Math 2.0

Mechanics of particles and systems Particle systems and celestial mechanics Problem

Two-body problem

Builds on Newton's law of universal gravitationCentre of massReduced massKepler problemn-body problemConic section

Problem

Let G>0G > 0 be the gravitational constant and let m1,m2>0m_1, m_2 > 0 be masses. Let the positions x1(t),x2(t)∈ℝ3x_1(t), x_2(t) \in \R^3 satisfy Newton's law of universal gravitation m1x1′′=Gm1m2(x2−x1)∥x2−x1∥3,m2x2′′=Gm1m2(x1−x2)∥x1−x2∥3.m_1 x_1'' = \frac{G m_1 m_2 (x_2 - x_1)}{\norm{x_2 - x_1}^3}, \qquad m_2 x_2'' = \frac{G m_1 m_2 (x_1 - x_2)}{\norm{x_1 - x_2}^3}. Given initial positions with x1(0)≠x2(0)x_1(0) \neq x_2(0) and initial velocities, determine the motion of the two bodies.

Remarks

The problem is solved completely. Let M=m1+m2M = m_1 + m_2, let R=(m1x1+m2x2)/MR = (m_1 x_1 + m_2 x_2) / M be the centre of mass and let r=x2−x1r = x_2 - x_1 be the relative position. Adding the two equations gives R′′=0R'' = 0, so the centre of mass moves on a straight line at constant speed. Subtracting them gives r′′=−GM r∥r∥3,r'' = - \frac{G M \, r}{\norm{r}^3}, which is the Kepler problem with k=GMk = G M. Equivalently, μr′′=−Gm1m2 r/∥r∥3\mu r'' = - G m_1 m_2 \, r / \norm{r}^3, where μ=m1m2/M\mu = m_1 m_2 / M is the reduced mass. So rr moves on a conic section with a focus at 00, or on a line through 00 when the angular momentum is zero. Then x1=R−m2Mrx_1 = R - \frac{m_2}{M} r and x2=R+m1Mrx_2 = R + \frac{m_1}{M} r, so each body moves on a scaled copy of the relative orbit around the centre of mass.

The two-body problem is the case n=2n = 2 of the n-body problem, while for n≥3n \ge 3 there is no general solution by elementary formulas. Newton solved it in the Principia of 1687.

Reading path

To understand this page from the ground up, read these 55 pages in order. The foundations come first.

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

Where this leads

15 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Two-body_problem
About this page
TagCSVPJAA75E
URIm2://CSVPJAA75E
Kindproblem
WritingW0 stub
MSC70F05, 70F15
StatusThis copy of the site has no build data. Run m2 status CSVPJAA75E.
Aboutn-body problem, Kepler problem
UsesNewton's law of universal gravitation, Centre of mass, Reduced mass, Kepler problem, n-body problem, Conic section
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Changed3 November 2026

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