N-dimensional isotropic oscillator (original) (raw)
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Integrable perturbations of the N-dimensional isotropic oscillator
Physics Letters A, 2010
Two new families of completely integrable perturbations of the N-dimensional isotropic harmonic oscillator Hamiltonian are presented. Such perturbations depend on arbitrary functions and N free parameters and their integrals of motion are explicitly constructed by making use of an underlying h 6-coalgebra symmetry. Several known integrable Hamiltonians in low dimensions are obtained as particular specializations of the general results here presented. An alternative route for the integrability of all these systems is provided by a suitable canonical transformation which, in turn, opens the possibility of adding (N − 1) 'Rosochatius' terms that preserve the complete integrability of all these models.
Symmetries and integrals of motion of a superintegrable deformed oscillator
Annals of Physics
The symmetry structure of twodimensional nonlinear isotropic oscillator, introduced in Physica D237 (2008) 505, is discussed. It is shown that it possesses three independent integrals of motion which can be chosen in such a way that they span SU (2), E(2) or SU (1, 1) algebras, depending on the value of total energy. They generate the infinitesimal canonical symmetry transformations; integrability of the latter is analyzed. The results are then generalized to the case of arbitrary number of degrees of freedom.
A new integrable anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane
Journal of Physics a-Mathematical and Theoretical, 2014
A new integrable generalization to the 2D sphere S 2 and to the hyperbolic space H 2 of the 2D Euclidean anisotropic oscillator Hamiltonian with Rosochatius (centrifugal) terms is presented, and its curved integral of the motion is shown to be quadratic in the momenta. In order to construct such a new integrable Hamiltonian H κ , we will make use of a group theoretical approach in which the curvature κ of the underlying space will be treated as an additional (contraction) parameter, and we will make extensive use of projective coordinates and their associated phase spaces. It turns out that when the oscillator parameters Ω 1 and Ω 2 are such that Ω 2 = 4Ω 1 , the system turns out to be the well-known superintegrable 1 : 2 oscillator on S 2 and H 2. Nevertheless, numerical integration of the trajectories of H κ suggests that for other values of the parameters Ω 1 and Ω 2 the system is not superintegrable. In this way, we support the conjecture that for each commensurate (and thus superintegrable) m : n Euclidean oscillator there exists a two-parametric family of curved integrable (but not superintegrable) oscillators that turns out to be superintegrable only when the parameters are tuned to the m : n commensurability condition.
arXiv (Cornell University), 2021
We study four particular 3-dimensional natural Hamiltonian systems defined in conformally Euclidean spaces. We prove their superintegrability and we obtain, in the four cases, the maximal number of functionally independent integrals of motion. The two first systems are related to the 3-dimensional isotropic oscillator and the superintegrability is quadratic. The third system is obtained as a continuous deformation of an oscillator with ratio of frequencies 1:1:2 and with three additional nonlinear terms of the form k 2 /x 2 , k 3 /y 2 and k 4 /z 2 , and the fourth system is obtained as a deformation of the Kepler Hamiltonian also with these three particular nonlinear terms. These third and fourth systems are superintegrable but with higher-order constants of motion. The four systems depend on a real parameter in such a way that they are continuous functions of the parameter (in a certain domain of the parameter) and in the limit of such parameter going to zero the Euclidean dynamics is recovered.
Journal of Physics A: Mathematical and Theoretical
We study four particular 3-dimensional natural Hamiltonian systems defined in conformally Euclidean spaces. We prove their superintegrability and we obtain, in the four cases, the maximal number of functionally independent integrals of motion. The two first systems are related to the 3-dimensional isotropic oscillator and the superintegrability is quadratic. The third system is obtained as a continuous deformation of an oscillator with ratio of frequencies 1:1:2 and with three additional nonlinear terms of the form k 2 /x 2 , k 3 /y 2 and k 4 /z 2 , and the fourth system is obtained as a deformation of the Kepler Hamiltonian also with these three particular nonlinear terms. These third and fourth systems are superintegrable but with higher-order constants of motion. The four systems depend on a real parameter in such a way that they are continuous functions of the parameter (in a certain domain of the parameter) and in the limit of such parameter going to zero the Euclidean dynamics is recovered.
A Continuum of Hamiltonian Structures for the Two-Dimensional Isotropic Harmonic Oscillator
International Journal of Pure and Apllied Mathematics, 2013
We show the existence of a continuum of Hamiltonian structures for the two-dimensional isotropic harmonic oscillator. In particular, a continuum of Hamiltonian structures having noncommutative coordinates is presented. A study of the symmetries of these structures is performed and their physical plausibility is discussed.
N -dimensional integrability from two-photon coalgebra symmetry
Journal of Physics A: Mathematical and Theoretical, 2009
A wide class of Hamiltonian systems with N degrees of freedom and endowed with, at least, (N −2) functionally independent integrals of motion in involution is constructed by making use of the two-photon Lie-Poisson coalgebra (h 6 , ∆). The set of (N − 2) constants of the motion is shown to be a universal one for all these Hamiltonians, irrespectively of the dependence of the latter on several arbitrary functions and N free parameters. Within this large class of quasi-integrable N-dimensional Hamiltonians, new families of completely integrable systems are identified by finding explicitly a new independent integral I through the analysis of the sub-coalgebra structure of h 6. In particular, new completely integrable Ndimensional Hamiltonians describing natural systems, geodesic flows and static electromagnetic Hamiltonians are presented.
A note on the non-integrability of some Hamiltonian systems with a homogeneous potential
Methods and Applications of Analysis, 2001
We obtain a non-integrability result on Hamiltonian Systems with a homogeneous potential with an arbitrary number of degrees of freedom which generalizes a Yoshida's Theorem . Except for the cases when the degree of homogeneity of the potential is equal to two or minus two, only a discrete set of families of these type of potentials are compatible with the complete integrability condition. We illustrate this result with two examples: the collinear problem of three bodies and a highly symmetrical family introduced by Umeno ([6]).
Integrable and superintegrable extensions of the rational Calogero–Moser model in three dimensions
Journal of Physics A: Mathematical and Theoretical
We consider a class of Hamiltonian systems in 3 degrees of freedom, with a particular type of quadratic integral and which includes the rational Calogero–Moser system as a particular case. For the general class, we introduce separation coordinates to find the general separable (and therefore Liouville integrable) system, with two quadratic integrals. This gives a coupling of the Calogero–Moser system with a large class of potentials, generalising the series of potentials which are separable in parabolic coordinates. Particular cases are superintegrable, including Kepler and a resonant oscillator. The initial calculations of the paper are concerned with the flat (Cartesian type) kinetic energy, but in section 5, we introduce a conformal factor φ to H and extend the two quadratic integrals to this case. All the previous results are generalised to this case. We then introduce some two and three dimensional symmetry algebras of the Kinetic energy (Killing vectors), which restrict the co...