Teresinha de Jesus STUCHI

Departamento de Fisica Matematica
CCMN Instituto de Fisica..- Ilha do Fundao, CT, BLOCO A -3o.andar
Caixa Postal 68.528
CEP 21945-970 - Rio de Janeiro - RJ, 
BRASIL
TSTUCHI@IF.UFRJ.BR
 

Center Stable/Unstable Manifolds and the protect destruction of KAM tori in the $2D$ Hill problem

Abstract : The classical Hill Problem (HP) is a simplified version of the Restricted Three Body Problem (RTBP) where the distance of the two massive bodies is made infinite through the use of Hill's variables and a limiting procedure. In this way Hill's problem is obtained as zero--th order approximation of a power series in the mass ratio $\mu^{1/3}$. A Levi-Civita canonical regularization is used to obtain a polynomial Hamiltonian which is very convenient for numerical calculations. Also it conveys a general framework for many researchers of other areas than Celestial Mechanics. The final form of the Hamiltonian is: \begin{eqnarray} \nonumber

H(Q_1,Q_2,P_1,P_2)=\frac {1} {2} (Q_1^2+Q_2^2+P_1^2+P_2^2)+ 2({Q_1}^2+{Q_2}^2)(Q_2P_1-Q_1P_2) \cr \quad-4({Q_1}^6-3{Q_1}^4{Q_2}^2-3{Q_1}^2{Q_2}^4+{Q_2}^6). \end{eqnarray}\noindent

Up to fourth degree we have the Kepler problem in a rotating frame, which is integrable. The perturbation of degree six is due to the massive body and numerical evidence shows that it is non integrable. A rigorous proof of non--integrability is easier to obtain in the $3D$ case, but it is skipped on this report. In this approximation we retain only two of the three collinear Lagrangian equilibria. These points are usually denoted as $L_1$ and $L_2$ and, in this case, they are symmetric with respect to the origin.

As it is well known the $L_1$ and $L_2$ points are of saddle--center type. The centre manifold is foliated by symmetric periodic orbits known as Lyapunov orbits. From them we obtain, through numerical globalization, the stable/unstable manifolds of $L_1$ and $L_2$. A convenient Poincar\'e section exhibits the relation of these manifolds and the remaining KAM structure. The geometrical relation between this structure and the center--stable/unstable manifolds is our main concern in this report.

An interesting result is the persistence of KAM structure, even for large values of the energy, confined by the stable and unstable manifolds.