For science, theoretical or applied, to significantly advance, researchers must use the most appropriate mathematical methods. A century and a half elapsed between Newton's development of the calculus and Laplace's development of celestial mechanics. One cannot imagine the latter without the former. Today, more than three-quarters of a century has elapsed since the birth of stochastic systems t...

Integrability, one of the classic issues in galactic dynamics and in general in celestial mechanics, is here revisited in a Riemannian geometric framework, where newtonian motions are seen as geodesics of suitable “mechanical” manifolds. The existence of constants of motion that entail integrability is associated with the existence of Killing tensor fields on the mechanical manifolds. Such tens...

We present a new analysis of the dynamics of the planetary system around the pulsar B1257+12. A semi-analytical theory of perturbation between terrestrial-mass planets B and C is developed and applied to improve multi-orbit timing formula for this object. We use numerical simulations of the pulse arrival times for PSR B1257+12 to demonstrate that our new timing model can serve as a toll to dete...

The spin-orbit problem in celestial mechanics describes the motion of an oblate satellite moving on a Keplerian orbit around a primary body. We apply the conjugate points criterion for the nonexistence of rotational invariant tori. We treat both the conservative case and a case including a dissipative effect modeling a tidal torque generated by internal nonrigidity. As a by-product of the conju...

The existence of planets in stellar binary (and higher order) systems has now been confirmed by many observations. The stability of planetary orbits in these systems has been extensively studied, but no precise stability criteria have so far been introduced. Therefore, there is an urgent need for developing stringent mathematical criteria that allow us to precisely determine whether a planetary...

We investigated the apsidal motion for the multi-planet systems. In the simulations, we found that the two planets of HD 37124, HD 12661, 47 Uma and HD 82943 separately undergo apsidal alignment or antialignment. But the companions of GJ 876 and υ And are only in apsidal lock about 0. Moreover, we obtained the criteria with Laplace-Lagrange secular theory to discern whether a pair of planets fo...

Reading the Principia [1] it is easy to realize that Newton, without being aware of the fact, has corrected the law of the fall of bodies formulated by Galilei. In the present paper we examine the consequences of such a fact and show that they affect the entire Celestial Mechanics, being related to the very small 43” per century of the perihelion shift of Mercury as well as to the substantial 5...

To characterize the dynamical stability of an asteroid orbit, the calculation of its Lyapunov-Time TL is a widely used mean in celestial mechanics. In the present work we investigated the effects of the used computer hardware and integration method on the outcome of such stability computations. We showed that for some asteroids the change of the employed numerical method can change the obtained...

We examine the corecollapse times of isolated, twomass-component star clusters using Fokker-Planck models. With initial condition of Plummer models, we find that the corecollapse times of clusters with M1/M2 ≫ 1 are well correlated with (N1/N2) (m1/m2) trh, where M1/M2 and m1/m2 are the light to heavy component total and individual mass ratios, respectively, N1/N2 is the number ratio, and trh i...