Proceedings of the Celestial Mechanics Conference: The search for asymmetric periodic orbits in the restricted problem of three bodies

. The paper contains a discussion of periodic orbits in the restricted problem by K. Schwarzschild's method with numerical applications using Leverrier's developments of the coefficients in the disturbing functions. The problem is restricted to the periodic orbits in the vicinity of commensurabilities of the first order for which (p + I)n - pn' = (Tn, with I small compared to unity, n being the mean motion of the small mass, n' that of Jupiter. These developments show that asymmetric periodic orbits exist for p = + I, = + 2, while an indication is found that an asymmetric periodic orbit also exists for p = + 3 but for a higher value of the eccentricity than for which the Leverrier expression converges numerically. For orbits with negative values of p, (n > n') no asymmetric periodic orbits were found-in agreement with a result obtained by S. C. Van Veen. Numerical integrations with an IBM computer were used to obtain numerical verifications. The results for a sym- metrical and an asymmetrical periodic solution obtained by this procedure for the case p = + I are given in Table II.