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The Kolmogorov–Arnold–Moser (KAM) theorem is a result in dynamical systems about the persistence of quasiperiodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the perturbation theory of classical mechanics.
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In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation vector diverge at a rate given by
Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mechanics and classical chaos?" The correspondence principle states that classical mechanics is the classical limit of quantum mechanics, specifically in the limit as the ratio of Planck's constant to the action of the system tends to zero. If this is true, then there must be quantum mechanisms underlying classical chaos. If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, how can exponential sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle limit of quantum mechanics?
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can be studied in both Hamiltonian mechanics and dynamical systems theory.
In physics, symmetry breaking is a phenomenon where a disordered but symmetric state collapses into an ordered, but less symmetric state. This collapse is often one of many possible bifurcations that a particle can take as it approaches a lower energy state. Due to the many possibilities, an observer may assume the result of the collapse to be arbitrary. This phenomenon is fundamental to quantum field theory (QFT), and further, contemporary understandings of physics. Specifically, it plays a central role in the Glashow–Weinberg–Salam model which forms part of the Standard model modelling the electroweak sector.
In quantum field theory, a nonlinear σ model describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. The non-linear σ-model was introduced by Gell-Mann & Lévy, who named it after a field corresponding to a spinless meson called σ in their model. This article deals primarily with the quantization of the non-linear sigma model; please refer to the base article on the sigma model for general definitions and classical (non-quantum) formulations and results.
In mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations. They are widely used in nonlinear dynamics, molecular dynamics, discrete element methods, accelerator physics, plasma physics, quantum physics, and celestial mechanics.
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations occur in both continuous systems and discrete systems.
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals that its motion is confined to a submanifold of much smaller dimensionality than that of its phase space.
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In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a -dimensional symplectic manifold for which the following conditions hold:
The Nekhoroshev estimates are an important result in the theory of Hamiltonian systems concerning the long-time stability of solutions of integrable systems under a small perturbation of the Hamiltonian. The first paper on the subject was written by Nikolay Nekhoroshev in 1971.
In applied mathematics, Arnold diffusion is the phenomenon of instability of integrable Hamiltonian systems. The phenomenon is named after Vladimir Arnold who was the first to publish a result in the field in 1964. More precisely, Arnold diffusion refers to results asserting the existence of solutions to nearly integrable Hamiltonian systems that exhibit a significant change in the action variables.
In physics, the n-body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, and visible stars. In the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem. The n-body problem in general relativity is considerably more difficult to solve due to additional factors like time and space distortions.
Lars Håkan Eliasson is a Swedish mathematician.
Zhihong "Jeff" Xia is a Chinese-American mathematician.
Luigi Chierchia is an Italian mathematician, specializing in nonlinear differential equations, mathematical physics, and dynamical systems.
DmitriiValerevich Treschev is a Russian mathematician and mathematical physicist, specializing in dynamical systems of classical mechanics.
References
- ↑ "Antonio Giorgilli, Prof. ordinario di Fisica Matematica". Università deli Studi di Milano.
- 1 2 "Curriculum accademico di Antonio Giorgilli". Università degli Studi di Milano.
- ↑ Giorgilli, Antonio (1998). "On the problem of stability for near to integrable Hamiltonian systems". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. III. pp. 143–152.
