By Prund A. H.

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**Additional info for A Vapore-Free Vacuum Seal(en)(2s)**

**Example text**

First of all we rewrite the Hamiltonian H in (100) in a form suitable to apply the KAM Theorem of [P¨o1]. Introducing the coordinate y ∈ Rn around each torus T (J0 ) in the usual way J = J0 + y, the Hamiltonian H can be developed as H(y, ψ, z, z; ω) := H(J0 + y, ψ, z, z) = N + P (109) where N := hε (J0 )+ hε (J0 ) · y+ εΩ(J0 )zz and P = P (y, ψ, z, z; J0 ) = P1 + P2 + P3 + P4 with P1 := hε (J0 + y) − hε (J0 ) − hε (J0 ) · y = O(|y|2 ), P2 := ε Ω(J0 + y) − Ω(J0 ) zz = O(ε|y||z||z|), P3 := εg(J0 + y, z, z) = O ε(|z| + |z|)3 , P4 := ε3 f (J0 + y, x, z, z) = O(ε3 ) .

5, with n = m = 2. Let also Ω := εΩ(Jε ), M := 2 ∂J hε (Jε ) and M := 12 − eiεΩ(Jε )T . Notice that, if 0 < εT ≤ 1/c1 is small enough (namely c1 is large enough), then |M−1 | is bounded by a constant (independent on ε). We look for T -periodic solutions of the Hamiltonian system (108) of the form ζ = ζˆψ0 + (J, ψ, z, z) . Set −ε4 ∂ψ f ( ) J −∂J2 hε (Jε ) J + ∂J hε (Jε + J) − ∂J hε (Jε ) + ε∂J Ω(Jε + J)z · z+ , P ψ = + ε∂J g + ε4 ∂J f ( ) z 4 i ε Ω(Jε + J) − Ω(Jε ) · z + ε∂z g + ε ∂z f ( ) where the star above denotes, for short, := (Jε + J, ψ0 + ωt + ψ, z, z).

There exists a symplectic set of variables20 (I0 , ϕ0 , p0 , q0 ) ∈ R2 × T2 × R2 × R2 20 With symplectic form dI0 ∧ dϕ0 + dp0 ∧ dq0 . 30 where I0 = Λ∗ ∈ I and I ⊂ L ⊂ R2 is a suitable two-dimensional open cube centered at Λ∗0 , such that the Hamiltonian of the spatial planetary three-body problem takes the form H0 (I0 , ϕ0 , p0 , q0 ) = h0 (I0 ) + f0 (I0 , ϕ0 , p0 , q0 ) , (95) with 1 2 κi mi 3 1 , κi := , 2 2 2 i=1 I0i ε m0 (m0 + mi ) f0 := εf1 (I0 , p0 , q0 ) + εf2 (I0 , ϕ0 , p0 , q0 ) , h0 := − κ ¯3 < κi < 1 , 2 2 2 Ωj (I0 )(p20j + q0j ) + f1 (I0 , p0 , q0 ) , f1 := f1,0 (I0 ) + j=1 T2 f2 dϕ0 = 0 , sup |f1 | ≤ const|(p0 , q0 )|4 .