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Additional info for On the stability of properly-degenerate hamiltonian systems with two degrees of freedom

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J. Appl. Math. Mech. 45, No. : An exponential estimate of the time of stability of nearly–integrable Hamiltonian systems. Russ. Math. Surv. 32, No. : Nekhoroshev estimates for quasi–convex Hamiltonian Systems Math.

In41 W 1 (E0 ) = { χ− (E0 , q), with q ∈ M(E0 ) } ◦ and W 2 (E0 ) = { π − χ− (E0 , q), with q ∈ M(E0 ) } everything is analytic, Let us check what happens on42 ∂W1 (E0 ) ∩ ∂W2 (E0 ). Let us suppose (to fix ideas) that Q0 ∈ W1 (E0 ). Let Qn → Q0 . Then we have to check that (pn , qn ) := φ−1 (P0 , Qn ) → φ−1 (P0 , Q0 ) =: (p0 , q0 ). If Qn ∈ W1 (E0 ), then, continuity in q0√comes from the definition of χ− . Also, since pn ∈ C+ √ and Img(E0 , qn ) < 0, we have pn := 2 g(E0 , qn ) → 2 g(E0 , q0 ) =: p0 .

Finally, there is a unique p∗ ∈ C+ ∪ {0} such that E ∗ = E(p∗ , q ∗ ). Fix E. Observe that χ+ (E, q) is 2π-periodic and that χ+ (E, 0) = 0, χ+ (E, ±π) = ±π. Let us first consider the case E = E1 ∈ R. In such a case38 χ+ (E1 , (−π, π)) = (−π, π), χ+ (E1 , (0, ±i∞)) = (0, ±is+ (E1 )), χ+ (E1 , (±π, ±π ± iψ0 (E1 ))) = (±π, ±π ± is+ (E1 )), where s+ (E1 ) := √ 2( dP + −1 ) dE ∞ dψ E1 + ε(1 + cosh ψ) 0 . + In fact, it is χ+ (E1 , M+ p (E1 )) = Ts+ (E1 ) . Let, now, E = E1 + iE2 ∈ E . In this case, χ+ (E, Tσ ∩ M+ p (E)) ⊇ Ts( E,σ) where s+ (E, σ) := min Imχ+ (E, t + iσ) , t∈(−π,π) and, as in the case E = E1 , s+ (E1 , σ) = Imχ+ (E1 , iσ) := √ 2( dP + −1 ) dE σ dψ E1 + ε(1 + cosh ψ) 0 .

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