Integrable Hamiltonian Systems Research Articles

Integrable systems of the ellipsoidal, paraboloidal and conical type with magnetic field

We construct integrable Hamiltonian systems with magnetic fields of the ellipsoidal, paraboloidal and conical type, i.e. systems that generalize natural Hamiltonians separating in the respective coordinate systems to include nonvanishing magnetic field. In the ellipsoidal and paraboloidal case each this classification results in three one–parameter families of systems, each involving one arbitrary function of a single variable and a parameter specifying the strength of the magnetic field of the given fully determined form. In the conical case the results are more involved, there are two one–parameter families like in the other cases and one class which is less restrictive and so far resists full classification.

A modified parameterization method for invariant Lagrangian tori for partially integrable Hamiltonian systems

In this paper we present an a-posteriori KAM theorem for the existence of an (n−d)-parameter family of d-dimensional isotropic invariant tori with Diophantine frequency vector ω∈Rd, of type (γ,τ), for n degrees of freedom Hamiltonian systems with (n−d) independent first integrals in involution. If the first integrals induce a Hamiltonian action of the (n−d)-dimensional torus, then we can produce n-dimensional Lagrangian tori with frequency vector of the form (ω,ωp), with ωp∈Rn−d. In the light of the parameterization method, we design a (modified) quasi-Newton method for the invariance equation of the parameterization of the torus, whose proof of convergence from an initial approximation, and under appropriate non-degeneracy conditions, is the object of this paper. We present the results in the analytic category, so the initial torus is real-analytic in a certain complex strip of width ρ, and the corresponding error in the functional equation is ɛ. We heavily use geometric properties and the so called automatic reducibility to deal directly with the functional equation and get convergence if γ−2ρ−2τ−1ɛ is small enough, in contrast with most of KAM results based on the parameterization method, that get convergence if γ−4ρ−4τɛ is small enough. The approach is suitable to perform computer assisted proofs.

Global properties of generic real–analytic nearly–integrable Hamiltonian systems

We introduce a new class Gsn of generic real analytic potentials on Tn and study global analytic properties of natural nearly–integrable Hamiltonians 12|y|2+εf(x), with potential f∈Gsn, on the phase space M=B×Tn with B a given ball in Rn. The phase space M can be covered by three sets: a ‘non–resonant’ set, which is filled up to an exponentially small set of measure e−cK (where K is the maximal size of resonances considered) by primary maximal KAM tori; a ‘simply resonant set’ of measure εKa and a third set of measure εKb which is ‘non perturbative’, in the sense that the H–dynamics on it can be described by a natural system which is not nearly–integrable. We then focus on the simply resonant set – the dynamics of which is particularly interesting (e.g., for Arnol'd diffusion, or the existence of secondary tori) – and show that on such a set the secular (averaged) 1 degree–of–freedom Hamiltonians (labeled by the resonance index k∈Zn) can be put into a universal form (which we call ‘Generic Standard Form’), whose main analytic properties are controlled by only one parameter, which is uniform in the resonance label k.

Solutions to the wave equation for commuting flows of dispersionless PDEs

Motivated by the viewpoint of integrable systems, we study commuting flows of 2-component quasilinear equations, reducing to investigate the solutions of the wave equation with non-constant speed. In this paper, we apply the reduction procedure of differential constraints to obtain a complete set of solutions of such an equation for some fixed velocities a2(u,v). As a result, we present some examples of Hamiltonian integrable systems (as the shallow water equations) with relative symmetries, conserved quantities and solutions.

On parametric generalizations of the Kardar-Parisi-Zhang equation and their integrability

There are analyzed two physically reasonable generalizations of the Kardar-Parisi-Zhang equation describing the spin glasses growth models and possessing important from physical point of view properties. The first one proved to be a completely integrable Hamiltonian dynamical system with an infinite hierarchy of commuting to each other conservation laws, and the second one proved to be linearized modulo some nonlinear constraints, imposed on its solutions.

Quasi-periodic solutions of a discrete integrable equation with a finite-dimensional integrable symplectic structure

Through the paper, we research the quasi-periodic solutions to a semi-discrete hierarchy which has Hamiltonian structure and integrable symplectic map. Firstly, a semi-discrete hierarchy is derived by use of the discrete zero-curvature representation and then its integrability is proved under the Liouville condition. Using the binary nonlinearization approach, the integrable symplectic map and finite-dimensional Hamiltonian system of the hierarchy are obtained. Moreover, the trigonal curve Kq−1 is denoted through the characteristic polynomials for the Lax pair, as well as the related Baker–Akhiezer function and meromorphic function are introduced, from which the asymptotic properties and divisors of the two functions mentioned above are analyzed. Finally, we introduce the three kinds of Abel differentials and straighten out of corresponding continuous and discrete flows, the quasi-periodic solutions of the equations are received via the Riemann theta function.

Concatenation of Two Different Integrable Systems is Not Integrable

Aims/Objectives: Nonlinear, completely integrable Hamiltonian systems representing charged particle motion in external electromagnetic fields hold promise for models of novel intensity frontier particle accelerators. The main reason is the combination of large regions of stable orbits with damping of collective instabilities by conservative relaxation. Intensity frontier particle accelerators are essential for discovery science in particle and nuclear physics, and a host of industrial and security applications.Study Design: Mathematical proof.Methodology: Tools of Hamiltonian dynamics and differential geometry.Results: Realistic system lattices include additional sections or inserts that themselves may be integrable (such as linear optics, phase trombones, thin lenses, kicks, etc.). However, in general the full system fails to remain integrable.Conclusion: The non-integrability proof is presented and some consequences of integrability failure are explored.

Polynomial Entropy and Polynomial Torsion for Fibered Systems

Given a continuous fibered dynamical system, we first introduce the notion of polynomial torsion of a fiber,which measures the “infinitesimal variation” of the dynamics between the fiber and the neighboring ones.This gives rise to an (upper semicontinous) torsion function,defined on the base of the system, which is a newC^{0} (fiber) conjugacy invariant. We prove that the polynomial entropy of the system is the supremum ofthe torsion of its fibers, which yields a new insight into the creation of polynomial entropy in fibered systems.We examine the relevance of these results in the context of integrable Hamiltoniansystems or diffeomorphisms, with the particular cases of C^{0}-integrable twist maps on the annulus and geodesic flows.Finally, we bound from below the polynomial entropy of ell-modal interval maps in terms of their lap number and answer a question by Gomes and Carneiro.

Convexity of Singular Affine Structures and Toric-Focus Integrable Hamiltonian Systems

This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), analogous to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus singularities. We obtain a series of convexity results, both positive and negative, for such singular integral affine base spaces. In particular, near a focus singular point, they are locally convex and the local-global convexity principle still applies. They are also globally convex under some natural additional conditions. However, when the monodromy is sufficiently large, the local-global convexity principle breaks down and the base spaces can be globally non-convex, even for compact manifolds. As a surprising example, we construct a 2-dimensional “integral affine black hole”, which is locally convex but for which a straight ray from the center can never escape.

Geometric Scattering Monodromy

In this paper we give geometric conditions so that the integral mapping of a Liouville integrable Hamiltonian system with a focus-focus equilibrium point has scattering monodromy. Using a complex version of the Morse lemma, we show that scattering monodromy is the same as the scattering monodromy of the standard focus-focus system.

Symplectic and inverse spectral geometry of integrable systems: A glimpse and open problems

We first give a glimpse of finite dimensional classical integrable Hamiltonian systems from the point of view of symplectic geometry and briefly discuss their quantum counterparts, with an emphasis on recent progress on inverse spectral geometry. Then we propose several open problems about the geometry, topology and dynamics of these systems. The problems are largely motivated by the works of a number of authors, including Arnold, Atiyah, Colin de Verdière, Delzant, Duistermaat, Eliasson, Fomenko, Guillemin, Kolmogorov, Kostant, Moser and Sternberg.

Stochastic averaging for a completely integrable Hamiltonian system with fractional Brownian motion

This paper proposes an effective approximation result for the behavior of a small transversal perturbation to a completely integrable stochastic Hamiltonian system on a symplectic manifold. We derive an averaged stochastic differential equations (SDEs) in the action space for the action component of the perturbed system, where the averaged drift coefficient is characterized by the averages of that of the action component with respect to the invariant measure of the unperturbed system on the corresponding invariant manifolds. Then, the averaging principle is shown to be valid such that the action component of the perturbed system converges to the solution of averaged SDEs in the mean square sense.

Arnold diffusion for nearly integrable Hamiltonian systems

Arnold diffusion for nearly integrable Hamiltonian systems

Family of nonstandard integrable and superintegrable classical Hamiltonian systems in non-vanishing magnetic fields

In this paper, we present the construction of all nonstandard integrable systems in magnetic fields whose integrals have leading order structure corresponding to the case (i) of theorem 1 in Marchesiello and Šnobl (2022 J. Phys. A: Math. Theor. 55 145203). We find that the resulting systems can be written as one family with several parameters. For certain limits of these parameters the system belongs to intersections with already known standard systems separating in Cartesian and / or cylindrical coordinates and the number of independent integrals of motion increases, thus the system becomes minimally superintegrable. These results generalize the particular example presented in section 3 of Marchesiello and Šnobl (2022 J. Phys. A: Math. Theor. 55 145203).

Integrability and trajectory confinement in -symmetric waveguide arrays

We consider -symmetric ring-like arrays of optical waveguides with purely nonlinear gain and loss. Regardless of the value of the gain–loss coefficient, these systems are protected from spontaneous -symmetry breaking. If the nonhermitian part of the array matrix has cross-compensating structure, the total power in such a system remains bounded—or even constant—at all times. We identify two-, three-, and four-waveguide arrays with cross-compensatory nonlinear gain and loss that constitute completely integrable Hamiltonian systems.

Resonance conjecture via weak KAM theory

Poincaré established the problem how much of the stability mechanism of integrable Hamiltonian systems can persist under small perturbations, which he called “the fundamental problem of dynamics”. This paper deals with the fundamental problem in general resonant case. We give a weak KAM type result that for each y in the g (with rank m0)-resonant surface, the nearly integrable Hamiltonian system has at least m0+1 weak KAM solutions associated with relative equilibria.

On the geometry of Liouville foliations

The paper studies the geometry of Liouville foliation generated by integrable Hamiltonian system. It is shown that regular leaves are two-dimensional surface of zero Gaussian curvature and zero Gaussian torsion.

Billiards and integrable systems

The survey is devoted to the class of integrable Hamiltonian systems and the class of integrable billiard systems and to the recent results of the authors and their students on the problem of comparison of these classes from the point of view of leafwise homeomorphy of their Liouville foliations. The key tool here are billiards on piecewise planar CW-complexes - topological billiards and billiard books - introduced by Vedyushkina. A construction of the class of evolutionary (force) billiards, introduced recently by Fomenko, is presented, enabling one to model a system in several non-singular energy ranges by a single billiard system, and the use of this class for geodesic flows on two-dimensional surfaces and some systems in mechanics is demonstrated. Some other integrable generalizations of classical billiard systems, including billiards with potentials, billiards in magnetic fields, and billiards with slipping, are discussed. Billiard books with Hooke potentials glued of planar confocal or circular tables, model four-dimensional semilocal singularities of Liouville foliations for integrable systems that contain non-degenerate equilibria. Considering the intersections of several confocal quadrics in $\mathbb{R}^n$ results in a generalization of the Jacobi-Chasles theorem. Bibliography: 144 titles.

On the algebraic properties of exponentially stable integrable hamiltonian systems

Steepness is a geometric property which, together with complex-analyticity, is needed in order to insure stability of a near-integrable hamiltonian system over exponentially long times. Following a strategy developed by Nekhoro-shev, we construct sufficient algebraic conditions for steepness for a given function that involve algebraic equations on its derivatives up to order five. The underlying analysis suggests some interesting considerations on the gener-icity of steepness and represents a first step towards the construction of sufficient conditions for steepness involving the derivatives of the studied function up to an arbitrary order.

On a Family of Integrable Hamiltonian Systems

On a Family of Integrable Hamiltonian Systems