Related papers: Jacobi fields of completely integrable Hamiltonian…
We construct a large family of evidently integrable Hamiltonian systems which are generalizations of the KM system. The Hamiltonian vector field is homogeneous cubic but in a number of cases a simple change of variables transforms such a…
We consider a natural Hamiltonian system of $n$ degrees of freedom with a homogeneous potential. Such system is called partially integrable if it admits $1<l<n$ independent and commuting first integrals, and it is called super-integrable if…
We study several aspects of the regular deformations of completely integrable systems. Namely, we prove the existence of a Hamiltonian normal form for these deformations and we show the necessary and sufficient conditions a perturbation has…
Recently, M. de Le\'on el al. ([8]) have developed a geometrical description of Hamilton-Jacobi theory for multisymplectic field theory. In our paper we analyse in the same spirit a special kind of field theories which are gauge field…
In this work we discuss the natural appearance of the Generalized Brackets in systems with non-involutive (equivalent to second class) constraints in the Hamilton-Jacobi formalism. We show how a consistent geometric interpretation of the…
We develop a geometric framework for the exact integration of Hamiltonian systems based on triangular closure relations among a finite family of functions. Unlike Liouville-Arnold integrability and its noncommutative generalizations, the…
The non-commutative integrability (NCI) is a property fulfilled by some Hamiltonian systems that ensures, among other things, the exact solvability of their corresponding equations of motion. The latter means that an "explicit formula" for…
In this paper we introduce a notion of integrability in the non autonomous sense. For the cases of 1 + 1/2 degrees of freedom and quadratic homogeneous Hamiltonians of 2 + 1/2 degrees of freedom we prove that this notion is equivalent to…
We study integrable and superintegrable systems with magnetic field possessing quadratic integrals of motion on the three-dimensional Euclidean space. In contrast with the case without vector potential, the corresponding integrals may no…
We introduce a family of exactly-solvable two-dimensional Hamiltonians whose wave functions are given in terms of Laguerre and exceptional Jacobi polynomials. The Hamiltonians contain purely quantum terms which vanish in the classical limit…
The problem of the motion of a charged particle in an electric dipole field is used to illustrate that the Hamilton-Jacobi method does not necessarily give all solutions to the equations of motion of a mechanical system. The mathematical…
Near-integrability is usually associated with smooth small perturbations of smooth integrable systems. Studying integrable mechanical Hamiltonian flows with impacts that respect the symmetries of the integrable structure provides an…
We state two recent results concerning the linearization of integrable systems on generalised Jacobians. Then we apply this to the (complexified) spherical pendulum.
A novel Hamiltonian system in n dimensions which admits the maximal number 2n-1 of functionally independent, quadratic first integrals is presented. This system turns out to be the first example of a maximally superintegrable Hamiltonian on…
In the paper, we consider a completely integrable Hamiltonian system with three degrees of freedom found by V.V.Sokolov and A.V.Tsiganov. This system is known as the generalized two-field gyrostat. For the case of only gyroscopic forces…
The classical Liouvile integrability means that there exist $n$ independent first integrals in involution for $2n$-dimensional phase space. However, in the infinite-dimensional case, an infinite number of independent first integrals in…
By Poissonization of Jacobi structures on real three-dimensional Lie groups $\mathbf{G}$ and using the realizations of their Lie algebras, we obtain integrable bi-Hamiltonian systems on $\mathbf{G}\otimes \mathbb{R}$.
This paper is devoted to the study of symplectic manifolds and their connection with Hamiltonian dynamical systems. We review some properties and operations on these manifolds and see how they intervene when studying the complete…
In this second paper of the series we specify general theory developed in the first paper. Here we study the structure of Jacobi fields in the case of an analytic system and piece-wise analytic control. Moreover, we consider only…
It is known that, if a point in $R^n$ is driven by a bounded below potential $V$, whose gradient is always in a closed convex cone which contains no lines, then the velocity has a finite limit as time goes to $+\infty$. The components of…