Motion on a given surface: potentials producing geodesic lines as trajectories

In the light of inverse problem of dynamics, we consider the motion of a material point on an arbitrary two-dimensional surface, submersed in OE3. We study twodimensional potentials which produce a mono-parametric family of geodesic lines as trajectories. We establish a new, non-linear partial differential condition for the potential function V = V(u, v). With the aid of this condition, we examine if a given potential produces a family of geodesic lines on a certain surface or not. On the other hand, we can check if a given family of regular orbits is indeed a family of geodesic lines on a certain surface and then find the potential function V = V(u, v) which gives rise to this family of orbits. Special cases are also studied and pertinent examples are worked out

Planar resonant periodic orbits in Kuiper belt dynamics

In the framework of the planar restricted three body problem we study a considerable number of resonances associated to the Kuiper Belt dynamics and located between 30 and 48 a.u. Our study is based on the computation of resonant periodic orbits and their stability. Stable periodic orbits are surrounded by regular librations in phase space and in such domains the capture of trans-Neptunian object is possible. All the periodic orbits found are symmetric and there is evidence for the existence of asymmetric ones only in few cases. In the present work first, second and third order resonances are under consideration. In the planar circular case we found that most of the periodic orbits are stable. The families of periodic orbits are temporarily interrupted by collisions but they continue up to relatively large values of the Jacobi constant and highly eccentric regular motion exists for all cases. In the elliptic problem and for a particular eccentricity value of the primary bodies the periodic orbits are isolated. The corresponding families, where they belong to, bifurcate from specific periodic orbits of the circular problem and seem to continue up to the rectilinear problem. Both stable and unstable orbits are obtained for each case. In the elliptic problem the unstable orbits found are associated with narrow chaotic domains in phase space. The evolution of the orbits, which are located in such chaotic domains, seems to be practically regular and bounded for long time intervals.Comment: preprint, 20 pages, 10 figure

Three-dimensional potentials producing families of straight lines (FSL)

We identify a given two-parametric family of regular orbits in the 3-D Cartesian space by two functions α and β. Then, from the inverse-problem viewpoint, we find three necessary and sufficient conditions that the functions α and β must satisfy when the given family is a two-parametric family of straight lines (FSL) and is actually created by a potential V. Some pertinent theorems are shown and several examples are worked out

Semi-Separable Potentials as Solutions to the 3D Inverse Problem of Newtonian Dynamics

We study the motion of a test particle in a conservative force-field. Our aim is to find three-dimensional potentials with symmetrical properties, i.e., V(x,y,z)=P(x,y)+Q(z), or, V(x,y,z)=P(x2+y2)+Q(z) and V(x,y,z)=P(x,y)Q(z), where P and Q are arbitrary C2-functions, which are characterized as semi-separable and they produce a pre-assigned two-parametric family of orbits f(x,y,z) = c1, g(x,y,z) = c2 (c1, c2 = const) in 3D space. There exist two linear PDEs which are the basic equations of the Inverse Problem of Newtonian Dynamics and are satisfied by these potentials. Pertinent examples are presented for all the cases. Two-dimensional potentials are also included into our study. Families of straight lines is a special category of curves in 3D space and are examined separately

Real and Complex Potentials as Solutions to Planar Inverse Problem of Newtonian Dynamics

We study the motion of a test particle in a conservative force field. In the framework of the 2D inverse problem of Newtonian dynamics, we find 2D potentials that produce a preassigned monoparametric family of regular orbits f(x,y)=c on the xy-plane (where c is the parameter of the family of orbits). This family of orbits can be represented by the “slope function” γ=fyfx uniquely. A new methodology is applied to the basic equation of the planar inverse problem in order to find potentials of a special form, i.e., V=F(x+y)+G(x−y), V=F(x+iy)+G(x−iy) and V=P(x)+Q(y), and polynomial ones. According to this methodology, we impose differential conditions on the family of orbits f(x,y) = c. If they are satisfied, such a potential exists and it is found analytically. For known families of curves, e.g., circles, parabolas, hyperbolas, etc., we find potentials that are compatible with them. We offer pertinent examples that cover all the cases. The case of families of straight lines is referred to

Dynamical evolution of bodies in resonant regions in the outer solar system

The subject of the present phd. thesis is the dynamical evolution of small bodies which are located in a belt, beyond the orbit of neptune and is called Kuiper-Belt. In the first part of this work we computed families of planar and three-dimensional periodic orbits and studied their stability using the restricted 3-body problem with sun and neptune as primaries. We emphasized at 1/2, 2/3, 3/4 mean motion resonances. In the second part we constructed a symplectic mapping model in order to study the dynamical evolution of small bodies near the 3/4 resonance. In the third part we made a comparative study of the dynamics at the 2/3 & 3/4 resonances. Using a spectral analysis method of trajectories, we constructed maps which show regular and chaotic motion in planar and spatial problem. The results were used for the explanation of the present distribution of small bodies at 2/3 & 3/4 resonances.Αντικείμενο της παρούσης διατριβής αποτελεί η δυναμική εξέλιξη σωμάτων που βρίσκονται σε μια ζώνη πέρα από την τροχιά του Ποσειδώνα και καλείται ζώνη Kuiper. Στο πρώτο μέρος της εργασίας υπολογίσαμε οικογένειες επιπέδων και τρισδιάστατων περιοδικών τροχιών χρησιμοποιώντας το μοντέλο του περιορισμένου προβλήματος των 3-σωμάτων με τον ήλιο και τον Ποσειδώνα ως πρωτεύοντα. Μελετήσαμε επίσης την ευστάθεια τους. Δόθηκε ιδιαίτερη έμφαση στους συντονισμούς μέσης κίνησης 1/2, 2/3 & 3/4. Τα αποτελέσματα χρησίμευσαν στον καθορισμό της τοπολογίας του χώρου των φάσεων. Στο δεύτερο μέρος κατασκευάσαμε ένα συμπλεκτικό μοντέλο απεικόνισης με σκοπό να μελετήσουμε τη δυναμική εξέλιξη μικρών σωμάτων κοντά στο συντονισμό 3/4. Εξετάσαμε πολλούς τύπους τροχιών με διαφορετική αρχική φάση. Στο τρίτο μέρος κάναμε μια συγκριτική μελέτη της δυναμικής στους συντονισμούς 2/3 & 3/4. Χρησιμοποιώντας μια μέθοδο φασματικής ανάλυσης τροχιών, κατασκευάσαμε χάρτες που δείχνουν περιοχές κανονικής και χαοτικής κίνησης στο χώρο φάσης. Τα αποτελέσματα χρησίμευσαν στην εξήγηση της κατανομής σωμάτων στους συντονισμούς αυτούς