-body problem
Same as
Changed3 November 2026

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