A Manifold of Periodic Orbits

Abstract

Abstract : In the restricted problem of three bodies, when the mass ratio is such that the characteristic exponents at L sub 4 are equal in pair, the triangular equilibrium is a point of ramification in the analytical manifold of periodic orbits emanating from L sub 4: the branch L sub 4 superscript s of short period orbits can be continued through L sub 4 by the branch L sub 4 superscript l of long period orbits, and this real analytical continuation is unique. The branch L sub 4 superscript l ends with an orbit traveled twice which is an element of L sub 4 superscript s. The branch L sub 4 superscript s meets its mirror image L sub 5 superscript s of short period orbits around L sub 5 on a symmetric orbit which is also an element of the branch L sub 3 of periodic solutions emanating from the collinear equilibrium L sub 3. Around L sub 4, the branches L sub 4 superscript l and L sub 4 superscript s are connected by bridges B(pL,qS) of periodic orbits which start from a long period orbit traveled p times and end with a short period one traveled q times. We have completely explored the bridges B(2L,3S), B(3L,4S) and B(4L,5S). (Author)