Natural Hamiltonian Systems Research Articles

On particular integrability in classical mechanics

In this study the notion of particular integrability in Classical Mechanics, introduced in Turbiner (2013 J. Phys. A: Math. Theor. 46 025203), is revisited within the formalism of symplectic geometry. A particular integral I is a function not necessarily conserved in the whole phase space T∗Q but when restricted to a certain invariant subspace W⊆T∗Q it becomes a Liouville first integral. For natural Hamiltonian systems, it is demonstrated that such a function I allows us to construct a lower dimensional Hamiltonian in W . This symmetry reduction is intimately related with a phenomenon beyond separation of variables and it is based on an adaptive application of the classical results due to Lie and Liouville on integrability. Three physically relevant systems are used to illustrate the underlying key aspects of the symplectic theory approach to particular integrability: (I) the integrable central-force problem, (II) the chaotic two-body Coulomb system in a constant magnetic field as well as (III) the N-body system.

Non-integrability and chaos for natural Hamiltonian systems with a random potential

Consider the ensemble of Gaussian random potentials {VL(q)}L=1∞ on the d-dimensional torus where, essentially, VL(q) is a real-valued trigonometric polynomial of degree L whose coefficients are independent standard normal variables. Our main result ensures that, with a probability tending to 1 as L→∞, the dynamical system associated with the natural Hamiltonian function defined by this random potential, HL:=12|p|2+VL(q), exhibits a number of chaotic regions which coexist with a positive-volume set of invariant tori. In particular, these systems are typically neither integrable with non-degenerate first integrals nor ergodic. An analogous result for random natural Hamiltonian systems defined on the cotangent bundle of an arbitrary compact Riemannian manifold is presented too.

Separation of Variables and Superintegrability on Riemannian Coverings

We introduce Stäckel separable coordinates on the covering manifolds $M_k$, where $k$ is a rational parameter, of certain constant-curvature Riemannian manifolds with the structure of warped manifold. These covering manifolds appear implicitly in literature as connected with superintegrable systems with polynomial in the momenta first integrals of arbitrarily high degree, such as the Tremblay-Turbiner-Winternitz system. We study here for the first time multiseparability and superintegrability of natural Hamiltonian systems on these manifolds and see how these properties depend on the parameter $k$.

Geodesic flows of c-projectively equivalent metrics are quantum integrable

Given two c-projectively equivalent metrics on a Kähler manifold, we show that canoncially constructed Poisson-commuting integrals of motion of the geodesic flow, linear and quadratic in momenta, also commute as quantum operators. The methods employed here also provide a proof of a similar statement in the case of projective equivalence. We also investigate the addition of potentials, i.e. the generalization to natural Hamiltonian systems. We show that commuting operators lead to separation of variables for Schrödinger’s equation.

On two-dimensional Hamiltonian systems with sixth-order integrals of motion

We obtain 21 two-dimensional natural Hamiltonian systems with sextic invariants, which are polynomials of the sixth order in momenta. Following to Bertrand, Darboux, and Drach these results of the standard brute force experiments can be applied to construct a new mathematical theory.

Superintegrability of three-dimensional Hamiltonian systems with conformally Euclidean metrics. Oscillator-related and Kepler-related systems

We study four particular three-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 three-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.

On the integrability of new examples of two-dimensional Hamiltonian systems in curved spaces

In this work, we inspect the integrability of a natural Hamiltonian system interpreted physically as the motion of a particle in the Euclidean plane under the effect of conservative forces derived from a certain type of a non-homogeneous potential. We announce the necessary conditions for its integrability by using the differential Galois theorem. We present three examples to clarify the applicability of the obtained results is easy and efficacious. Some of these examples restore the previous results in the literature, and one of them gives a new integrable case describing a generalization of the well-known Swinging Atwood machine.

On the Detuned $$2\!{:}\!4$$ Resonance

We consider families of Hamiltonian systems in two degrees of freedom with an equilibrium in 1:2 resonance. Under detuning, this "Fermi resonance" typically leads to normal modes losing their stability through period-doubling bifurcations. For cubic potentials this concerns the short axial orbits and in galactic dynamics the resulting stable periodic orbits are called "banana" orbits. Galactic potentials are symmetric with respect to the co-ordinate planes whence the potential -- and the normal form -- both have no cubic terms. This $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetry turns the 1:2 resonance into a higher order resonance and one therefore also speaks of the 2:4 resonance. In this paper we study the 2:4 resonance in its own right, not restricted to natural Hamiltonian systems where $H = T + V$ would consist of kinetic and (positional) potential energy. The short axial orbit then turns out to be dynamically stable everywhere except at a simultaneous bifurcation of banana and "anti-banana" orbits, while it is now the long axial orbit that loses and regains stability through two successive period-doubling bifurcations.

Kovalevskaya Exponents, Weak Painlevé Property and Integrability for Quasi-homogeneous Differential Systems

We present some necessary conditions for quasi-homogeneous differential systems to be completely integrable via Kovalevskaya exponents. Then, as an application, we give a new link between the weak-Painleve property and the algebraical integrability for polynomial differential systems. Additionally, we also formulate stronger theorems in terms of Kovalevskaya exponents for homogeneous Newton systems, a special class of quasi-homogeneous systems, which gives its necessary conditions for B-integrability and complete integrability. A consequence is that the nonrational Kovalevskaya exponents imply the nonexistence of Darboux first integrals for two-dimensional natural homogeneous polynomial Hamiltonian systems, which relates the singularity structure to the Darboux theory of integrability.

Лиувиллева классификация интегрируемых геодезических потоков на проективной плоскости в потенциальном поле

Получена лиувиллева классификация натуральной гамильтоновой системы на проективной плоскости с метрикой вращения и линейным интегралом. Вычислены все инварианты Фоменко — Цишанга (т. е. меченые молекулы) системы.

Bi-homogeneity and integrability of rational potentials

In this paper we consider natural Hamiltonian systems with two degrees of freedom for which Hamiltonian function has the form H=12(p12+p22)+V(q1,q2) and potential V(q1,q2) is a rational function. Necessary conditions for the integrability of such systems are deduced from integrability of dominate term of the potential which usually is appropriately chosen homogeneous term of V. We show that introducing weights compatible with the canonical structure one can find new dominant terms which can give new necessary conditions for integrability. To deduce them we investigate integrability of a family of bi-homogeneous potentials which depend on two integer parameters. Unexpectedly systems with these potentials can be reduced to the Lotka–Volterra quadratic planar vector field. Then theorem of Jean Moulin Ollagnier, allows us to make complete classification of integrable cases. Moreover, the reduction is used for explicit integration of an exceptional case with a polynomial first integral of degree 4 in momenta.

Multiple orthogonal geodesic chords in nonconvex Riemannian disks using obstacles

We use nonsmooth critical point theory and the theory of geodesics with obstacle to show a multiplicity result about orthogonal geodesic chords in a Riemannian manifold (with boundary) which is homeomorphic to an N-disk. This applies to brake orbits in a potential well of a natural Hamiltonian system, providing a further step towards the proof of a celebrated conjecture by Seifert (Math Z 51:197–216, 1948).

On certain integrable and superintegrable weight-homogeneous Hamiltonian systems

In this paper we deal with integrability and superintegrability of a certain class of 2-dimensional, weight-homogeneous generalizations of natural Hamiltonian systems in polar coordinates. For this purpose, we use a version of the so-called direct method of searching for polynomial first integrals of such systems. As results we found certain integrable as well as superintegrable cases.

Integrability analysis of natural Hamiltonian systems in curved spaces

In this work we perform integrability analysis of natural Hamiltonian systems with two degrees of freedom governed by a metric defining the infinitesimal linear element dl2=12[a(r)−1dr2+b(r)−1dφ2], where a(r) and b(r) are arbitrary meromorphic functions of variable r. We formulate necessary integrability conditions for such systems in a framework of differential Galois theory. We present several examples which show that the obtained conditions are simple for applications and effective.

Extended Hamiltonians and shift, ladder functions and operators

In recent years, many natural Hamiltonian systems, classical and quantum, with constants of motion of high degree, or symmetry operators of high order, have been found and studied. Most of these Hamiltonians, in the classical case, can be included in the family of extended Hamiltonians, geometrically characterized by the structure of warped manifold of their configuration manifold. For the extended Hamiltonians, the characteristic constants of motion of high degree are polynomial in the momenta of determined form. We consider here a different form of the constants of motion, based on the factorization procedure developed for systems of two degrees of freedom by S. Kuru, J. Negro and others. We show that an important subclass of the extended Hamiltonians, with arbitrary dimension, admits factorized constants of motion and we determine their expression. The classical constants can be polynomial or non-polynomial in the momenta, but the factorization procedure allows, in a type of extended Hamiltonians, their quantization via shift and ladder operators, for systems of any finite dimension.

Topological approach to the generalized -centre problem

This paper considers a natural Hamiltonian system with two degrees of freedom and Hamiltonian . The configuration space is a closed surface (for non-compact certain conditions at infinity are required). It is well known that if the potential energy has Newtonian singularities, then the system is not integrable and has positive topological entropy on the energy level . This result is generalized here to the case when the potential energy has several singular points of type . Let , , and let be the number of singular points with . It is proved that if 2\\chi(M), \\end{equation*} ?> then the system has a compact chaotic invariant set of collision-free trajectories on any energy level . This result is purely topological: no analytical properties of the potential energy are used except the presence of singularities. The proofs are based on the generalized Levi-Civita regularization and elementary topology of coverings. As an example, the plane -centre problem is considered. Bibliography: 29 titles.

Modified Laplace-Beltrami quantization of natural Hamiltonian systems with quadratic constants of motion

It is natural to investigate if the quantization of integrable or superintegrable classical Hamiltonian systems is still integrable or superintegrable. We study here this problem in the case of natural Hamiltonians with constants of motion quadratic in the momenta. The procedure of quantization here considered transforms the Hamiltonian into the Laplace-Beltrami operator plus a scalar potential. In order to transform the constants of motion into symmetry operators of the quantum Hamiltonian, additional scalar potentials, known as quantum corrections, must be introduced, depending on the Riemannian structure of the manifold. We give here a complete geometric characterization of the quantum corrections necessary for the case considered. In particular, Stäckel systems are studied in detail. Examples in conformally and non-conformally flat manifolds are given.

Energy dependent integrability

We review the conditions for separability of 2-dimensional natural Hamiltonian systems. We examine the possibility that the separability condition is satisfied on a given energy hypersurface only (weak integrability) and derive the additional requirement necessary to have separability at arbitrary values of the Hamiltonian (strong integrability). We give some new examples of systems admitting separating coordinates whose relation with the original ones explicitly depends on energy and provide a list of separable potentials discussing the nature of conserved quantities they admit.

Classification of explicit three-stage symplectic difference schemes for the numerical solution of natural Hamiltonian systems: A comparative study of the accuracy of high-order schemes on molecular dynamics problems

The natural Hamiltonian systems (systems with separable Hamiltonians) are considered. The variety of explicit three-stage symplectic schemes is described. A classification of the third-order accurate schemes is given. All fourth-order schemes are found (there are seven of them). It is proved that there are no fifth-order schemes. The schemes with improved properties, such as invertibility and optimality with respect to the phase error, are listed. Numerical results that demonstrate the properties of these schemes are presented, and their comparative analysis with respect to the accuracy–efficiency criterion is given. The disbalance of total energy is used as the accuracy criterion.

Separability in Riemannian Manifolds

An outline of the basic Riemannian structures underlying the separation of variables in the Hamilton-Jacobi equation of natural Hamiltonian systems.