5 research outputs found
Linear stability analysis of resonant periodic motions in the restricted three-body problem
The equations of the restricted three-body problem describe the motion of a
massless particle under the influence of two primaries of masses and
, , that circle each other with period equal to
. When , the problem admits orbits for the massless particle that
are ellipses of eccentricity with the primary of mass 1 located at one of
the focii. If the period is a rational multiple of , denoted ,
some of these orbits perturb to periodic motions for . For typical
values of and , two resonant periodic motions are obtained for . We show that the characteristic multipliers of both these motions are given
by expressions of the form in the limit . The coefficient is analytic in at and
C(e,p,q)=O(e^{\abs{p-q}}). The coefficients in front of e^{\abs{p-q}},
obtained when is expanded in powers of for the two resonant
periodic motions, sum to zero. Typically, if one of the two resonant periodic
motions is of elliptic type the other is of hyperbolic type. We give similar
results for retrograde periodic motions and discuss periodic motions that
nearly collide with the primary of mass
Uniform distribution of integral points on 3-dimensional spheres via modular forms
AbstractThe problem of the asymptotic distribution of integral points on a sequence of expanding spheres x2 + y2 + z2 = mi [first penetrated by Linnik with his ergodic method] is shown to be attackable by analytic methods, especially via the theory of modular forms. Success depends on a new estimate for the Fourier coeff. of cusp forms of half integral dimension on the theta group, which we obtain after improving upon previously known estimates of the associated Kloostermann sums