Papers by Harry Varvoglis
Proceedings of the First Greek-Austrian Workshop on Extrasolar Planetary Systems
New Developments in the Dynamics of Planetary Systems, 2001
The study of the motion of a point mass, moving in the gravitational field of two fixed attractin... more The study of the motion of a point mass, moving in the gravitational field of two fixed attracting centers, is a problem first posed by Euler in the 18th century, as an intermediate step towards the solution of the famous three-body problem. Euler himself, in a series of three papers (Euler 1766a, 1766b and 1767), was able to integrate the equations of motion for the two-dimensional (2-D) case, i.e. the case where the point mass moves on a plane containing the two attracting centers. Almost a century later Jacobi (1842) showed that the corresponding potential of the full 3-D case is separable in prolate spheroidal coordinates. Another century later Erikson and Hill (1949) found, in explicit form, the third integral of motion of the full three-dimensional (3-D) case (besides the total energy and the angular momentum about the axis passing through the two centers). Since then the problem has been considered as a non-exciting example of a separable potential and it is included, as such, in many textbooks of Theoretical Mechanics.
General Relativity and Gravitation, 1997
T he int eract ion of ch arged part icles, m oving in a uniform m agn et ic ® eld, w ith a plane ... more T he int eract ion of ch arged part icles, m oving in a uniform m agn et ic ® eld, w ith a plane polarized grav it at ional wave is considered using t he Fokker± P lanck± Kolm ogorov (f p k) ap proach. B y using a st och ast icity criterion, we det erm ine t he ex act locat ions in phase space, where reson an ce overlapping occu rs. We invest igat e the diOE usion of orbits aroun d each prim ary reson ance of order m by deriving gen eral an aly t ical ex pressions for an eOE ect ive diOE usion coe cient. A solution of t he corresp onding diOE usion equat ion (Fokker± P lanck equ at ion) for t he st at ic case is found. Num erical integrat ion of the full equ at ions of m ot ion an d subsequent calculat ion of the diOE usion coe cient veri® es t he analyt ical resu lt s. KE Y W ORDS : Fokker± P lanck equat ion w ith m agn et ic ® eld
Non-linear interaction of a gravitational wave with a distribution of particles
Astronomy and Astrophysics
The propagation of a gravitational wave in the non-empty interstellar space, in the presence of a... more The propagation of a gravitational wave in the non-empty interstellar space, in the presence of a uniform magnetic field, is investigated. It is found that the type of interaction of the wave with the interstellar plasma depends on the direction of propagation of the gravitational wave with respect to the magnetic field. In the oblique and parallel propagation cases the results are consistent with the already studied propagation in empty space (Kleidis et al. 1993). In the oblique case the interaction may be chaotic, leading to a diffusive acceleration of the plasma particles. In the parallel case the particles may be trapped in a resonance and accelerated to "infinite" energies. In the quasiparallel case, however (i.e. propagation at a small angle with respect to the magnetic field) a new type of interaction is possible, consisting of a combination of chaotic and resonant interactions.
Conditions on the stability of the external space solutions in a higher-dimensional quadratic theory of gravity
Journal of Mathematical Physics, 1996
ABSTRACT By using Lyapounov's direct method the authors examine the conditions under whic... more ABSTRACT By using Lyapounov's direct method the authors examine the conditions under which stable solutions to the field equations for the scale function of the external space may be derived in the context of a five-dimensional quadratic theory of gravity. They show that the time evolution of the distance, in a diagram t-R, between the solution to the field equations and a neighbouring one is determined, in the linear approximation, in terms of a second-order linear differential equation. Asking for bounded solutions of this equation the authors arrive at a stability criterion for the external scale function solutions, indicating that there exist three types of cosmological evolution of the visible universe which are linearly stable at all times. These are (1) the Milne model, (2) the spatially flat Friedmann radiation solution, and (3) the De Sitter inflationary solution.
Kinetic Theory of Gases
SpringerReference
Classical and Quantum Gravity, 1996
We study the resonant interaction of charged particles with a gravitational wave propagating in t... more We study the resonant interaction of charged particles with a gravitational wave propagating in the non-empty interstellar space in the presence of a uniform magnetic field. It is found that this interaction can be cast in the form of a parametric resonance problem which, besides the main resonance, allows for the existence of many secondary ones. Each of them is associated with a non-zero resonant width, depending on the amplitude of the wave and the energy density of the interstellar plasma. Numerical estimates of the particles' energisation and the ensuing damping of the wave are given.
Astronomy and Astrophysics, 1993
The interaction of a charged particle, moving in a uniform magnetic field, with a gravitational w... more The interaction of a charged particle, moving in a uniform magnetic field, with a gravitational wave is considered for two types of wave polarization (linear and circular) and for an arbitrary direction of propagation, with respect to the magnetic field. It is found that, in the oblique and perpendicular propagation cases, the motion of the particle is chaotic. An asymptotic criterion is calculated, which determines, for both types of polarization, the stochasticity threshold in terms of the wave amplitude and frequency as well as the initial momentum of the particle. We also show that, under a resonant condition, we can find the exact solutions of the geodesics in the case of parallel propagation
If an asteroid is located in a mean motion resonance with Jupiter, its orbital elements, especial... more If an asteroid is located in a mean motion resonance with Jupiter, its orbital elements, especially the eccentricity, e, can be transported to Jupiter-crossing values due to chaotic motion. For resonances closer to Jupiter, such as those placed in the outer asteroid belt (defined here by 3.45AU ≤ a ≤ 3.90AU), a large fraction of orbits is expected to be chaotic while, at the same time, the eccentricity value needed to cross the orbit of Jupiter is small (e ∼ 0.3). In the absence of mechanisms which can provide ‘shortcuts’ to high values of e (such as resonant periodic orbits; see Tsiganis et al., 2001), the ‘random walk’-like manner by which the eccentricity grows resembles a diffusion process. Murray & Holman (1997; hereafter MH97) constructed an analytical theory for this ‘slow chaos’ in outer-belt mean motion resonances, in the framework of the planar elliptic restricted three-body problem (hereafter ERTBP). Their calculations were based on an averaged Hamiltonian with 2 degrees ...
The CTMC method as part of the study of classical chaotic Hamiltonian systems
Journal of Physics B: Atomic, Molecular and Optical Physics, 1995
Degenerate Dynamical Systems and the Disappearance of (K.A.M.-Type) Integrals of Motion
International Astronomical Union Colloquium, 1983
ABSTRACT
AIP Conference Proceedings, 2006
When one thinks of the solar system, he has usually in mind the picture based on the solution of ... more When one thinks of the solar system, he has usually in mind the picture based on the solution of the two-body problem approximation presented by Newton, namely the ordered clockwork motion of planets on fixed, non-intersecting orbits around the Sun. However, already by the end of the 18th century this picture was proven to be wrong. As discussed by Laplace and Lagrange (for a modern approach see or [2]), the interaction between the various planets leads to secular changes in their orbits, which nevertheless were believed to be corrections of higher order to the Keplerian elliptical motion.
Large orbital eccentricities and close encounters at the 2:1 resonance of a dynamical system modelling asteroidal motion
Astron Astrophys, 1993
In a recent work (Varvoglis 1991) we proposed the use of a model dynamical system as a fast and e... more In a recent work (Varvoglis 1991) we proposed the use of a model dynamical system as a fast and effective method to study the motion of asteroids in the main asteroidal belt. This dynamical system is, essentially, a modified planar elliptical restricted three-body problem, in which the variation of Jupiter's eccentricity, caused by the perturbation of Saturn, is taken explicitly into account. Here we show that, near the 2:1 resonance, the (osculating) eccentricity, e, of the trajectories of this dynamical system may attain high values (e ≥ 0.6) and that a fictitious asteroid, following such a "high-e" trajectory, may undergo a close encounter with Jupiter and be subsequently removed from the 2:1 resonance region.
Stochasticity in dynamical systems and the curvature of the associated riemannian manifold
Astrophysics and Space Science, 1985
It is shown that no conflicts need to arise between results on the stochastic properties of a dyn... more It is shown that no conflicts need to arise between results on the stochastic properties of a dynamical system obtained through the method of the surface of section mapping and results obtained through the Hedlund-Hopf-Lobachevsky-Hadamard theorem.
Lecture Notes in Physics, Nov 30, 2005
All laws that describe the time evolution of a continuous system are given in the form of differe... more All laws that describe the time evolution of a continuous system are given in the form of differential equations, ordinary (if the law involves one independent variable) or partial (if the law involves two or more independent variables). Historically the first law of this type was Newton's second law of motion. Since then Dynamics, as it is customary to name the branch of Mechanics that studies the motion of a body as the result of a force acting on it, has become the “typical„ case that comes into one's mind when a system of ordinary differential equation is given, although this system might as well describe any other system, e.g. physical, chemical, biological, financial etc. In particular the study of “conservative„ dynamical systems, i.e. systems of ordinary differential equations that originate from a time-independent Hamiltonian function, has become a thoroughly developed area, because of the fact that mechanical energy is very often conserved, although many other physical phenomena, beyond motion, can be described by Hamiltonian systems as well. In what follows we will restrict ourselves exactly to the study of Hamiltonian systems, as typical dynamical systems that find applications in many scientific disciplines.
Lecture Notes in Physics, 2005
All laws that describe the time evolution of a continuous system are given in the form of differe... more All laws that describe the time evolution of a continuous system are given in the form of differential equations, ordinary (if the law involves one independent variable) or partial (if the law involves two or more independent variables). Historically the first law of this type was Newton's second law of motion. Since then Dynamics, as it is customary to name the branch of Mechanics that studies the motion of a body as the result of a force acting on it, has become the "typical" case that comes into one's mind when a system of ordinary differential equation is given, although this system might as well describe any other system, e.g. physical, chemical, biological, financial etc. In particular the study of "conservative" dynamical systems, i.e. systems of ordinary differential equations that originate from a time-independent Hamiltonian function, has become a thoroughly developed area, because of the fact that mechanical energy is very often conserved, although many other physical phenomena, beyond motion, can be described by Hamiltonian systems as well. In what follows we will restrict ourselves exactly to the study of Hamiltonian systems, as typical dynamical systems that find applications in many scientific disciplines. The method used traditionally in the study of a conservative dynamical system, whose Hamiltonian leads to a complex set of differential equations, is to follow a sequence of approximations. In the beginning, we try to solve exactly the equations derived from a simplified form of the Hamiltonian, the so-called zero-order approximation, omitting complex or non-linear terms. Then we revert to the full Hamiltonian and consider the omitted terms as a "perturbation" of the zero order approximation. We then modify the initial solution by a sequence of "corrections", which are expected to describe better and better the real system. This method is based on the implicit assumption that the sequence of successive approximations converges to the "real" solution. Since the method is based on the fact that the solution of the zero order dynamical system is known exactly, it is closely connected to the theory of integrable, or else regular dynamical systems. By this term, we denote the Hamiltonian dynamical systems whose differential equations of motion can be solved at least by quadratures, i.e. they can be expressed as integrals of functions of one
Journal de Physique, 1985
2014 Le comportement ordonné ou chaotique des trajectoires de particules chargées dans le champ m... more 2014 Le comportement ordonné ou chaotique des trajectoires de particules chargées dans le champ magnétique statique du réacteur thermonucléaire Astron est étudié numériquement. Malgré le fait que la fonction hamiltonienne correspondante est de classe C°, ce qui entraîne que ce système dynamique ne possède pas de véritables tores invariants dans l'espace de phase pour aucune valeur numérique h de la fonction hamiltonienne, aucun signe de comportement chaotique n'a été constaté pour des valeurs de h modérées et pour des intervalles de temps d'importance physique. Nous calculons une intégrale formelle du mouvement qui peut, dans certains cas, décrire d'une manière satisfaisante le comportement ordonné des trajectoires. Abstract. 2014 The ordered or chaotic behaviour of charged particle trajectories in the static magnetic field of the Astron thermonuclear reactor is numerically investigated. Despite the fact that the corresponding Hamiltonian function is of class C°, from which it follows that this dynamical system does not possess true phase space invariant tori for any numerical value h of the Hamiltonian function, no sign of chaotic behaviour is detected for moderate values of h and for time intervals of physical significance. A formal integral of motion is calculated that can, in certain cases, describe in a satisfactory way the ordered trajectory behaviour. Tome 46 No 4 AVRIL 1985 LE JOURNAL DE PHYSIQUE J. Physique 46 (1985) 495-502 AVRIL 1985, Classification Physics Abstracts 03.20 -41.70
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Papers by Harry Varvoglis