Analytical perturbative theories of motion in highly inhomogeneous gravitational fields : Ariadna AO/1-6790/11/NL/CBI

In this report we show that modern computer performances and state-of-the-art algebraic manipulator software are sufficiently developed to carry out our generalised analytical perturbative theory. This report addresses three technical aspects to develop a general perturbative theory and illustrates its power by applying it to investigate the inhomogeneous gravitational fields of asteroids

Methods of algebraic manipulation in perturbation theory

We give a short introduction to the methods of representing polynomial and trigonometric series that are often used in Celestial Mechanics. A few applications are also illustrated.Comment: 37 pages, 10 figure

Domains of analyticity of Lindstedt expansions of KAM tori in dissipative perturbations of Hamiltonian systems

Many problems in Physics are described by dynamical systems that are conformally symplectic (e.g., mechanical systems with a friction proportional to the velocity, variational problems with a small discount or thermostated systems). Conformally symplectic systems are characterized by the property that they transform a symplectic form into a multiple of itself. The limit of small dissipation, which is the object of the present study, is particularly interesting. We provide all details for maps, but we present also the modifications needed to obtain a direct proof for the case of differential equations. We consider a family of conformally symplectic maps $f_{\mu, \epsilon}$ defined on a $2d$-dimensional symplectic manifold $\mathcal M$ with exact symplectic form $\Omega$; we assume that $f_{\mu,\epsilon}$ satisfies $f_{\mu,\epsilon}^*\Omega=\lambda(\epsilon) \Omega$. We assume that the family depends on a $d$-dimensional parameter $\mu$ (called drift) and also on a small scalar parameter $\epsilon$. Furthermore, we assume that the conformal factor $\lambda$ depends on $\epsilon$, in such a way that for $\epsilon=0$ we have $\lambda(0)=1$ (the symplectic case). We study the domains of analyticity in $\epsilon$ near $\epsilon=0$ of perturbative expansions (Lindstedt series) of the parameterization of the quasi--periodic orbits of frequency $\omega$ (assumed to be Diophantine) and of the parameter $\mu$. Notice that this is a singular perturbation, since any friction (no matter how small) reduces the set of quasi-periodic solutions in the system. We prove that the Lindstedt series are analytic in a domain in the complex $\epsilon$ plane, which is obtained by taking from a ball centered at zero a sequence of smaller balls with center along smooth lines going through the origin. The radii of the excluded balls decrease faster than any power of the distance of the center to the origin

Физика космоса : труды 44-й международной студенческой научной конференции, 2-6 февраля 2015 г.

В сборнике представлены доклады и сообщения студенческой научной конференции, которая ежегодно проводится в Астрономической обсерватории Уральского федерального университета. Цель конференции — обобщить достижения в области астрономии и астрофизики и способствовать формированию навыков и способностей молодых исследователей. Сборник предназначен для профессиональных астрономов и физиков, студентов и аспирантов соответствующих специальностей

Физика космоса : труды 43-й международной студенческой научной конференции, 3-7 февраля 2014 г.

В сборнике представлены доклады и сообщения студенческой научной конференции, которая ежегодно проводится в Астрономической обсерватории Уральского федерального университета. Цель конференции обобщить достижения в области астрономии и астрофизики и способствовать формированию навыков и способностей молодых исследователей. Сборник предназначен для профессиональных астрономов и физиков, студентов и аспирантов соответствующих специальностей