On the Kolmogorov Set for many body problems

Abstract

We study existence and estimate the measure of the invariant set of the Hamiltonian system, known as many-body problem, governed by Hplt(µ y, x) = 1 iN | yi|2 2 ~mi - ~mi ^mi |xi| + µ 1 i<jN y i yj ¯m0 - ¯mi ¯mj |xi - xj| , (0.1) where y, x (R3 )N and ^mi := ¯m0 + µ ¯mi , ~mi := ¯m0 ¯mi ¯m0 + µ ¯mi which correspons to the gravitational interaction of the 1 + N masses ¯m0, µ ¯m1, , µ ¯mN , from the point of view of Arnol'd's Fundamental Theorem for properly degenerate system, which we refine as theorem 1. Assume that (ft0) H(I, , p, q) = h(I) + f(I, , p, q) is realanalytic on I x T¯n x B2^n r (0), with I an open, bounded, connected subset of R¯n ; (ft1) h is a diffeomorphism of an open neighborhood of I, with non degenerate Jacobian = 2 h on such neighborhood; (ft2) the mean perturbation ¯f(I, p, q) := 1 (2)¯n T ¯n f(I, , p, q)d has the form ¯f = ¯f = f0(I) + 1 im i(I)Ji + 1 2 1 i, jN Aij(I)JiJj + o4 + o6 with Ji := p2 i + q2 i 2 and o4/|(p, q)|4 0; (ft3) = (1, , ^n) is 4-non resonant, i.e. , |(I) k| = 0 for any I ¯I, k = (k1, , km) Zm : 1 im |ki| 4 (0.2) (ft4) A is non singular on I, i.e. , detA(I) = 0 for any I ¯I . 1 Then, for any sufficiently large b > 0, there exist r, 0 < c <1 < C such that, for any 0 < r < r and 0 < < c(log r-1 )-2b (0.3) a set V (, r) I x T¯n x B2^n ¯r (0) of the same measure of I x T¯n x B2^n r (0) and an invariant set K(, r) V (, r) ("Kolmogorov set") with measure meas K (, r) 1 - C1/2 (log r-1 b - Cr1/2 m eas V (, r)

(0.4) consisting of n = ¯n + ^n-dimensional invariant tori where the motion is analytically conju- gated to Tn + t. The tori frequencies may be chosen ( r5/2 , )-Diophantine. We apply th1 to the plane Problem and, using the set of Boigey-Deprit variables, to the spatial problem. Using only a partial reduction, we prove (inductively) that, for N 3, the many body problem verifies the assuptions of theorem 1 allowing to find KAM tori with 3N - 1 Diophantine frequencies and estimating the measure of the invariant set as prescribed by equation (0.4), where represents the masses and r eccentricities and co-inclinations. We also prove that, reducing furtherly the system, in the range of small eccentricities and inclinations, the secular perturbation can be put into the form ¯fplt = ^f0 plt(, G) + 1 iN si(, G) 2 i + 2 i 2 + 1 iN-2 zi(, G) p2 i + q2 i 2 + O(3) where the first Birkhoff invariants (s1, , sN ), (z1, , zN-2)do not satisfy any linear relation. This allows us to apply an analytic first order properly KAM Theory such as the one developed in Chierchia Pusateri 2008 without any modification of the Hamiltonian, hence gaining some information on the motion of the system (on the KAM of the tori and their amount of irrationality).