Related papers: Dynamical symmetries of the Kepler problem

This article reviews the role of hidden symmetries of dynamics in the study of physical systems, from the basic concepts of symmetries in phase space to the forefront of current research. Such symmetries emerge naturally in the description…

It is shown that several Hamiltonian systems possessing dynamical or hidden symmetries can be realized within the framework of Nambu's generalized mechanics. Among such systems are the SU(n)-isotropic harmonic oscillator and the…

We study a long-recognised but under-appreciated symmetry called "dynamical similarity" and illustrate its relevance to many important conceptual problems in fundamental physics. Dynamical similarities are general transformations of a…

We study the nonholonomic motion of a point particle on the Heisenberg group around the fixed "sun" whose potential is given by the fundamental solution of the sub-Laplacian. Unlike arXiv:1212.2713 where the variational problem is studied…

Simple examples are used to introduce and examine symmetries of open quantum dynamics that can be described by unitary operators. For the Hamiltonian dynamics of an entire closed system, the symmetry takes the expected form which, when the…

It is shown that all spherical symmetric potentials are capable of producing dynamical symmetries in classical one-body motions, thanks to the inevitable existence of symmetry axes associated with turning points for corresponding…

Isochrone potentials, as defined by Michel H\'enon in the fifties, are spherically symmetric potentials within which a particle orbits with a radial period that is independent of its angular momentum. Isochrone potentials encompass the…

The Kepler-Heisenberg problem is that of determining the motion of a planet around a sun in the sub-Riemannian Heisenberg group. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the…

Here we provide an overview of what is known, and what is not known, about an interesting dynamical system known as the Kepler-Heisenberg problem. The main idea is to pose a version of the classical Kepler problem of planetary motion, but…

We split the generic conformal mechanical system into a "radial" and an "angular" part, where the latter is defined as the Hamiltonian system on the orbit of the conformal group, with the Casimir function in the role of the Hamiltonian. We…

Posing Kepler's problem of motion around a fixed "sun" requires the geometric mechanician to choose a metric and a Laplacian. The metric provides the kinetic energy. The fundamental solution to the Laplacian (with delta source at the "sun")…

Many theories of physical interest, which admit a Hamiltonian description, exhibit symmetries under a particular class of non - strictly canonical transformation, known as dynamical similarities. The presence of such symmetries allows a…

After a brief survey of the definition and the properties of Lambda-symmetries in the general context of dynamical systems, the notion of "Lambda-constant of motion'' for Hamiltonian equations is introduced. If the Hamiltonian problem is…

This work presents the basic elements of the formalism involved in the treatment of Hamiltonian dynamical systems with symmetry and the geometrical description of collective motion.

The additional hidden symmetry of the Coulomb-Kepler problem is reviewed in classical as well as in quantum mechanics. The main purpose is to elucidate the role of this kind of symmetries in the reduction of physical problems, to show…

For each simple euclidean Jordan algebra $V$ of rank $\rho$ and degree $\delta$, we introduce a family of classical dynamic problems. These dynamical problems all share the characteristic features of the Kepler problem for planetary…

The action of the quantum mechanical volume operator, introduced in connection with a symmetric representation of the three-body problem and recently recognized to play a fundamental role in discretized quantum gravity models, can be given…

The problem of Kepler dynamics on a conformable Poisson manifold is addressed. The Hamiltonian function is defined and the related Hamiltonian vector field governing the dynamics is derived, which leads to a modified Newton second law.…

In this thesis, we study the one parameter point transformations which leave invariant the differential equations. In particular we study the Lie and the Noether point symmetries of second order differential equations. We establish a new…

To study discrete dynamical systems of different types --- deterministic, statistical and quantum --- we develop various approaches. We introduce the concept of a system of discrete relations on an abstract simplicial complex and develop…