Coupling Constant Metamorphosis Research Articles

Integrability of Hamiltonian systems with gyroscopic term

We study the integrability of 2D Hamiltonian systems H_mu =frac{1}{2}(p_1^2+p_2^2) + omega (p_1q_2-p_2q_1) -tfrac{mu }{r}+ V(q_1,q_2), where r^2=q_1^2+q_2^2, and potential V(q_1,q_2) is a homogeneous rational function of integer degree k. The main result states that under very general assumptions: mu omega ne 0, |k|>2 and V(1,mathrm {i})ne 0 or V(-1,mathrm {i})ne 0, the system is not integrable. It was obtained by combining the Levi-Civita regularization, differential Galois methods, and the so-called coupling constant metamorphosis transformation. The proof that the regularized version of the problem is not integrable contains the most important theoretical results, which can be applied to study integrability of other problems.

Coupling-constant metamorphosis in [formula omitted]-invariant systems

It is shown that, if the energy in the Schwarzian mechanics (SM) is equal to the coupling constant in the de Alfaro-Fubini-Furlan (dAFF) model, there exists a link between these two systems. In particular, the equation of motion, sl(2,R)-symmetry transformations and the corresponding conserved charges of SM can be derived from those of dAFF model by applying a coordinate transformation of a special type, while the general solution of dAFF system maps to the velocity function of SM. It is also demonstrated that the Hamiltonian of SM can be obtained from the Hamiltonian of dAFF model by applying coupling-constant metamorphosis and the oxidation procedure.

Stäckel Equivalence of Non-Degenerate Superintegrable Systems, and Invariant Quadrics

A non-degenerate second-order maximally conformally superintegrable system in dimension 2 naturally gives rise to a quadric with position dependent coefficients. It is shown how the system's Stäckel class can be obtained from this associated quadric.The Stäckel class of a second-order maximally conformally superintegrable system is its equivalence class under Stäckel transformations, i.e., under coupling-constant metamorphosis.

Projectively equivalent 2-dimensional superintegrable systems with projective symmetries

This paper combines two classical theories, namely metric projective differential geometry and superintegrability. We study superintegrable systems on 2-dimensional geometries that share the same geodesics, viewed as unparametrized curves. We give a definition of projective equivalence of such systems, which may be considered the projective analog of (conformal) Stäckel equivalence (coupling constant metamorphosis). Then, we discuss the transformation behavior for projectively equivalent superintegrable systems and find that the potential on a projectively equivalent geometry can be reconstructed from a characteristic vector field. Moreover, potentials of projectively equivalent Hamiltonians follow a linear superimposition rule. The techniques are applied to several examples. In particular, we use them to classify, up to Stäckel equivalence, the superintegrable systems on geometries with one, non-trivial projective symmetry.

Schwarzian derivative treatment of the quantum second-order supersymmetry anomaly, and coupling-constant metamorphosis

A canonical quantization scheme applied to a classical supersymmetric system with quadratic in momentum supercharges gives rise to a quantum anomaly problem described by a specific term to be quadratic in Planck constant. We reveal a close relationship between the anomaly and the Schwarzian derivative, and specify a quantization prescription which generates the anomaly-free supersymmetric quantum system with second order supercharges. We also discuss the phenomenon of a coupling-constant metamorphosis that associates quantum systems with the first-order supersymmetry to the systems with the second-order supercharges.

Recurrence approach and higher rank cubic algebras for the N-dimensional superintegrable systems

By applying the recurrence approach and coupling constant metamorphosis, we construct higher order integrals of motion for the Stackel equivalents of the N-dimensional superintegrable Kepler–Coulomb model with non-central terms and the double singular oscillators of type (). We show how the integrals of motion generate higher rank cubic algebra with structure constants involving Casimir operators of the Lie algebras L1 and L2. The realizations of the cubic algebras in terms of deformed oscillators enable us to construct finite dimensional unitary representations and derive the degenerate energy spectra of the corresponding superintegrable systems.

Extended Hamiltonians, Coupling-Constant Metamorphosis and the Post-Winternitz System

The coupling-constant metamorphosis is applied to modified extended Hamiltonians and sufficient conditions are found in order that the transformed high-degree first integral of the transformed Hamiltonian is determined by the same algorithm which computes the corresponding first integral of the original extended Hamiltonian. As examples, we consider the Post-Winternitz system and the 2D caged anisotropic oscillator.

Null lifts and projective dynamics

We describe natural Hamiltonian systems using projective geometry. The null lift procedure endows the tangent bundle with a projective structure where the null Hamiltonian is identified with a projective conic and induces a Weyl geometry. Projective transformations generate a set of known and new dualities between Hamiltonian systems, as for example the phenomenon of coupling-constant metamorphosis. We conclude outlining how this construction can be extended to the quantum case for Eisenhart–Duval lifts.

Coupling constant metamorphosis as an integrability-preserving transformation for general finite-dimensional dynamical systems and ODEs

In the present Letter we extend the multiparameter coupling constant metamorphosis, also known as the generalized Stäckel transform, from Hamiltonian dynamical systems to general finite-dimensional dynamical systems and ODEs. This transform interchanges the values of integrals of motion with the parameters these integrals depend on but leaves the phase space coordinates intact. Sufficient conditions under which the transformation in question preserves integrability and a simple formula relating the solutions of the original system to those of the transformed one are given.

Coupling constant metamorphosis and Nth-order symmetries in classical and quantum mechanics

We review the fundamentals of coupling constant metamorphosis (CCM) and the Stäckel transform, and apply them to map integrable and superintegrable systems of all orders into other such systems on different manifolds. In general, CCM does not preserve the order of constants of the motion or even take polynomials in the momenta to polynomials in the momenta. We study specializations of these actions which preserve polynomials and also the structure of the symmetry algebras in both the classical and quantum cases. We give several examples of non-constant curvature third- and fourth-order superintegrable systems in two space dimensions obtained via CCM, with some details on the structure of the symmetry algebras preserved by the transform action.

Generalized Stäckel transform and reciprocal transformations for finite-dimensional integrable systems

We present a multiparameter generalization of the Stäckel transform (the latter is also known as the coupling-constant metamorphosis) and show that under certain conditions this generalized Stäckel transform preserves Liouville integrability, noncommutative integrability and superintegrability. The corresponding transformation for the equations of motion proves to be nothing but a reciprocal transformation of a special form, and we investigate the properties of this reciprocal transformation. Finally, we show that the Hamiltonians of the systems possessing separation curves of apparently very different form can be related through a suitably chosen generalized Stäckel transform.

Equivalence of superintegrable systems in two dimensions

In two dimensions, all nondegenerate superintegrable systems having constants quadratic in the momenta possess a quadratic algebra. In this paper, it is shown how the quadratic algebra can be used to classify all such systems into seven classes that are preserved by coupling constant metamorphosis.

Second order superintegrable systems in conformally flat spaces. IV. The classical 3D Stäckel transform and 3D classification theory

This article is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems, both classical and quantum, in conformally flat spaces. In the first part of the article we study the Stäckel transform (or coupling constant metamorphosis) as an invertible mapping between classical superintegrable systems on different three-dimensional spaces. We show first that all superintegrable systems with nondegenerate potentials are multiseparable and then that each such system on any conformally flat space is Stäckel equivalent to a system on a constant curvature space. In the second part of the article we classify all the superintegrable systems that admit separation in generic coordinates. We find that there are eight families of these systems.

Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform

This paper is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems, both classical and quantum, in conformally flat spaces. Here we study the Stäckel transform (or coupling constant metamorphosis) as an invertible mapping between classical superintegrable systems on different spaces. Through the use of this tool we derive and classify for the first time all two-dimensional (2D) superintegrable systems. The underlying spaces are exactly those derived by Koenigs in his remarkable paper giving all 2D manifolds (with zero potential) that admit at least three second order symmetries. Our derivation is very simple and quite distinct. We also show that every superintegrable system is the Stäckel transform of a superintegrable system on a constant curvature space.

Superintegrable systems in Darboux spaces

Almost all research on superintegrable potentials concerns spaces of constant curvature. In this paper we find by exhaustive calculation, all superintegrable potentials in the four Darboux spaces of revolution that have at least two integrals of motion quadratic in the momenta, in addition to the Hamiltonian. These are two-dimensional spaces of nonconstant curvature. It turns out that all of these potentials are equivalent to superintegrable potentials in complex Euclidean 2-space or on the complex 2-sphere, via “coupling constant metamorphosis” (or equivalently, via Stäckel multiplier transformations). We present a table of the results.

Lax pair tensors in arbitrary dimensions

A recipe is presented for obtaining Lax tensors for any n-dimensional Hamiltonian system admitting a Lax representation of dimension n. Our approach is to use the Jacobi geometry and coupling-constant metamorphosis to obtain a geometric Lax formulation. We also exploit the results to construct integrable spacetimes, satisfying the weak energy condition.

On the ℏ2correction terms in quantum integrability

The authors study the question of quantum integrability for two-dimensional Hamiltonian systems with special attention on the 2 correction terms in the potential. A class of Hamiltonians of type H=1/2(px2 + py2)+ v4y4 + y2(f2(x) + α 6) + f0(x) with a second invariant of type I = px4 + A(x,y)px2 + B(x,y)pxpy + C(x,y) py2 + D(x,y) is considered. The general solution for f2 involves elliptic integrals. For quantum integrability the potential must be modified with 2-dependent terms. They construct a point transformation which, after coupling constant metamorphosis, takes the Hamiltonian to a new quantum integrable Hamiltonian for which no correction terms are necessary. The new system does not in general have a flat space kinetic part.

Coupling-Constant Metamorphosis and Duality between Integrable Hamiltonian Systems

We introduce a noncanonical ("new-time") transformation which exchanges the roles of a coupling constant and the energy in Hamiltonian systems while preserving integrability. In this way we can construct new integrable systems and, for example, explain the observed duality between the H\'enon-Heiles and Holt models. It is shown that the transformation can sometimes connect weak- and full-Painlev\'e Hamiltonians. We also discuss quantum integrability and find the origin of the deformation $\ensuremath{-}\frac{5}{72}{\ensuremath{\hbar}}^{2}{x}^{\ensuremath{-}2}$.