A simple dynamical model is employed to study the possible orbital evolution of scattered planets and phase plane analysis is used to classify the parameter space and solutions. Our results reconfirm that there is always an increase in eccentricity when the planet was scattered to migrate outward when the initial eccentricity is zero. Applying our study on the Solar System and considering the e...

We have constructed families of possible orbits for a collection of stars located within 0.5” of Sgr A*. These are constrained by observed stellar positions and angular proper motions as listed in Ghez et al. (1998). Reconstructing these orbits allows us to make useful predictions for the likely positions of these stars in future observations. We also show that some of these stars may exhibit s...

We illustrate a completely analytic approach to Mel’nikov theory, which is based on a suitable extension of a classical method, and which is parallel and – at least in part – complementary to the standard procedure. This approach can be also applied to some “degenerate” situations, as to the case of nonhyperbolic unstable points, or of critical points located at the infinity (thus giving rise t...

Kepler’s equations are considered as central to Celestial Mechanics since their solutions permit ’to find the position of a body for a given time’. To obtain a kinematical deduction of them, only one new concept is introduced with the name of ”maccel”. By assuming the simultaneous conservation of the maccel and of the angular momentum, it is shown that the position vector always describes a con...

The evolution of the lunar spin axis is studied. Prior work has assumed that the inclination of the lunar orbit is constant and that the node regresses uniformly. This work takes into account the nonconstant inclination and nonuniform regression of the node as determined from averaged models of the motion of the lunar orbit. The resulting dynamics is considerably more rich, exhibiting additiona...

In this expository and resources chapter we review selected aspects of the mathematics of dynamical systems, stability, and chaos, within a historical framework that draws together two threads of its early development: celestial mechanics and control theory, and focussing on qualitative theory. From this perspective we show how concepts of stability enable us to classify dynamical equations and...

Using simple geometrical arguments, we paint an overview of the variety of resonant structures a single planet with moderate eccentricity (e . 0.6) can create in a dynamically cold, optically thin dust disk. This overview may serve as a key for interpreting images of perturbed debris disks and inferring the dynamical properties of the planets responsible for the perturbations. We compare the re...