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The lectures in this 2005 book are intended to bring young researchers to the current frontier of knowledge in geometrical mechanics and dynamical systems. They succinctly cover an unparalleled range of topics from the basic concepts of... more
The lectures in this 2005 book are intended to bring young researchers to the current frontier of knowledge in geometrical mechanics and dynamical systems. They succinctly cover an unparalleled range of topics from the basic concepts of symplectic and Poisson geometry, through integrable systems, KAM theory, fluid dynamics, and symmetric bifurcation theory. The lectures are based on summer schools for graduate students and postdocs and provide complementary and contrasting viewpoints of key topics: the authors cut through an overwhelming amount of literature to show young mathematicians how to get to the core of the various subjects and thereby enable them to embark on research careers.
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Lie symmetric cotangent bundle systems with free and proper actions
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The two full body problem concerns the dynamics of two spatially extended rigid bodies (e.g. rocky asteroids) subject to mutual gravitational interaction. In this note we deduce the EulerPoincaré and Hamiltonian equations of motion using... more
The two full body problem concerns the dynamics of two spatially extended rigid bodies (e.g. rocky asteroids) subject to mutual gravitational interaction. In this note we deduce the EulerPoincaré and Hamiltonian equations of motion using the geometric mechanics formalism.
Using geometric mechanics methods, we examine aspects of the dynamics ofnmass points inR4with a general pairwise potential. We investigate the central force problem, set up then-body problem and discuss certain properties of relative... more
Using geometric mechanics methods, we examine aspects of the dynamics ofnmass points inR4with a general pairwise potential. We investigate the central force problem, set up then-body problem and discuss certain properties of relative equilibria. We describe regularn-gons inR4, and when the masses are equal we determine the invariant manifold of motions with regularn-gon configurations. In the casen = 3, we reduce the dynamics to a 6 d.f. system and we show that for generic potentials and momenta, relative equilibria with equilateral configuration are unstable.This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.
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We study a classical model of isosceles triatomic “A-B-A” molecules. The atoms, considered mass points, interact mutually via a generic repulsive-attractive binary potential. First we show that the steady states, or relative equilibria... more
We study a classical model of isosceles triatomic “A-B-A” molecules. The atoms, considered mass points, interact mutually via a generic repulsive-attractive binary potential. First we show that the steady states, or relative equilibria (RE), corresponding to rotations about the molecule symmetry axis may be determined qualitatively assuming the knowledge of (1) the shape of the binary interaction potential, (2) the equilibrium diatomic distances (i.e., the equilibrium bond length) of the A-A and A-B molecules, and (3) the distance at which the RE of the diatomic A-A molecule ceases to exist. No analytic expression for the interaction potentials is needed. Second we determine the stability of the isosceles RE modulo rotations using geometric mechanics methods and using Lennard-Jones diatomic potentials. As a by-product, we verify the qualitative results on RE existence and bifurcation. For isosceles RE, we employ the Reduced Energy-Momentum method presented by Marsden [Lectures in Me...
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Within the class of canonical polynomial Hamiltonian systems anti-symmetric under phase-space involutions, we generalise some results on the existence of Darboux polynomial and rational first integrals for “kinetic plus potential” systems... more
Within the class of canonical polynomial Hamiltonian systems anti-symmetric under phase-space involutions, we generalise some results on the existence of Darboux polynomial and rational first integrals for “kinetic plus potential” systems to general systems.
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ABSTRACT Using Bismut's approach to stochastic canonical systems, this paper applies stochastic methods to the concrete case of the perturbed two-body problem.
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ABSTRACT
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The motion of a material point of unit mass in a field determined by a generalized HénonHeiles potential U= Aq12+ Bq22+ Cq12q2+ Dq23, with (q1, q2)= standard Cartesian coordinates and (A, B, C, D)∈(0,∞) 2× R2, is addressed for two limit... more
The motion of a material point of unit mass in a field determined by a generalized HénonHeiles potential U= Aq12+ Bq22+ Cq12q2+ Dq23, with (q1, q2)= standard Cartesian coordinates and (A, B, C, D)∈(0,∞) 2× R2, is addressed for two limit situations: collision ...
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In this dissertation we analyse from a qualitative standpoint motion in a quasihomogeneous potential field: we offer a complete description of the flow associated with the two-body problem in quasihomogeneous field, obtain necessary and... more
In this dissertation we analyse from a qualitative standpoint motion in a quasihomogeneous potential field: we offer a complete description of the flow associated with the two-body problem in quasihomogeneous field, obtain necessary and sufficient ...
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Title: Small stochastic perturbations of integrable canonical systems. Authors: Stoica,Gheorghe; Stoica, Cristina. Publication: Romanian Astronomical Journal (1210-5168), Vol. 6, No. 2, p. 170 - 177 (1996). Publication Date: 00/1996.... more
Title: Small stochastic perturbations of integrable canonical systems. Authors: Stoica,Gheorghe; Stoica, Cristina. Publication: Romanian Astronomical Journal (1210-5168), Vol. 6, No. 2, p. 170 - 177 (1996). Publication Date: 00/1996. Origin: ARI. ...
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... VASILE MIOC Astronomical Institute of the Romanian Academy, Astronomical Observatory Bucharest, Str ... 1995; Stoica and Mioc 1996b,d; Mioc and Stoica 1997) are to be added to the above ... using (6) and (7), and denoting by rcr = (C2... more
... VASILE MIOC Astronomical Institute of the Romanian Academy, Astronomical Observatory Bucharest, Str ... 1995; Stoica and Mioc 1996b,d; Mioc and Stoica 1997) are to be added to the above ... using (6) and (7), and denoting by rcr = (C2 − 2B)/A the point where dVC (r)/dr = 0, and ...
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... VASILE MIOC Astronomical Institute of the Romanian Academy, Astronomical Observatory Cluj-Napoca, Str. ... 166 C. STOICA AND V. MIOC ... r1 r2 r3 if they are all real) for the roots of the equation f(r) = 0, and r'1,... more
... VASILE MIOC Astronomical Institute of the Romanian Academy, Astronomical Observatory Cluj-Napoca, Str. ... 166 C. STOICA AND V. MIOC ... r1 r2 r3 if they are all real) for the roots of the equation f(r) = 0, and r'1, r'2 (r'1 r'2 if they are real) for the roots of the equation df(r)=dr = 0. An ...
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The present paper offers an alternative point of view of block regularization for the motion of a particle in a central potential field of the form-xa, where x is the distance between the particle and the source and a some positive real... more
The present paper offers an alternative point of view of block regularization for the motion of a particle in a central potential field of the form-xa, where x is the distance between the particle and the source and a some positive real number.