Francesco Borghero

On Szebehely's problem for holonomic systems involving generalized potential functions

We consider the problem of finding the generalized potential function V = Ui(q1, q2,..., qn)qi + U(q1, q2,...;qn) compatible with prescribed dynamical trajectories of a holonomic system. We obtain conditions necessary for the existence of solutions to the problem: these can be cast into a system of n − 1 first order nonlinear partial differential equations in the unknown functions U1, U2,...;, Un, U. In particular we study dynamical systems with two degrees of freedom. Using ‘adapted’ coordinates on the configuration manifold M2 we obtain, for potential function U(q1, q2), a classic first kind of Abel ordinary differential equation. Moreover, we show that, in special cases of dynamical interest, such an equation can be solved by quadrature. In particular we establish, for ordinary potential functions, a classical formula obtained in different way by Joukowsky for a particle moving on a surface.

A two-dimensional inverse problem of geometrical optics

In the framework of geometrical optics we consider the inverse problem consisting of obtaining refractive indexes n = n(x, y) of a two-dimensional transparent heterogeneous isotropic (dispersive or not) medium from a known (observed or given) family f(x, y) = c0 of planar light rays of a definite colour. We establish a first-order linear partial differential equation relating the assigned family of light rays with all possible refractive indexes compatible with this family. Using this equation we derive certain criteria to check whether a given family of rays can be traced in the presence of a refractive index, which we assume in advance to be either radial or homogeneous of any degree m. We give appropriate examples for the two special cases and also an example for the general case.

Isoenergetic Families of Planar Orbits Generated by Homogeneous Potentials

In the light of the inverse problem of dynamics, we study in some detail the monoparametric isoenergetic families of planar orbits f (x, y) = c, created by homogeneous potentials V (x, y). For any preassigned family of orbits and for any degree of homogeneity m of the potential, we offer the criteria which the family has to satisfy so that it can result from such a potential. When the criteria are fulfilled, Szebehely’s first order partial differential equation for the unknown potential V (x, y) is substantially simplified and, in most of the cases, can be solved to completion and uniqueness.

Family boundary curves for holonomic systems with two degrees of freedom

The notion of the family boundary curves (FBC) is introduced for that version of the inverse problem of Lagrangian dynamics which deals with the determination of the potential V(u,v) under which a given monoparametric family of curves f(u,v)=c, on the configuration manifold (M2,g) of a conservative holonomic system with n=2 degrees of freedom, can be described as dynamical trajectories (orbits) of the representative point. It is shown that, in general, curves of the family f(u,v)=c generated by a class of potentials V(u,v) are actual orbits only in a subregion of the region where they are defined as geometrical entities. (In general, the FBC are distinct from the well known zero velocity curves (ZVC), the later referring to orbits of the same constant energy). If, however, the holonomic system is subject to non-conservative generalized forces, it is shown that we can always find many pairs of such forces (Q1,Q2) giving rise to any family of trajectories lying in any pre-assigned (open or closed) region of the configuration space. Three examples are presented to account both for conservative and non-conservative forces.

Two solvable problems of planar geometrical optics

In the framework of geometrical optics we consider a two-dimensional transparent inhomogeneous isotropic medium (dispersive or not). We show that (i) for any family belonging to a certain class of planar monoparametric families of monochromatic light rays given in the form f(x,y)=c of any definite color and satisfying a differential condition, all the refractive index profiles n=n(x,y) allowing for the creation of the given family can be found analytically (inverse problem) and that (ii) for any member of a class of two-dimensional refractive index profiles n=n(x,y) satisfying a differential condition, all the compatible families of light rays can be found analytically (direct problem). We present appropriate examples.

An exact macroscopic extended model with many moments for ultrarelativistic gases

Extended Thermodynamics is a very important theory: for example, it predicts hyperbolicity, finite speeds of propagation waves as well as continuous dependence on initial data. Therefore, it constitutes a significative improvement of ordinary thermodynamics. Here its methods are applied to the case of an arbitrary, but fixed, number of moments. The kinetic approach has already been developed in literature; then, the macroscopic approach is here considered and the constitutive functions appearing in the balance equations are determined up to whatever order with respect to thermodynamical equilibrium. The results of the kinetic approach are a particular case of the present ones.

A new formulation of the two-dimensional inverse problem of dynamics

Taking the parameter b to vary along a monoparametric family of planar curves, given in the form , ( being the parameter along each specific curve of the family), we derive two equations to formulate the inverse problem of dynamics and find all potentials creating, for adequate initial conditions, the given family. One of these equations offers the total energy on each specific orbit traced under a known potential, the other equation relates merely potentials and orbital data. This later equation lends itself to series expansion solutions for small values of the parameter b. Two applications to isotach and geometrically similar orbits are discussed as special cases and two examples are given to demonstrate the efficiency and the indispensability of the new equations.

Wave speeds in the macroscopic extended model for ultrarelativistic gases

An exact macroscopic extended model for ultrarelativistic gases, with an arbitrary number of moments, is present in the literature. Here we exploit equations determining wave speeds for that model. We find interesting results; for example, the whole system for their determination can be divided into independent subsystems and some, but not all, wave speeds are expressed by rational numbers. Moreover, the extraordinary property that these wave speeds for the macroscopic model are the same of those in the kinetic model, is proved.

Refractive-index distributions generating as light rays a given family of curves lying on a surface

In the framework of geometrical optics, we consider the inverse problem consisting in obtaining refractive-index distributions n=n(u,v) of a two-dimensional transparent inhomogeneous isotropic medium from a known family f(u,v)=c of monochromatic light rays, lying on a given regular surface. Using some basic concepts of differential geometry, we establish a first-order linear partial differential equation relating the assigned family of light rays with all possible refractive-index profiles compatible with this family. In particular, we study the refractive-index distribution producing, as light rays, a given family of geodesic lines on some remarkable surfaces. We give appropriate examples to explain the theory.

Three-dimensional inverse problem of geometrical optics: a mathematical comparison between Fermat's principle and the eikonal equation.

In the framework of geometrical optics, we consider the following inverse problem: given a two-parameter family of curves (congruence) (i.e., f(x,y,z)=c1,g(x,y,z)=c2), construct the refractive-index distribution function n=n(x,y,z) of a 3D continuous transparent inhomogeneous isotropic medium, allowing for the creation of the given congruence as a family of monochromatic light rays. We solve this problem by following two different procedures: 1. By applying Fermats principle, we establish a system of two first-order linear nonhomogeneous PDEs in the unique unknown function n=n(x,y,z) relating the assigned congruence of rays with all possible refractive-index profiles compatible with this family. Moreover, we furnish analytical proof that the family of rays must be a normal congruence. 2. By applying the eikonal equation, we establish a second system of two first-order linear homogeneous PDEs whose solutions give the equation S(x,y,z)=const. of the geometric wavefronts and, consequently, all pertinent refractive-index distribution functions n=n(x,y,z). Finally, we make a comparison between the two procedures described above, discussing appropriate examples having exact solutions.