Trojan Orbits. II. Birkhoff's Normalization

摘要

In the Restricted Problem of Three Bodies for the system Sun-Jupiter, all terms of short and long period up to degree 13 are eliminated from the Hamiltonian expanded in power series in the neighborhood of an equilateral center of libration. The normalizing canonical transformation expresses the Cartesian phase variables as double Fourier series in two angle coordinates whose coefficients are power series in the square roots of two action momenta. In the normalized part of the transformed Hamiltonian, the angle coordinates are ignorable. It is shown how such a Birkhoff's normalization can be implemented automatically on a computer. As a first application, the characteristic exponents along the singular families of long and short period orbits originating at the equilibrium are developed as power series of the orbital parameter. (Author)