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An introduction to the classical three-body problem

5. Restricted three-body problem

The restricted three-body problem is a simplified version where one of the masses m₃ is assumed much smaller than the primaries m₁ and m₂. Thus the primaries move in fixed Keplerian orbits which are not affected by m₃, as in the Sun-Earth-Moon system. The planar case has a 4-dimensional phase space with one known conserved quantity, the Jacobi integral.

E = ½m₃ṙ² + ½m₃r²φ̇² − Gm₃(m₁/r₁ + m₂/r₂) ≡ T + Veff   (15)

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