Arnold Rom

The symbolic manipulation of poisson series

Poisson series of three variables are manageable symbolically through a set of formal subroutines written partially in the IBM 7094 machine language, but to be called in the FORTRAN language for use in FORTRAN programs. An effort has been made to supply those operations which are most required by Celestial Mechanics. The routines are entirely self-contained subroutines and require only standard FORTRAN input-output units 5 and 6; they are designed to avoid waste and overflow of core storage space.

TROJAN ORBITS. II. BIRKHOFF'S NORMALIZATION,

Abstract 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 here how such a Birkhoffs 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.

Asymptotic representation of the cycle of Van der Pol's equation for small damping coefficients

RésuméLe cycle limite de léquation de Van der Pol est développé en série potentielle selon le coefficient damortissement ε; les coefficients de la série sont des sommes trigonométriques finies ayant le temps pour argument. Du fait que ces développements sont produits sur ordinateur de manière complètement automatique, on a pu les pousser jusquà une puissance élevée de ε. Aussi, même pour des valeurs de ε aussi, élevées que 1.75, les estimations de lamplitude et de la période fournie par les séries ne sécartent pas de plus de 10−3 des valeurs correctes; ces dernières ont été calculées en intégrant par séries de puissances récurrentes léquation de Van der Pol et léquation aux variations qui lui est associée.

NATURAL ORBITS OF THE FIRST KIND IN THE RESTRICTED PROBLEM

Abstract Lindstedts method is applied to expand analytically the natural families of periodic orbits of the first kind in the restricted problem of three bodies. The mass ratio is kept as a literal variable; the series proceed in the powers of Hills ratio of periods. They cover all four cases at once, namely, the direct or retrograde orbits for either inferior planets or for satellites. When the mass ratio is given a numerical value, the expansions contain fewer terms and thus can be carried up to degree 17 within a 32K core of an IBM 7094. For the system Sun-Jupiter, the initial conditions provided by the series have been corrected to yield the beginnings of all four natural families, and the characteristic exponents have been computed. Comparison between the values computed out of the series and their improvement by numerical integrations and successive variational corrections show that, within an accuracy of one part in ten thousand, the series represent moderately well even the orbit of J VIII; the retrograde planetary orbits are fairly well covered up to 90% of the distance from Sun-Jupiter, whereas the direct planetary orbits are covered only up to 60% of that distance.