Über periodische Bahnen im eingeschränkten Dreikörperproblem und in der Nähe der Kommensurabilitäten vom Typus (k + 1)/k

Abstract

Giittingen &lit 18 .\bbiltlungcn. (Eingegangen 1968 Mai 24)In eincr Gemcinschaftsarbeit fiihrten die Autoren im Winter 1966/67 numerische ITntersuchungen iibcr Form und Verhalten periodischer Bahnen des problbme restreint im Sonne-Jupiter-System und in der Nahe der Kommensurabilitaten vom Typus (k + x)/k durch, insbesondere fur die beiden Fallc 211 (Hekuba-Liicke) und 3/2 (Hilda-Gruppe). Dic Rcchnungen wurden mit einer schnellen IBhf-Rechenanlage ausgefiihrt und ergaben neue Einsichten in die Genealogic drr periodischen L6sungen bei groI3er Annlherung an die kritischen mittleren Bewegungen. Dariiber hinaus versuchte I-. CARPENTER cine hier nicht ausfiihrlich auseinandergesetzte Thcorie, die fur Bahncn mit nicht zu groDer Exzentrizitat konvergente trigonometrische Entwicklungen ermoglicht, wobei er seine gemeinsam init P. MUSEN erarbeitete Theoric dcr allgemeinen St6rungen in rechtwinkligen Koordinaten benutzte.The authors, collaborating during the winter 1966/67, performed numerical investigations on form and character of periodic solutions of the restricted problem of three bodies in the Sun-Jupitcr-System and in the neighbourhood of the resonances of the (k + r)/k type. The principal part of these computations which wcre achieved using a fast IBMcomputer refers to the cases 211 (Hecuba-gap) and 3/2 (Hilda-group). The results gave new insight into the genealogy of periodic orbits in the regions which do not contain solutions of POINCARO'S first gcnre. Moreover, L. CARPENTER tried to develop an analytical theory (here not explicitely described) which leads to convergent trigonometric series for the coordinates of periodic orbits with restricted eccentricities, using the method of general perturbations in rectangular coordinates recently published by P. MUSEN and himself.

Periodic Orbits in Trigonometric Series

“…The method described in this study has been applied in the computation of periodic orbits in the restricted problem of four bodies by Kolenkiewicz andCarpenter (1967 and and in the restricted problem of three bodies by Carpenter and Stumpff (1968). Further applications are being made by Deprit and Carpenter for locating elements of many new families or orbits in the problem of three bodies.…”