Classical
and Celestial Mechanics
I.I.Kosenko, V.N.Tkhai
e-mai: kosenko@ccas.ru
There
are not a lot of conferences and Symposiums one can find in Russian scientific
society like one on Classical and Celestial Mechanics held in Velikie Luki.
According to tradition it has been performed last summer from 15 to 20 August
2001. At the beginning Symposium arose as a result of joint efforts of
investigators from Moscow and Saint-Petersburg. In general latter ones were the
representatives of Saint-Petersburg schools in field of celestial mechanics.
Simposium venue was selected in a natural way:
it located in approximately equal distances from two cities cited above.
Velikie Luki is peaceful and comfortable town in North West of Russia, South of
Pskov region. Registration and sessions were performed in camp “Sunny”
relocated in pine forest on the beach of picturesque glacial lake Balasdygn.
Conditions created are promoted effective work and rest. All resources need to
perform the Symposium were provided by Joint-Stock Holding Company ELVO,
President B.N.Karakaev, and ZETO Limited Company, Director General
N.N.Koslovsky. Symposium sponsors have organized the rest of participants in
natural environment and unforgettable trips to Pushkin Hills and museums of
Sofia Kovalevskaya and Ivan Vinogradov.
Symposium organizers are: Russian Academy of
Sciences, Branch of Machine-Building, Mechanics, and Control Processes,
Computing Center of RAS, Academy of Cosmonautics, Moscow State University,
Moscow State Aviation Institute, Moscow State Academy of Instrument Making and
Computer Science, International Informatization Academy. Symposium chairman is
V.V.Rumyantsev, Academician of RAS. Co-chairmen are: P.S.Krasil’nikov, V. N.
Tkhai. Scientific secretary is I.I.Kosenko. In Symposium work take a part
participants from Algeria, Brazil, Georgia, Italy, Russia, Ukraine, France.
Symposium was performed with financial support
of Russia Fund for Basic Researches, grant 01-0110101. Also special financial
support was provided for young participants.
Symposium was dedicated to 80th
birthday of Academician V.V.Rumyantsev.
Scientific program was including: plenary
sessions (A), mini-Symposium on joint
sessions of two sections (B),
mini-Symposium in frame of classical mechanics section (C), mini-Symposium in frame of celestial mechanics section (D), and computer presentation of PMM
journal electronic version Web site. Time-limit was as follow: in category A were 40 minutes per lecture, in
category B were 30 minutes per
lecture, in categories C and D were 30 minutes per message. Then let
us give more detailed structure of cited sessions categories.
In class B
were included the following mini-Symposium: Methods of Classical and Celestial
Mechanics – B1, and Selected Problems
of Classical and Celestial Mechanics – B2.
In class C were included the
mini-Symposium: Analytical Mechanics – C1,
Theory of Stability and Bifurcations – C2,
Regular and Chaotic Dynamics – C3,
Oscillations of Mechanical Systems – C4,
Dynamics of Rigid and/or Deformable Bodies – C5. Correspondingly in class D
were included the mini-Symposium: The Problems of Three and N Bodies – D1, Periodic and Almost Periodic Orbits, Resonances – D2, Investigations on Dynamics of Solar
System Planets – D3, Dynamics of Rotational
Motion of Celestial Bodies – D4,
Dynamics of Orbital Systems – D5.
Two lectures have performed by the order of
Scientific Committee namely: A.V.Karapetyan (Moscow), S.Ya.Stepanov (Moscow),
R.S.Sulikashwili (Tbilisi), “The 80-th Anniversary of Academician
V.V.Rumyantsev”; Yu.P.Gupalo (Moscow), “The Journal Prikladnaya Matematika i
Mekhanika as a Booster for Advances in Mechanics”.
Below we will give short abstracts of lectures
from A and B categories.
A.S.Andreev (Ulyanovsk). “Stability with
Respect to Part of the Variables: Some Results and Perspectives of Their
Development”. The observation of the results concerning investigation of the
stability with respect to part of the variables on the base of Lyapunov
functions method and of limit equations one from the viewpoint of perspective
of their usage and development is introduced.
A.S.Andreev (Ulyanovsk), E.B.Kim (Ulyanovsk),
C.Risito (Parma). “On the Stability of Generalized Stationary and
Quasistationary Motions”. In the work sufficient conditions of unconditional
stability of generalized steady and quasi steady motions are obtained. Some
examples are considered.
V.V.Beletsky (Moscow), M.L.Pivovarov (Moscow),
A.A.Savchenko (Moscow). “Regular and Chaotic Attitude Motion of
Dumb-Bell-Spacecraft”. Attitude motion of orbiting tether connected two-body
system is considered. The model of dumb-bell-spacecraft in elliptic orbit is
used. Regular and chaotic motions of the system with tense tether are studied.
S.V.Bolotin (Moscow). “Chaotic Trajectories of
the Second Species for the Restricted 3-Body Problem”. For the n-center problem of one particle moving
in the potential of attracting centers of small mass fixed in an arbitrary
smooth potential and magnetic field. Large subshifts of solutions of this type
for the circular restricted 3-body problem of celestial mechanics are obtained.
A.D.Bruno (Moscow). “Families of Periodic
Solutions to the Beletsky Equation”. The survey of results of two parametric
families of periodic solutions constructing is given. The structure of families
of symmetric and asymmetric solutions is considered. Data on resonant rotations
of bodies in Solar System are compared with computed families of periodic solutions
to the Beletsky equation.
B.M.Darinskii (Voronezh), Yu.I.Sapronov
(Voronezh). “Bifurcations of Extremals, Phase Transitions, and Characteristic CW-complexes”. Investigation of extremals
of smooth functional in smooth Banach manifold often can be reduced to similar
problem of analysis of extremals of key function (in a finite-dimensional manifold
of key parameters). For spread in theory of crystals singularity of n-dimensional pleat type (defined by a
quartic part of Taylor decomposition of key function) the rather complete lists
of bif-decompositions had been received
at n £ 3.
G.Cantarelli (Cagliari). “Stability of the
Origin of Scleronomic Systems, I”. A holonomic scleronomic mechanical systems
with bilateral and frictionless time independent constraints are considered.
A.V.Karapetyan (Moscow). “Steady Motions of
Mechanical Systems”. Problems of the existence and stability of steady motions
of mechanical systems with first integrals are discussed. Recently, such the
theory was modified for the existence and stability problems of steady motions
of dynamical systems with a non-increasing energy function and first integrals
(in particular, for conservative and dissipative mechanical systems with
symmetry).
V.V.Kozlov (Moscow). “General Vortex Theory,
Changeable System Dynamics and the Lie Groups”. Problems, connected with the
motion of changeable mechanical systems are discussed. It mean that the
internal forces change mass geometry of the system. Equations of motion are
reduced to the non-autonomous first order system of ordinary differential equations
on the Lie group. The hydrodynamic analogy for the phase flow of system derived
is used. General results are applied to the "falling cat" problem and
the problem of the motion of the body with rigid surface in the liquid.
I.I.Kosenko (Moscow). “Methods to Describe
Impacts in Dynamics of Tethered Satellite Systems”. The toolkit of
computational procedures to construct various models of tethered satellite
system dynamics is built. Two approaches are under consideration. The first one
is the simulation of successive impacts in unrestricted form when orbits are
not necessary to be Keplerian. The second approach is to use the method of
reflection in dynamics with impacts.
P.S.Krasil'nikov (Moscow). “The Generalized
Classical Method of the Construction of V-Functions
from the First Integrals”. The new heuristic method generalizing the classical
construction of V-function from first
integrals is described. It is shown the most of the investigated stability
problems from classical mechanics are covered and ordered by new method. By
this method, some algebraic unsolvable stability problems of ordinary
differential equations are investigated.
A.L.Kunitsyn (Moscow). “On the Libration Points
of the Photogravitational 3-Body Problem”. The positions of relative
equilibrium and their stability in the restricted photogravitational 3-body
problem when a passively gravitating particle is not only attracted by the main
bodies but is subjected also to the light pressure from each of the primeries
is considered. The review of the results on the determination, clasification
and stability investigations of the possible libration points both in the
circular and the elliptic case obtained by different authors are presented.
Conditions of stability obtained previously are analyzed with the new point of
view permitting to get more natural visual representation.
A.P. Markeev (Moscow). “Nonlocal Problem of
Stability of Periodic Motions of Hamiltonian Systems”. Periodic motion, which
is orbital unstable because of the presence in the system of a resonance of the
second or third order is assumed to be exist. Despite of the instability, the
trajectories of the disturbed motion can remain in the limited vicinity of
trajectories of the undisturbed periodic motion for all moments of time. Some
interesting applications for description of instabilities in asteroid belt are
considered.
V.M.Matrosov (Moscow), I.A.Finogenko (Irkutsk).
“Analytical Dynamics of Multibody Systems with Dry Friction”. Analytical
dynamics, developed by the authors, of Multibody Systems with dry friction in
kinematic pairs of one degree of freedom is presented. The results clarify
known paradoxes and derive the general theory of mechanical systems with
friction motion.
V.V.Rumyantsev (Moscow). “Routh's Equations and
Variational Principles”. Holonomic mechanical systems in Routh's variables with
equations of motion both of Lagrangian and Hamiltonian type are considered. The
variational principles of d'Alembert-Lagrange, Hamilton-Ostrogradsky, Hamilton
(in the third form), as well as Hölder's principle and one of the minimal
action in Lagrange and Hamilton forms are given in Routh's variables.
A.S.Sumbatov (Moscow). “On the Quasi-Static
Model of Motion of a Particle along the Plane with Dry Friction”. 3-dimensional
motion of a particle subjected to elastic and damped forces is considered. The
particle moves along a plane and can leave it. The force of dry (Coulomb)
friction acts on the particle from this plane. Within the framework of the
quasi-static statement of the problem the conditions of the existence and
uniqueness of motions coherent with the Coulomb friction law are examined.
Unlike another considerations of this problem the most general case of
permanent positive matrix of the coefficients of elasticity is treated.
V.N.Tkhai (Moscow). “Reversible Dynamical
Models in Celestial Mechanics”. The reversible problems of celestial mechanics
are investigated. Three-body problem, satellite in the plane of the elliptic
orbit and tethered satellite system are analyzed in details.
D.P.Chevallier (Paris). “Objectivity and
Dynamics of Generalized Rigid Body”. Importance of the objectivity (or
frame-indifference) principle in mechanics of continua was pointed out by
W. Noll. However, its application to dynamics is still questioned in
relation with objectivity of inertial forces and torques. It is proved that
most of the features of dynamics of ordinary or generalized rigid body are
actually mathematical consequences of the principle of objectivity of inertial
forces (within a suitable meaning) and of geometry (displacement group).
Totally numbers of messages distributed over
mini-Symposium were as follow: C1 –
6, C2 – 11, C3 – 8, C4 – 5, C5 – 14, D1 – 3, D2 – 6, D3 – 4, D4 – 3, D5 – 5. Work in
sessions performed in harmony by all participants. Lectures and messages
followed by discussions sometimes with excitement.