A global view of the isochronous trajectory families by Hamilton's hodographic transformation

An overall view of the characteristic features of the isochronous trajectory families is presented. The trajectories are assumed Keplerian, isoenergetic, and satisfying Lambert's conditions, that is the sum of radial distances of the two termini, and their linear separation are both constant. A geometric approach is adopted by using Hamilton's hodographic transformation. The connection between Hamilton's hodographic isochronism and the Lambert's theorem is briefly reviewed, and the interrelations among the Hamiltonian set of parameters and the Lambertian sets are developed. The totality of the isochronous trajectory families is then viewed in the velocity space. Various Lambertian invariants as well as some essential noninvariants are examined, and relevant formulas derived. The salient features of some particular isochronous families are then summarized; and Hamilton's time integral is briefly treated. The analysis is essentially two-dimensional, with its extension to the three-dimensional case finally discussed.