First Integrals of Extended Hamiltonians in n+ 1 Dimensions Generated by Powers of an Operator⋆ (original) (raw)
Related papers
2011
In previous papers we determined necessary and sufficient conditions for the existence of a class of natural Hamiltonians with non-trivial first integrals of arbitrarily high degree in the momenta. Such Hamiltonians were characterized as (n+1)-dimensional extensions of n-dimensional Hamiltonians on constant-curvature (pseudo-)Riemannian manifolds Q. In this paper, we generalize that approach in various directions, we obtain an explicit expression for the first integrals, holding on the more general case of Hamiltonians on Poisson manifolds, and show how the construction of above is made possible by the existence on Q of particular conformal Killing tensors or, equivalently, particular conformal master symmetries of the geodesic equations. Finally, we consider the problem of Laplace-Beltrami quantization of these first integrals when they are of second-degree.
First integrals and Darboux polynomials of natural polynomial Hamiltonian systems
Physics Letters A, 2010
In this Letter we study some aspects of the relationship between the existence of Darboux polynomials and additional polynomial first integrals in natural polynomial Hamiltonian systems with a finite number of degrees of freedom. More precisely, first we improve results of the paper of Maciejewski and Przybylska [A.J. Maciejewski, M. Przybylska, Phys. Lett. A 326 and after we answer two open questions presented in that paper.
Extensions of natural Hamiltonians
2013
Given an n-dimensional natural Hamiltonian L on a Riemannian or pseudo-Riemannian manifold, we call "extension" of L the n+1 dimensional Hamiltonian H = 1 2 p 2 u + α(u)L + β(u) with new canonically conjugated coordinates (u, pu). For suitable L, the functions α and β can be chosen depending on any natural number m such that H admits an extra polynomial first integral in the momenta of degree m, explicitly determined in the form of the mth power of a differential operator applied to a certain function of coordinates and momenta. In particular, if L is maximally superintegrable (MS) then H is MS also. Therefore, the extension procedure allows the creation of new superintegrable systems from old ones. For m=2, the extra first integral generated by the extension procedure determines a second-order symmetry operator of a Laplace-Beltrami quantization of H, modified by taking in account the curvature of the configuration manifold. The extension procedure can be applied to several Hamiltonian systems, including the three-body Calogero and Wolfes systems (without harmonic term), the Tremblay-Turbiner-Winternitz system and n-dimensional anisotropic harmonic oscillators. We propose here a short review of the known results of the theory and some previews of new ones.
Integrability of Hamiltonians with polynomial potentials
Journal of Computational and Applied Mathematics, 2003
In this work, we study dynamical systems with polynomial potentials-such as those of Henon-Heiles, Yang-Mills and various generalizations-by means of the nonintegrability theory developed by the authors. All these problems have also been investigated by using other theories like those proposed by Ziglin, Yoshida, Morales or the Painlevà e analysis. In the examples considered, our method allows us to reproduce with quite less work or even to improve the results obtained by other authors.
Polynomial and Rational First Integrals for Non–Autonomous Polynomial Hamiltonian Systems
Dynamic Systems and Applications, 2020
Known results on the existence of polynomial and rational first integrals for autonomous polynomial Hamiltonian systems are extended to non-autonomous polynomial Hamiltonian systems invariant under an involution. The key tool for proving these results is the existence of Darboux polynomials for the nonautonomous polynomial Hamiltonian systems.
On integrals of the third degree in momenta
1999
Consider a Riemannian metric on a surface, and let the geodesic ow of the metric have a second integral that is a third degree polynomial in the momenta. Then we can naturally construct a vector eld on the surface. We show that the vector eld preserves the volume of the surface, and therefore is a Hamiltonian vector eld. As examples we treat the Goryachev-Chaplygin top, the Toda lattice and the Calogero-Moser system, and construct their global Hamiltonians. We show that the simplest choice of Hamiltonian leads to the Toda lattice.
On Integrals of Third Degree in Momenta
2002
Consider a Riemannian metric on a surface, and let the geodesic flow of the metric have a second integral that is a third degree polynomial in the momenta. Then we can naturally construct a vector field on the surface. We show that the vector field preserves the volume of the surface, and therefore is a Hamiltonian vector field. As examples we treat the Goryachev-Chaplygin top, the Toda lattice and the Calogero-Moser system, and construct their global Hamiltonians. We show that the simplest choice of Hamiltonian leads to the Toda lattice.
Extensions of Hamiltonian systems dependent on a rational parameter
2013
The technique of "extension" allows to build (n+1)-dimensional Hamiltonian systems with a non-trivial polynomial in the momenta first integral of any given degree starting from a n-dimensional Hamiltonian satisfying some additional properties. Until now, the application of the method was restricted to integer values of a certain fundamental parameter determining the degree of the additional first integral. In this article we show how this technique can be generalized to any rational value of the same parameter. Several examples are given, among them the anisotropic oscillator and a special case of the Tremblay-Turbiner-Winternitz system.
Non-integrability of some hamiltonians with rational potentials
Discrete and Continuous Dynamical Systems - Series B, 2008
In this work we compute the families of classical Hamiltonians in two degrees of freedom in which the Normal Variational Equation around an invariant plane falls in Schrödinger type with polynomial or trigonometrical potential. We analyze the integrability of Normal Variational Equation in Liouvillian sense using the Kovacic's algorithm. We compute all Galois groups of Schrödinger type equations with polynomial potential. We also introduce a method of algebrization that transforms equations with transcendental coefficients in equations with rational coefficients without changing essentially the Galoisian structure of the equation. We obtain Galoisian obstructions to existence of a rational first integral of the original Hamiltonian via Morales-Ramis theory.
Rational integrals of 2-dimensional geodesic flows: New examples
Journal of Geometry and Physics, 2021
This paper is devoted to searching for Riemannian metrics on 2surfaces whose geodesic flows admit a rational in momenta first integral with a linear numerator and denominator. The explicit examples of metrics and such integrals are constructed. Few superintegrable systems are found having both a polynomial and a rational integrals which are functionally independent of the Hamiltonian.