Download On the stability of properly-degenerate hamiltonian systems by Biasco L., Chierchia L. PDF

April 4, 2017 | Programming | By admin | 0 Comments

By Biasco L., Chierchia L.

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

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Sixth step: angle variable (positive energy); estimate on analyticity radius. Fix E and observe that χ− (E, 0) = 0, χ− (E, ±ψ0 (E)) = ±π/2. Consider the case E = E1 ∈ R. As in the positive energy case we find χ− (E1 , (−ψ0 (E1 ), ψ0 (E1 ))) = (−π/2, π/2) and χ− (E1 , (0, ±i∞)) = (0, ±is− (E1 )), where s− (E1 ) := √ 2( ∞ dP − −1 ) dE dψ E1 + ε(1 + cosh ψ) 0 . 1 − In fact, χ− (E1 , M− p (E1 )) = { |Req| < π/2 } = 2 M. Let now, in general, E = E1 + iE2 ∈ E . Then, − χ− (E, Tσ ∩ M− p (E)) ⊇ Ts− (E,σ) ∩ Mp (E) , s− (E, σ) := where min Imχ− (E, t + iσ).

Sixth step: angle variable (positive energy); estimate on analyticity radius. Fix E and observe that χ− (E, 0) = 0, χ− (E, ±ψ0 (E)) = ±π/2. Consider the case E = E1 ∈ R. As in the positive energy case we find χ− (E1 , (−ψ0 (E1 ), ψ0 (E1 ))) = (−π/2, π/2) and χ− (E1 , (0, ±i∞)) = (0, ±is− (E1 )), where s− (E1 ) := √ 2( ∞ dP − −1 ) dE dψ E1 + ε(1 + cosh ψ) 0 . 1 − In fact, χ− (E1 , M− p (E1 )) = { |Req| < π/2 } = 2 M. Let now, in general, E = E1 + iE2 ∈ E . Then, − χ− (E, Tσ ∩ M− p (E)) ⊇ Ts− (E,σ) ∩ Mp (E) , s− (E, σ) := where min Imχ− (E, t + iσ).

0 0 0 42 It 31 = √ √ − 2 g(E0 , q¯n ) → − 2 g(E0 , q0 ) = −¯ p0 = p0 . The continuity for fixed Q0 and Pn → P0 is checked similarly. Sixth step: angle variable (positive energy); estimate on analyticity radius. Fix E and observe that χ− (E, 0) = 0, χ− (E, ±ψ0 (E)) = ±π/2. Consider the case E = E1 ∈ R. As in the positive energy case we find χ− (E1 , (−ψ0 (E1 ), ψ0 (E1 ))) = (−π/2, π/2) and χ− (E1 , (0, ±i∞)) = (0, ±is− (E1 )), where s− (E1 ) := √ 2( ∞ dP − −1 ) dE dψ E1 + ε(1 + cosh ψ) 0 . 1 − In fact, χ− (E1 , M− p (E1 )) = { |Req| < π/2 } = 2 M.

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