Dino Boccaletti

Quantitative predictions with detuned normal forms

The phase-space structure of two families of galactic potentials is approximated with a resonant detuned normal form. The normal form series is obtained by a Lie transform of the series expansion around the minimum of the original Hamiltonian. Attention is focused on the quantitative predictive ability of the normal form. We find analytical expressions for bifurcations of periodic orbits and compare them with other analytical approaches and with numerical results. The predictions are quite reliable even outside the convergence radius of the perturbation and we analyze this result using resummation techniques of asymptotic series.

Periodic orbits in the logarithmic potential

Analytic methods to investigate periodic orbits in galactic potentials. To evaluate the quality of the approximation of periodic orbits in the logarithmic potential constructed using perturbation theory based on Hamiltonian normal forms. The solutions of the equations of motion corresponding to periodic orbits are obtained as series expansions computed by inverting the normalizing canonical transformation. To improve the convergence of the series a resummation based on a continued fraction may be performed. This method is analogous to that looking for approximate rational solutions (Prendergast method). It is shown that with a normal form truncated at the lowest order incorporating the relevant resonance it is possible to construct quite accurate solutions both for normal modes and periodic orbits in general position.

Stability of axial orbits in galactic potentials

We investigate the dynamics in a galactic potential with two reflection symmetries. The phase-space structure of the real system is approximated with a resonant detuned normal form constructed with the method based on the Lie transform. Attention is focused on the stability properties of the axial periodic orbits that play an important role in galactic models. Using energy and ellipticity as parameters, we find analytical expressions of bifurcations and compare them with numerical results available in the literature.

Geometry of Minkowski space-time

Introduction.- Hyperbolic Numbers.- Geometrical Representation of Hyperbolic Numbers.- Trigonometry in the Hyperbolic (Minkowski) Plane.- Equilateral Hyperbolas and Triangles in the Hyperbolic Plane.- The Motions in Minkowski Space-Time (Twin Paradox).- Some Final Considerations.

Integrating the geodesic equations in the Schwarzschild and Kerr space-times using Beltrami's "geometrical" method

We revisit a little known theorem due to Beltrami, through which the integration of the geodesic equations of a curved manifold is accomplished by a method which, even if inspired by the Hamilton-Jacobi method, is purely geometric. The application of this theorem to the Schwarzschild and Kerr metrics leads straightforwardly to the general solution of their geodesic equations. This way of dealing with the problem is, in our opinion, very much in keeping with the geometric spirit of general relativity. In fact, thanks to this theorem we can integrate the geodesic equations by a geometrical method and then verify that the classical conservation laws follow from these equations.

Chaos in N-body systems

Abstract Here is a selection of applications of what is now called theory of dynamical systems in galactic dynamics and N-body systems. The study of chaotic motions in potentials used as a model for elliptical galaxies is a first example of these applications. The interest in this problem stems from the fact that there are now many theoretical and observational evidences that the overall potentials of galaxies are indeed non-integrable. There are classes of objects, for example small and intermediate luminosity elliptical galaxies, for which the presence of the famous third integral is not necessary or others in which we observe peculiarities in their photometry or kinematics. We address here some of these issues and their implications in modifying our current understanding of the structure and evolution of galaxies. More in general, there is the natural question of how the systems we see have settled to their present status and what would happen if some external cause perturbs it. This issue is related to the question of the stochasticity involved in the general N-body dynamics, especially when N is very large. An N-body dynamical system is definitely chaotic, as shown by several numerical investigations, at least for N not very large. However, this statement must be reconciled with the picture of non-collisional equilibrium of big systems. The second part of this review presents a survey of numerical experiments and an interpretation of the results obtained using standard chaoticity indicators.

Galileo and the Principle of Relativity

Nowadays, in any textbook of physics (in the part devoted to mechanics) one finds written that the laws of Newtonian mechanics are invariant under Galileo transformations, or, equivalently, that the equations of the mechanics are the same in all inertial reference systems.

The Theory of Adiabatic Invariants

In this chapter, devoted to the theory of adiabatic invariants and its applications in the field of astronomy, we try to point out how procedures, seemingly different and originating in different fields, in the end derive from the same concepts. What closely relates perturbation theory, which originated from the demand for suitable solutions to the problems of planetary motion, and the theory of adiabatic invariants, which was developed to give a more solid bases to the quantization rules of early quantum mechanics, is recourse to the averaging procedure. In both cases, the average is performed over a periodic motion having a period by far shorter than the time which characterizes the evolution of the physical system one is studying. Through the application of Noether’s theorem, together with the averaging method, one then sees how also the approximate invariance of the quantities, called the adiabatic invariants, is connected with the symmetry properties of the system. It is clear that one could, as is sometimes done, overturn the argument and start from the element recognized as the central one, i.e. the averaging procedure, and apply it to the different classes of problems. But it seemed to us more effective from a didactic point of view to follow the inverse path of progressively identifying the unifying element in the different problems belonging to apparently disparate fields of research. But at this point one cannot omit the fact that what we have called the unifying element is far from being rigorously proven. Just to quote one of the most authoritative “users” of the averaging principle, “We note that this principle is neither a theorem, an axiom, nor a definition, but rather a physical proposition, i.e., a vaguely formulated and, strictly speaking, untrue assertion. Such assertions are often fruitful sources of mathematical theorems.”1 The introduction of the use of methods based on the use of the Lie transform on the one hand has emphasized the link of the adiabatic invariants theory with the traditional perturbation theory, on the other hand has enabled people to resort to automatic computations in actual applications. To conclude, we once more call the attention of the reader to the universality of the paradigm of the perturbed oscillator which, by help of a transformation of variables, can also be applied to the case of a charged particle in a magnetic field.

The Motion of Heavy Bodies and the Trajectory of Projectiles

The study of law of fall of heavy bodies and the consequent treatment of the uniformly accelerated motion are, as we know, the fundamental part of Galileo’s mechanics. He has worked on it, so to speak uninterruptedly, during the Paduan period and after his return to Florence

The Theories of Motion in the Middle Ages and in the Renaissance

The period considered in this chapter (about ten centuries long) is that which, in its final part, is close to the time of Galileo and then, also in the light of the theories on the importance of some medieval works, it is essential to make clear in what context Galileo worked.