N-Body Problem

3 min read#systems#emergence#gravitation

Predicting the motion of many masses pulling on one another through gravity — the original many-body problem.

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N-Body Problem

The N-body problem asks a question as old as astronomy: given N masses, each pulling on every other through gravity, how do they move? Each body feels the summed attraction of all the rest, and in turn changes how it pulls on them. The state is the set of all positions and velocities; the rule is Newton's law of gravitation applied to every pair at once. It is the founding example of a many-body Dynamical System, and the place where the dream of perfect prediction first ran into trouble.

The force on body i is the vector sum over all the others:

\ddot{\mathbf r}_i = G \sum_{j \neq i} m_j \frac{\mathbf r_j - \mathbf r_i}{\lVert \mathbf r_j - \mathbf r_i \rVert^3}.

With N bodies that is a coupled system of N second-order vector equations — every body in every other body's equation. The coupling is what makes it hard, and what makes it interesting.

Two bodies are easy, three are not

For N = 2 the problem is completely solvable: Kepler's ellipses, parabolas, and hyperbolas, written down in closed form. The two-body motion is integrable — there are exactly as many conserved quantities (energy, momentum, angular momentum) as degrees of freedom, and they pin the motion onto simple curves.

Add a single body and that good fortune evaporates. The Three-Body Problem has no general closed-form solution, and for most initial conditions the motion is chaotic. The jump from two to three is one of the great surprises in the history of physics: difficulty does not grow gently with N, it arrives all at once.

Six gravitating masses. Watch how close encounters fling bodies onto new paths — the hallmark of strong coupling.

What survives the chaos

Even when individual orbits are unpredictable, the whole system obeys strict bookkeeping. Total momentum, total angular momentum, and total energy are conserved, and the center of mass drifts in a straight line forever. These conservation laws constrain the motion onto a lower-dimensional surface in State Space without ever taming it completely.

See also

Related topics

  • ChaosDeterministic systems whose long-term behavior is so sensitive to initial conditions that it becomes effectively unpredictable.
  • Three-Body ProblemThree gravitating masses — the smallest many-body system, and the birthplace of chaos in celestial mechanics.
  • Complex SystemsMany simple parts following local rules give rise to global behavior that none of the parts contains — the science of emergence.
  • Dynamical SystemA state together with a fixed rule that determines how that state evolves in time.
  • AttractorThe set a system settles onto after transients die away — a point, a loop, or something stranger.
  • State SpaceThe space of all possible states of a system, in which each point is a complete snapshot and motion traces a trajectory.

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