| Contents |
| KAM Theorem is a fundamental result
for Hamiltonian systems because it ensures the existence of a set, nowhere dense
but of positive measure, of points of the phase space which behave in a regular,
quasi-periodic way. We present a methodology for computer assisted proofs of the existence and the
KAM stability of an arbitrary periodic orbit for Hamiltonian systems.
In short, using interval arithmetics and rigorous ODE solvers we are able to get verified bounds for coefficients
of the Birkhoff normal form of a suitable Poincar\'e map. These estimates allow to check the KAM conditions.
We applied this algorithm to the 3-body problem and
we proved the KAM stability of the well-known figure eight orbit
and two selected orbits of the so called family of rotating Eights.
They are examples of the so called choreographic solutions
in which all bodies travel along the same curve with constant phase shift
(in case of rotating Eights in suitable rotating frame). |
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