Math 2.0

Mechanics of particles and systems Definition

Celestial mechanics

Builds on Newton's law of universal gravitationn-body problemKepler problemHamiltonian system

Definition

Celestial mechanics is the study of the motion of bodies that interact through gravity. Its standard model consists of point masses m1,…,mn>0m_1, \ldots, m_n > 0 at positions x1(t),…,xn(t)∈ℝ3x_1(t), \ldots, x_n(t) \in \R^3. They move according to Newton's law of universal gravitation, mi xi′′=∑j≠iGmimj(xj−xi)∥xj−xi∥3,i=1,…,n,m_i \, x_i'' = \sum_{j \neq i} \frac{G m_i m_j (x_j - x_i)}{\norm{x_j - x_i}^3}, \qquad i = 1, \ldots, n, where G>0G > 0 is the gravitational constant. The field also covers simplified models derived from this one, such as restricted problems in which some masses are negligible, and corrections for extended bodies.

Remarks

The equations above define the n-body problem, a Hamiltonian system. For n=2n = 2 they reduce to the Kepler problem, whose bounded orbits are ellipses. The motion of the Moon around the Earth under the pull of the Sun is a question of celestial mechanics. The fall of a stone through the air is not a typical question of the field. There gravity is a constant force, and air friction dominates the analysis. Relativistic corrections are treated in relativistic celestial mechanics, which uses general relativity instead of Newton's law.

Reading path

To understand this page from the ground up, read these 51 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. Partial derivative concept
  26. Inertial frame of reference concept
  27. Solution of an ordinary differential equation definition
  28. Continuity in R^n definition
  29. Differentiable function of several variables definition
  30. Newton's laws of motion statement
  31. Partial derivative definition
  32. Initial value problem definition
  33. Differentiable on an open set definition
  34. Gradient definition
  35. Higher partial derivatives definition
  36. Picard–Lindelöf hypotheses definition
  37. Iterated derivative definition
  38. Picard operator definition
  39. C^k function characterised by continuous derivatives statement
  40. Integral equation form of an initial value problem statement
  41. An iterate of the Picard operator is a contraction statement
  42. Vector field definition
  43. Picard–Lindelöf theorem statement
  44. Conservative vector field definition
  45. Classical mechanics definition
  46. Momentum definition
  47. Potential energy definition
  48. Angular momentum definition
  49. Central force definition
  50. Newton's law of universal gravitation statement
  51. Celestial mechanics definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Celestial_mechanics
About this page
Tag9Q1NZ7EC0J
URIm2://9Q1NZ7EC0J
Kinddefinition
WritingW0 stub
MSC70F15, 70F10, 85A04
DefinesCelestial mechanics
UsesNewton's law of universal gravitation, n-body problem, Kepler problem, Hamiltonian system
Changed3 November 2026

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