R. A. Broucke’s research while affiliated with University of California, Los Angeles and other places

This paper investigates the equinoctial orbit elements for the two-body problem, showing that the associated matrices are free from singularities for zero eccentricities and zero and ninety degree inclinations. The matrix of the partial derivatives of the position and velocity vectors with respect to the orbit elements is given explicitly, together with the matrix of inverse partial derivatives, in order to facilitate construction of the matrizant (state transition matrix) corresponding to these elements. The Lagrange and Poisson bracket matrices are also given. The application of the equinoctial orbit elements to general and special perturbations is discussed.

This article considers the two-dimensional elliptic restricted three-body problem, and in particular some of its aspects related to regularization and periodic collision orbits. The mechanism of regularization with Birkhoff coordinates and with the energy differential equation is described. Then the initial conditions for collision orbits are established. The theory is illustrated with the description of a new family of symmetric periodic collision orbits. It is shown how this family is related to the work of Moulton, Darwin and Stromgren in the circular restricted problem, and also to the Earth-Moon mass ratio. In the high eccentricity ranges, some relations with the triple collision problem are pointed out. The differential corrections have been made with an automatic search technique which is described in the appendix.

Matrizant of two body problem or integrable dynamic system for applications to computing perturbations and differential correction of orbits with observations

A systematic study has been made of periodic orbits in the two-dimensional, elliptic, restricted three-body problem. All ranges of eccentricities, from 0 to 1, and mass-ratios, from 0 to 1/2, have been investigated. Eleven hundred periodic orbits have been obtained. It is concluded that the elliptic problem behaves in a way which is completely different from the circular problem. The main difference is in the stability properties of the periodic orbits. Because of the nonexistence of the Jacobi integral (the elliptic problem is not conservative), the characteristic equation of the monodromy matrix does not have a pair of unit roots, in general. The stability is denned by two real numbers (stability indices) rather than one. Because of that, there are seven general classes of periodic orbits, according to their stability properties. The stability of the periodic orbits has been determined by numerically integrating the variational equations with a recurrent power series method. The results are in contrast with the circular problem, where there are only three classes of orbits (stability, even instability, and odd instability): in the elliptic problem there are one stable class and six unstable classes. The elliptic, restricted three-body problem can be considered as the prototype of all nonintegrable, nonconservative Hamiltonian systems, and in this paper, probably for the first time, a classification of the multipliers is given for these systems. © 1969 American Institute of Aeronautics and Astronautics, Inc., All rights reserved.