By Yang W.P., Du M.W.

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In order to remove this singularity (as ε → 0), we introduce new symplectic variables29 r (i) R(i) (i) , x = . 34) where 2 1 1 mi M i m1 m2 |X (i) |2 − (i) , H1tb := ε X (1) · X (2) − , 2 (1) |x | ε m0 |x − x(2) | i=1 2mi mi 1 mi mi := , Mi := 1 + . t. t. the symplectic form dX ˜ ∧ d˜ h(αX, x) with initial data flow governed by the Hamiltonian αβ 1 1 ˜ X(0) = α X(0) and x˜(0) = β x(0). 28), it follows that mi and Mi are bounded and bounded away from zero (uniformly in ε): κ ¯ ≤ mi ≤ 1 , 1 ≤ Mi ≤ 2 .

2 Reduction of the angular momentum The conservation of the total angular momentum allows to lower by two more units the number of degrees of freedom (“reduction of the angular momentum”). , motions with vanishing mutual inclinations. 40) is independent of ζ ∗ . More precisely, let rj∗ := p∗2j + q∗2j and r ∗ := (r1∗ , r2∗ ). 2), one has H1∗ (Λ∗ , η ∗ , (r1∗ cos ζ1∗ , r2∗ cos ζ2∗ ), λ∗ , ξ ∗ , (r1∗ sin ζ1∗ , r2∗ sin ζ2∗ )) = H1∗ (Λ∗ , η ∗ , r ∗ , λ∗ , ξ ∗ , (0, 0)). 43) In particular, Zj∗ := − rj∗ 2 2 are analytic integrals: {Zj∗ , H1∗ } = 0.

56) 2 , ¯ i are real. Furthermore, from the definition of showing, in particular, that Li and hence Ω (2) (1) Laplace coefficients, there follows that b3/2 (α) < b3/2 (α) for α ∈ (0, 1). , if the mutual inclination is sufficiently small), then, uniformly in Λ , inf Li > c¯ > 0 , inf L2 − L1 > c¯ > 0 . 58) ∗ which sends f1,2 into f1,2 := c L1 (Λ∗ ) · ((η1 )2 + (ξ1 )2 ) + L2 (Λ∗ ) · ((η2 )2 + (ξ2 )2 ) . 60) for a suitable function ˆ. 1 is completed. References [A63] Arnold, V. : Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat.