Mira-Cristiana Anisiu

PDEs in the Inverse Problem of Dynamics

The basic equations are exposed for the following version of the inverse problem of dynamics: determine the two-dimensional potential compatible with a given family of orbits, traced by a material point. If the potential is known in advance, a nonlinear equation is satisfied by the function representing the family of orbits. Its solutions are studied in the presence of additional information on the family. The possibility of programming the motion of a material point in a preassigned region of the plane is also considered.

Programmed motion for a class of families of planar orbits

Taking as a guide the case of the set of monoparametric families y = h (x )+c , for which Szebehelys equation can be solved by quadratures for the potential V (x ,y ) generating the given set of orbits, we propose the following programmed motion problem : can we manage so as to have members of the given set inside a preassigned domain T 2 of the xy plane? We come to understand that, among the various inequalities by means of which T can be ascribed, the simplest is b (x ,y ) 0 where, for each h (x ), the function b (x ,y ) is related to the kinetic energy of the moving point (equations (19)-(21)). We then proceed to show that, in general, if b (x ,y ) satisfies two conditions (equations (39) and (40)), the answer to our question is affirmative: on the grounds of the given appropriate b (x ,y ), a function h (x ) is found, associated with a certain potential V (x ,y ) creating members of the family y = h (x )+c inside the region b (x ,y ) 0. Some special cases which stem from the method are studied separately. The limitations and also the promising features of the method developed to face the above inverse problem are discussed.

Families of orbits compatible with galactic potentials

We find galactic potentials described by polynomial perturbations of harmonic oscillators, which are compatible with families of orbits xpy = const (p ≠ 0). To this aim we apply the techniques of the planar inverse problem of dynamics.

TWO-DIMENSIONAL INVERSE PROBLEM OF DYNAMICS FOR FAMILIES IN PARAMETRIC FORM

The two-dimensional inverse problem of dynamics is considered for nonconservative force fields, both in inertial and rotating frames. The families of curves are given in parametric form x = F(λ, b), y = G(λ, b), b varying along the monoparametric family of planar curves and λ being the parameter describing a specific curve. The special case of the force fields generated by a potential in an inertial field, already studied by Bozis and Borghero, is derived as well as the corresponding one in rotating frames.

The Planar Inverse Problem of Dynamics

We consider the following version of the inverse problem of Dynamics: given a monoparametric family of planar curves, find the force field, conservative or not, which determines a material point to move on the curves of that family.

Spatial Families of Orbits in 2D Conservative Fields

In the framework of the 3D inverse problem of dynamics, we establish the conditions which must be fulfilled by a spatial family of curves to possibly be described by a unit mass particle under the action of a 2D potential V = ν(y,z), and give a method to find the potential.

Construction of 3D potentials from a preassigned two-parametric family of orbits

One of the main problems of astrophysics is to determine the mean field potential of galaxies. The astronomical observations, as well as the numerical simulations, lead to the determination of families of star orbits in a galaxy. On this basis, using the tools of the inverse problem of dynamics, it is possible to find the gravitational potential which gives rise to such motions. The problem can be formulated in various ways; we consider here that the particle trajectories are given by a spatial two-parameter family of curves. From these trajectories we obtain certain functions known as orbital functions, and look for potentials of a special form. If the orbital functions satisfy some differential conditions, a step-by-step procedure offers the expression of the potential. Similar problems arise in thermodynamics and nuclear physics, where axially symmetric potentials are used as models for deformed nuclei.