Hamiltonian dynamics Research Papers - Academia.edu (original) (raw)

In this 607 page book, in Spanish, are described in clear and complete way several problems of statics, mechanics, kinematics, dynamics and analytical dynamics. Includes non conventional subjects like perturbation theory, Kepler problem... more

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- Mathematical Physics, Theoretical Physics, Hamiltonian dynamics, Chaos (Physics)

Hamilton’s principle is one of the great achievements of analytical mechanics. It offers a methodical manner of deriving equations motion for many systems, with the additional benefit that appropriate and correct boundary conditions are automatically produced as part of the derivation. It allows insight into the manner that the system is modeled, as any modelling assumptions are clear and the effects of changing basic system properties become apparent and are accounted for in a consistent manner. Simplifications may also be made and Hamilton’s principle can be used as the basis for an approximate solution. Classical mechanics dictates that Hamilton’s principle can only be used for systems that are always composed of the same particles. This has been more recently extended to include systems whose constitutent particles change with time, including open systems of changing mass. In this chapter, we review the principle and its extended version and show through application to examples how it can lead to insightful observations about the system being modelled.

- by Natalie Baddour
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- Hamiltonian dynamics

It is my great honour to welcome you on behalf of the Bureau of IUTAM to this Symposium on Hamiltonian dynamics, vortex structures and turbulence. The Symposium has been in preparation for two years, and I congratulate our hosts here at the Steklov Institute of the Russian Academy of Sciences for having prepared an excellent and wide-ranging programme, and for having succeeded in attracting such a distinguished gathering to debate problems in ﬂuid dynamics many of which have a long history, yet still today present many challenges of a fundamental nature. The letters IUTAM, as you all know, stand for the International Union of Theoretical and Applied Mechanics. This Union is one of the International Scientiﬁc Union members of ICSU, the International Council for Science, which this year celebrates its 75th anniversary. The roots of IUTAM itself go back to the early Congresses in Mechanics, the ﬁrst of which was held in Delft in the Netherlands, in 1924. IUTAM was formally established as an International Union at the 7th Congress, which was held in London in 1948. The 13th Congress of Theoretical and Applied Mechanics was held here in Moscow in 1972, under the Presidency of the great Mushkhelishvili. The most recent 21st Congress was held in Warsaw in 2004, and the next will be held in Adelaide, South Australia, in 2008.
Professor Keith Moﬀatt, Vice-President, IUTAM

- by Mikhail A Sokolovskiy and +2
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- Hamiltonian dynamics, Stability, Vortex dynamics, ocean eddies

In the 1830s, W. R. Hamilton established a formal analogy between optics and mechanics by constructing a mathematical equivalence between the extremum principles of ray optics (Fermat's principle) and corpuscular mechanics (Maupertuis's principle). Almost a century later, this optical-mechanical analogy played a central role in the development of wave mechanics. Schrödinger was well acquainted with Hamilton's analogy through earlier studies. From Schrödinger's research notebooks, we show how he used the analogy as a heuristic tool to develop de Broglie's ideas about matter waves and how the role of the analogy in his thinking changed from a heuristic tool into a formal constraint on possible wave equations. We argue that Schrödinger only understood the full impact of the optical-mechanical analogy during the preparation of his second communication on wave mechanics: Classical mechanics is an approximation to the new undulatory mechanics, just as ray optics is an approximation to wave optics. This completion of the analogy convinced Schrödinger to stick to a realist interpretation of the wave function, in opposition to the emerging mainstream. The transformations in Schrödinger's use of the optical-mechanical analogy can be traced in his research notebooks, which offer a much more complete picture of the development of wave mechanics than has been previously thought possible.

- by Christoph Lehner and +1
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- Foundations of Quantum Mechanics, Hamiltonian dynamics, Schrödinger
- by Berend Smit
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- Materials Science, Hamiltonian dynamics, Computer Simulation, Mathematical Sciences

II EDIZIONE:

Risposte alle domande di teoria del corso di fisica matematica (FISICA, UNIPD, II ANNO)

- by alessandro manta and +1
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- Mathematical Physics, Physics, Hamiltonian dynamics, Classical Physics

The integral equations for calculation of metric, equations of substance motion, as well as equations for gravitational and electromagnetic fields in covariant theory of gravitation are obtained by means of variation of action functional. In covariant form stress-energy tensor of gravitational field, strength tensor of gravitational field and 4-current of mass are determined. The meaning of the cosmological constant and its relation to the components of energy density in action functional are explained. The results obtained prove the validity of Mach's principle, assuming that gravitation effects are due to the flows of gravitons. The idea that metric can be entirely determined by variables describing fields’ properties is substantiated.

"François Beets, Michel Dupuis et Michel Weber (éditeurs), Alfred North Whitehead. De l’algèbre universelle à la théologie naturelle. Actes des Journées d’étude internationales tenues à l’Université de Liège les 11-12-13 octobre 2001. Publiés avec le concours du FNRS, Frankfurt / Paris / Lancaster, ontos verlag, Chromatiques whiteheadiennes II, 2004. (377 p. ; ISBN 3-937202-64-1 ; 79 €)
Les premières journées « Chromatiques » se sont donné pour objectif de faciliter une réflexion globale sur la trajectoire conceptuelle du philosophe et mathématicien britannique Alfred North Whitehead (1861–1947). Afin de mener le lecteur au cœur de l’ontologie organique de l’époque de Harvard, il est en effet urgent d’élucider Whitehead à partir de lui-même, de montrer — sans être victime d’une « illusion rétrospective » — la continuité qui s’atteste dans un développement idéel qui exploite cependant quelques notables « changements d’amure ». Les contributions au colloque, placées sous le signe du commerce avec les textes eux-mêmes, furent traversées par une double tension : d’une part, l’éclairement systématique d’un aspect technique d’une des époques spéculatives de l’auteur ; d’autre part, la mise en horizon de ce questionnement ponctuel à l’aide d’une perspective globale sur le cheminement spéculatif whiteheadien.
"

- by Michel Weber and +2
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- Algebra, Geometry And Topology, Metaphysics, Ontology

We use the global stochastic analysis tools introduced by P. A. Meyer and L. Schwartz to write down a stochastic generalization of the Hamilton equations on a Poisson manifold that, for exact symplectic manifolds, satisfy a natural critical action principle similar to the one encountered in classical mechanics. Several features and examples in relation with the solution semimartingales of these equations are presented.

- by juan ortega
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- Hamiltonian dynamics, Stochastic analysis, Symplectic geometry, Classical Mechanics
- by Paolo Caressa
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- Algebra, Hamiltonian dynamics, Meccanica
- by Antonio G
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- Hamiltonian dynamics, Celestial Mechanics, Dynamic Systems and Control
- by Vladimir Gerdjikov
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- Hamiltonian dynamics
- by marco pettini
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- Statistical Mechanics, Thermodynamics, Hamiltonian dynamics, Differential Geometry

In the framework of covariant theory of gravitation the Euler-Lagrange equations are written and equations of motion are determined by using the Lagrange function, in the case of small test particle and in the case of continuously distributed matter. From the Lagrangian transition to the Hamiltonian was done, which is expressed through three-dimensional generalized momentum in explicit form, and also is defined by the 4-velocity, scalar potentials and strengths of gravitational and electromagnetic fields, taking into account the metric. The definition of generalized 4-velocity, and the description of its application to the principle of least action and to Hamiltonian is done. The existence of a 4-vector of the Hamiltonian is assumed and the problem of mass is investigated. To characterize the properties of mass we introduce three different masses, one of which is connected with the rest energy, another is the observed mass, and the third mass is determined without taking into account the energy of macroscopic fields. It is shown that the action function has the physical meaning of the function describing the change of such intrinsic properties as the rate of proper time and rate of rise of phase angle in periodic processes.

- by marco pettini
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- Statistical Mechanics, Thermodynamics, Hamiltonian dynamics, Differential Geometry

O formalismo Hamiltoniano é uma importante ferramenta no estudo de problemas físicos e matemáticos. Sistemas físicos que envolvem pêndulos e molas são muito empregados em cursos de mecânica clássica como exemplos de aplicação dos formalismos estudados. Este trabalho tem por objetivos fazer uma breve introdução ao formalismo Hamiltoniano e mostrar de maneira mais detalhada a resolução, dentro deste formalismo, do problema de dois pêndulos acoplados por uma mola, encontrando seus modos normais de oscilação. São investigados também os invariantes adiabáticos deste sistema, quando diminuímos lentamente o comprimento de um dos fios de pêndulo, no limite do acoplamento fraco.
Palavras-Chave: Formalismo Hamiltoniano, Pêndulos Acoplados, Mola, Modos Normais de Oscilação, Invariantes Adiabáticos.

- by Romualdo Santos
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- Hamiltonian dynamics, Mecânica classica

In this article we derive the equations for a rotating stratified fluid governed by inviscid Euler–Boussinesq and primitive equations that account for the effects of the perturbations upon the mean. Our method is based on the concept of the geometric generalized Lagrangian mean recently introduced by Gilbert and Vanneste, combined with generalized Taylor and horizontal isotropy of fluctuations as turbulent closure hypotheses. The models we obtain arise as Euler–Poincaré equations and inherit from their parent systems conservation laws for energy and potential vorticity. They are structurally and geometrically similar to Euler–Boussinesq-α and primitive equations-α models, however feature a different regularizing second order operator.

- by Gualtiero Badin and +1
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- Applied Mathematics, Oceanography, Meteorology, Hamiltonian dynamics

Classical mechanics, relativity, electrodynamics and quantum mechanics are often depicted as separate realms of physics, each with its own formalism and notion. This remains unsatisfactory with respect to the unity of nature and to the necessary number of postulates. We uncover the intrinsic connection of these areas of physics and describe them using a common symplectic Hamiltonian formalism. Our approach is based on a proper distinction between variables and constants, i.e. on a basic but rigorous ontology of time and on a simple analysis of the conditions for measurements in physics. The result put the measurement problem of quantum mechanics and the Copenhagen interpretation of the quantum mechanical wavefunction into perspective. Based on this (onto-) logic spacetime can not be fundamental and we show how a geometric interpretation of symplectic dynamics emerges from the isomorphism between corresponding Lie algebra and the representation of a Clifford algebra. We derive the di...

The Hamiltonian dynamics associated with classical, planar, Heisenberg XY models is investigated for two- and three-dimensional lattices. In addition to the conventional signatures of phase transitions, here obtained through time averages of thermodynamical observables in place of ensemble averages, qualitatively different information is derived from the temperature dependence of Lyapunov exponents. A Riemannian geometrization of Newtonian dynamics suggests consideration of other observables of geometric meaning tightly related to the largest Lyapunov exponent. The numerical computation of these observables--unusual in the study of phase transitions--sheds light on the microscopic dynamical counterpart of thermodynamics, also pointing to the existence of some major change in the geometry of the mechanical manifolds at the thermodynamical transition. Through the microcanonical definition of the entropy, a relationship between thermodynamics and the extrinsic geometry of the constant ...

En este trabajo se presenta un estudio del modelo de regresión lineal del tipo y = Θx+e, donde el error tiene distribución Secante Hiperbólica Generalizada (SHG). El método para estimar los parámetros se obtienen mediante una configuración de máxima verosimilitud expresando las ecuaciones no lineales en forma lineal (Verosimilitud Modificada). Los estimadores resultantes son expresiones analíticas en términos de valores de la muestra y, por lo tanto, son fácilmente calculables. Mediante la aplicación de varios tipos de datos, se muestra la metodología descripta anterior, y se obtienen modelos plausibles frente a las verdaderas distribuciones subyacentes de los datos.

- by Revista Ingeniería y Ciencia and +2
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- Mathematics, Algebra, Algorithms, Physics

Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly non-trivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large N) region. The ordinates λn are the positive imaginary parts of the nontrivial zeta zeros in the critical line : sn = 1 2 + iλn. The latter results are consistent with the validity of the Bohr-Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly non-trivial functional form of the potential is found via the Bohr-Sommerfeld quantization formula using the full-fledged Riemann-von Mangoldt counting formula (without any truncations) for the number N (E) of zeroes in the critical strip with imaginary part greater than 0 and less than or equal to E.

- by Klee Irwin
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- Energy, Hamiltonian dynamics, Riemann zeta function, Spectrum
- by Alain Brizard
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- Hamiltonian dynamics, Modern physics, Magnetic field, Higher Order Thinking

The goal of the present account is to review our efforts to obtain and apply a “collective” Hamiltonian for a few, approximately decoupled, adiabatic degrees of freedom, starting from a Hamiltonian system with more or many more degrees of freedom. The approach is based on an analysis of the classical limit of quantum-mechanical problems. Initially, we study the classical problem within the framework of Hamiltonian dynamics and derive a fully self-consistent theory of large-amplitude collective motion with small velocities. We derive a measure for the quality of decoupling of the collective degree of freedom. We show for several simple examples, where the classical limit is obvious, that when decoupling is good, a quantization of the collective Hamiltonian leads to accurate descriptions of the low energy properties of the systems studied. In nuclear physics problems we construct the classical Hamiltonian by means of time-dependent mean-field theory, and we transcribe our formalism to this case. We report studies of a model for monopole vibrations, of 28Si with a realistic interaction, several qualitative models of heavier nuclei, and preliminary results for a more realistic approach to heavy nuclei. Other topics included are a nuclear Born–Oppenheimer approximation for an ab initio quantum theory and a theory of the transfer of energy between collective and noncollective degrees of freedom when the decoupling is not exact. The explicit account is based on the work of the authors, but a thorough survey of other work is included.

- by Niels Walet
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- Hamiltonian dynamics, Quantum Theory, Quantum Mechanics, Mathematical Sciences
- by marco pettini
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- Engineering, Statistical Mechanics, Hamiltonian dynamics, Mathematical Sciences

Any canonical quantum theory can be understood to arise from the compatibility of the statistical geometry of distinguishable observations with the canonical Poisson structure of Hamiltonian dynamics. This geometric perspective offers a novel, background independent non-perturbative formulation of quantum gravity. We invoke a quantum version of the equivalence principle, which requires both the statistical and symplectic geometries of canonical quantum

- by Djordje Minic
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- Quantum Gravity, Hamiltonian dynamics, Quantum Theory, Matrix Theory
- by Berend Smit
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- Materials Science, Hamiltonian dynamics, Computer Simulation, Mathematical Sciences

Abstract: We develop the argument that the Gibbs-von Neumann entropy is the appropriate statistical mechanical generalisation of the thermodynamic entropy, for macroscopic and microscopic systems, whether in thermal equilibrium or not, as a consequence of Hamiltonian dynamics. The mathematical treatment utilises well known results [Gib02, Tol38, Weh78, Par89], but most importantly, incorporates a variety of arguments on the phenomenological properties of thermal states [Szi25, TQ63, HK65, GB91] and of ...

- by O. Maroney
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- Quantum Physics, Statistical Mechanics, Thermodynamics, Hamiltonian dynamics
- by Yuri Kordyukov
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- Applied Mathematics, Hamiltonian dynamics, Differential Geometry, Pure Mathematics

In this thesis we investigate the existence and stability of periodic solutions of Hamiltonian systems with a discrete symmetry. The global existence of periodic motions can be proven using the classical techniques of the calculus of variations; our particular interest is in how the stability type of the solutions thus obtained can be determined analytically using solely the variational problem and the symmetries of the system - we make no use of numerical or perturbation techniques. Instead, we use a method introduced in [41] in the context of a special case of the three-body problem. Using techniques from symplectic geometry, and specifically the Maslov index for curves of Lagrangian subspaces along the minimizing trajectories, we verify conditions which preclude the existence of eigenvalues of the monodromy matrix on the unit circle.
We study the applicability of this method in two specific cases. Firstly, we consider another special case from celestial mechanics: the hip-hop solutions of the 2N-body problem.
This is a family of Z2-symmetric, periodic orbits which arise as collision-free minimizers of the Lagrangian action on a space of symmetric loops [14, 53]. Following a symplectic
reduction, it is shown that the hip-hop solutions are brake orbits which are generically hyperbolic on the reduced energy-momentum surface.
Secondly we consider a class of natural Hamiltonian systems of two degrees of freedom with a homogeneous potential function. The associated action functional is unbounded
above and below on the function space of symmetric curves, but saddle points can be located by minimization subject to a certain natural constraint of a type first considered by
Nehari [37,38]. Using the direct method of the calculus of variations, we prove the existence of symmetric solutions of both prescribed period and prescribed energy. In the latter case, we employ a variational principle of van Groesen [55] based upon a modification of the Jacobi functional, which has not been widely used in the literature. We then demonstrate that the (constrained) minimizers are again hyperbolic brake orbits; this is the first time the method has been applied to solutions which are not globally minimizing.

- by Mark Lewis
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- Ordinary Differential Equations, Dynamical Systems, Hamiltonian dynamics
- by Oksana Koltsova and +1
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- Applied Mathematics, Hamiltonian dynamics, Pure Mathematics

Contact geometry is the odd-dimensional analogue of symplectic geometry with certain manifolds of odd dimension. It is close to symplectic geometry and like the latter it originated in questions of classical and analytical mechanics. Contact geometry has, as does symplectic geometry, broad applications in mathematical physics, geometrical optics, classical mechanics, analytical mechanics, mechanical systems, thermodynamics, geometric quantization and applied mathematics such as control theory. On the other hand, one way of solving problems in classical mechanics is with the help of the Euler-Lagrange and the Hamilton equations. In this
study, Euler-Lagrange mechanical equations as representing the motion of the body were found on contact 5-manifolds. Also, closed solutions of the differential equations found in this study are solved by symbolic computation program.

Point-vortex models are presented for the generalized Euler equations, which are characterized by a fractional Laplacian relation between the active scalar and the stream function. Special focus is given to the case of the surface quasigeostrophic (SQG) equations, for which the existence of finite-time singularities is still a matter of debate. Point-vortex trajectories are expressed using Nambu dynamics. The formulation is based on a noncanonical bracket and allows for a geometrical interpretation of trajectories as intersections of level sets of the Hamiltonian and Casimir. Within this setting, we focus on the collapse of solutions for the three-point-vortex model. In particular, we show that for SQG the collapse can be either self-similar or non-self-similar. Self-similarity occurs only when the Hamiltonian is zero, while non-self-similarity appears for nonzero values of the same. For both cases, collapse is allowed for any choice of circulations within a permitted interval. These results differ strikingly from the classical point-vortex model, where collapse is self-similar for any value of the Hamiltonian, but the vortex circulations must satisfy a strict relationship. Results may also shed a light on the formation of singularities in the SQG partial differential equations, where the singularity is thought to be reached only in a self-similar way.

- by Gualtiero Badin and +1
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- Oceanography, Meteorology, Hamiltonian dynamics, Geophysical Fluid Dynamics

Nanocomposites of linear chain of ferroelectric-ferromagnetic crystal structure is considered. It is analyzed theoretically in the motion equation method on the pursuit of magnonic excitations,lattice vibration excitations and their interactions leading to a new collective mode of excitations,the electormagnons. In this particular work, it is observed that the magnetizations and polarizations are tunable in a given temperature ranges for some specific values of the coupling order parameter.

The Hamiltonian dynamics of classical planar Heisenberg model is numerically investigated in two and three dimensions. By considering the dynamics as a geodesic flow on a suitable Riemannian manifold, it is possible to analytically estimate the largest Lyapunov exponent in terms of some curvature fluctuations. The agreement between numerical and analytical values for Lyapunov exponents is very good in a wide range of temperatures. Moreover, in the three dimensional case, in correspondence with the second order phase transition, the curvature fluctuations exibit a singular behaviour which is reproduced in an abstract geometric model suggesting that the phase transition might correspond to a change in the topology of the manifold whose geodesics are the motions of the system.

- by Cecilia Clementi and +1
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- Hamiltonian dynamics, Chaotic Dynamics, Phase transition, Temperature Dependence
- by Harris McClamroch and +1
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- Cognitive Science, Applied Mathematics, Optimal Control, Hamiltonian dynamics

In some previous papers, a geometric description of Lagrangian Mechanics on Lie algebroids has been developed. In the present paper, we give a Hamiltonian description of Mechanics on Lie algebroids. In addition, we introduce the notion of a Lagrangian submanifold of a symplectic Lie algebroid and we prove that the Lagrangian (Hamiltonian) dynamics on Lie algebroids may be described in terms of Lagrangian submanifolds of symplectic Lie algebroids. The Lagrangian (Hamiltonian) formalism on Lie algebroids permits to deal with Lagrangian (Hamiltonian) functions not defined necessarily on tangent (cotangent) bundles. Thus, we may apply our results to the projection of Lagrangian (Hamiltonian) functions which are invariant under the action of a symmetry Lie group. As a consequence, we obtain that Lagrange-Poincare (Hamilton-Poincare) equations are the Euler-Lagrange (Hamilton) equations associated with the corresponding Atiyah algebroid. Moreover, we prove that Lagrange-Poincare (Hamilton-Poincare) equations are the local equations defining certain Lagrangian submanifolds of symplectic Atiyah algebroids.

- by Eduardo Martinez
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- Hamiltonian dynamics, Mathematical Sciences, Physical sciences, Time Dependent
- by I. Mamaev
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- Applied Mathematics, Mathematical Physics, Lie Algebra, Nonlinear dynamics

A Hamiitonisation for non-holonomic dynamical systems is developed. An example is given.

In the thermodynamic limit, systems with long-range interactions do not relax to equilibrium, but become
trapped in nonequilibrium stationary states. For a finite number of particles a nonequilibrium state has a finite
lifetime, so that eventually a system will relax to thermodynamic equilibrium. The time that a system remains
trapped in a quasistationary state (QSS) scales with the number of particles as N δ , with δ > 0, and diverges in
the thermodynamic limit. In this paper we will explore the role of chaotic dynamics on the time that a system
remains trapped in a QSS. We discover that chaos, measured by the Lyapunov exponents, favors faster relaxation
to equilibrium. Surprisingly, weak chaos favors faster relaxation than strong chaos.

- by Felipe Leite Antunes and +1
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- Hamiltonian dynamics, Dynamical systems and Chaos, Long-Range Interactions
- by I. Hoveijn
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- Hamiltonian dynamics, Pure Mathematics, Numerical Method, Case Study