Degenerate Hamiltonian Systems Research Articles

On the Role of Constraints and Degrees of Freedom in the Hamiltonian Formalism

Unfortunately, the Hamiltonian mechanics of degenerate Lagrangian systems is usually presented as a mere recipe of Dirac, with no explanation as to how it works. It then comes to discussing conjectures of whether all primary constraints correspond to gauge symmetries, and it goes all the way to absolutely wrong claims such as the statement that electrodynamics or gravity only have two physical components each, with others being spurious. One has to be very careful because non-dynamical, or constrained, does not mean unphysical. In this article, I give a pedagogical introduction to the degenerate Hamiltonian systems, showing both very simple mechanical examples and general arguments about how it works. For the familiar field theory models, I explain why the gauge freedom there “hits twice” in the sense of producing twice as many first-class constraints as gauge symmetries, and why primary, and only primary, constraints should be put into the total Hamiltonian.

Degenerate response tori in Hamiltonian systems with higher zero-average perturbation

Consider a normally degenerate Hamiltonian system with the following Hamiltonian H(θ,I,x,y,ϵ)=ω,I+λxn+1n+1+y22+ϵP(θ,x,y),(θ,I,x,y)∈Td×Rd+2, which is associated with the standard symplectic form dθ ∧ dI ∧ dx ∧ dy, where 0≠λ∈R and n > 1 is an integer. The existence of response tori for the degenerate Hamiltonian system has already been proved by Si and Yi [Nonlinearity 33, 6072–6098 (2020)] if [∂P(θ,0,0)∂x] satisfies some non-zero conditions, see condition (H) in the work of Si and Yi [Nonlinearity 33, 6072–6098 (2020)], where [·] denotes the average value of a continuous function on Td. However, when [∂P(θ,0,0)∂x]=0, no results were given by Si and Yi [Nonlinearity 33, 6072–6098 (2020)] for response tori of the above system. This paper attempts at carrying out this work in this direction. More precisely, with 2p < n, if P satisfies [∂jP(θ,0,0)∂xj]=0 for j = 1, 2, …, p and either λ−1[∂p+1P(θ,0,0)∂xp+1]<0 as n − p is even or λ−1[∂p+1P(θ,0,0)∂xp+1]≠0 as n − p is odd, we obtain the following results: (1) For λ̃<0 [see λ̃ in (2.1)] and ϵ sufficiently small, response tori exist for each ω satisfying a Brjuno-type non-resonant condition. (2) For λ̃>0 and ϵ* sufficiently small, there exists a Cantor set E∈(0,ϵ*) with almost full Lebesgue measure such that response tori exist for each ϵ∈E if ω satisfies a Diophantine condition. In the case where λ−1[∂p+1P(θ,0,0)∂xp+1]>0 and n − p is even, we prove that the system admits no response tori in most regions. The present paper is regarded as a continuation of work by Si and Yi [Nonlinearity 33, 6072–6098 (2020)].

Brouwer\u2019s satellite solution redux

Brouwer’s solution to the artificial satellite problem is revisited. We show that the complete Hamiltonian reduction is rather achieved in the plain Poincaré’s style, through a single canonical transformation, than using a sequence of partial reductions based on von Zeipel’s alternative for dealing with perturbed degenerate Hamiltonian systems. Beyond the theoretical interest of the new approach as regards the complete reduction of perturbed Keplerian motion, we also show that a solution based on a single set of corrections may yield computational benefits in the implementation of an analytic orbit propagator.

Moser's theorem for hyperbolic-type degenerate lower tori in Hamiltonian system

In this paper, we give a Moser-type theorem for C ℓ -smooth hyperbolic-type degenerate Hamiltonian system with the following Hamiltonian H = 〈 ω , y 〉 + 1 2 v 2 − u 2 d + P ( x , y , u , v ) , ( x , y , u , v ) ∈ T n × R n × R 2 , which is associated with the standard symplectic structure, with d ≥ 1 . Due to the difficulty coming from the degeneracy, our result is quite different from L. Chierchia and D. Qian's work [8] (non-degenerate case). An interesting phenomenon shown in degenerate case is the ℓ -regularity of above Hamiltonian system not only relies on the tori's dimension n but also strongly relies on the degenerate index d . Under arbitrary small perturbation P , we prove that if ℓ ≥ ( 5 d + 2 ) ( 8 τ + 3 ) , where τ > n − 1 , the above hyperbolic-type degenerate Hamiltonian system admits lower dimensional Diophantine tori which are proved to be of class C β for any β ≤ 8 τ + 2 . Our result can be seen a generalization of paper [42] from analytic case to C ℓ -smooth case and can also be seen a generalization of paper [8] from non-degenerate case to degenerate case.

The Cyclicity of Period Annulus of Degenerate Quadratic Hamiltonian Systems with Polycycles S(2) or S(3) Under Perturbations of Piecewise Smooth Polynomials

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the bifurcation of limit cycles for degenerate quadratic Hamilton systems with polycycles [Formula: see text] or [Formula: see text] under the perturbations of piecewise smooth polynomials with degree [Formula: see text]. Roughly speaking, for [Formula: see text], a polycycle [Formula: see text] is cyclically ordered collection of [Formula: see text] saddles together with orbits connecting them in specified order. The discontinuity is on the line [Formula: see text]. If the first order Melnikov function is not equal to zero identically, it is proved that the upper bounds of the number of limit cycles bifurcating from each of the period annuli with the boundary [Formula: see text] and [Formula: see text] are respectively [Formula: see text] and [Formula: see text] (taking into account the multiplicity).

Completely degenerate responsive tori in Hamiltonian systems**The first author was partially supported by a CSC scholarship and Fundamental Research Funds of Shandong University. The second author was partially supported by NSERC discovery Grant 1257749, PIMS CRG Grant, a faculty development grant from the University of Alberta, and a Scholarship from Jilin University.

We consider the existence of responsive tori for the completely degenerate Hamiltonian system with the following Hamiltonian which is associated with the standard symplectic structure, where λ = ±1 and n > 2, n ⩾ m ⩾ 2 are integers. With P satisfying certain non-degenerate conditions, we obtain the following results: (1) For λ = −1 and ϵ sufficiently small, responsive tori exist for each ω satisfying a weak non-resonant condition; (2) For λ = 1 and ϵ* sufficiently small, there exists a Cantor set with almost full Lebesgue measure such that responsive tori exist for each if ω satisfies a Diophantine condition. Non-existence of responsive tori are also discussed when P fails to satisfy the non-degenerate condition. Our results are directly applicable to the existence problem of quasi-periodic responsive solutions of degenerate harmonic oscillators.

On variants of Arnold conjecture

In this note we discuss the collection of statements known as Arnold conjecture for Hamiltonian diffeomorphisms of closed symplectic manifolds. We provide an overview of the homological, stable and strong versions of Arnold conjecture for non-degenerate Hamiltonian systems, a few versions of Arnold conjecture for possibly degenerate Hamiltonian systems, the degenerate version of Arnold conjecture for Hamiltonian homeomorphisms and Sandon’s version of Arnold conjecture for contactomorphisms.

Completely degenerate lower-dimensional invariant tori for Hamiltonian system

We study the persistence of lower-dimensional invariant tori for a nearly integrable completely degenerate Hamiltonian system. It is shown that the majority of unperturbed invariant tori can survive from the perturbations which are only assumed the smallness and smoothness.

Multistability and transition to chaos in the degenerate Hamiltonian system with weak nonlinear dissipative perturbation

The effect of small nonlinear dissipation on the dynamics of a system with the stochastic web which is linear oscillator driven by pulses is studied. The scenario of coexisting attractors evolution with the increase of nonlinear dissipation is revealed. It is shown that the period-doubling transition to chaos is possible only for the third-order resonance and only hard transitions can be seen for all other resonances.

LOCALIZING TRANSPORT BARRIERS IN DEGENERATE 3/2 D.O.F. HAMILTONIAN SYSTEMS WITH APPLICATIONS TO MAGNETIC CONFINEMENT FUSION

The existence of transport barriers in 3/2 degrees of freedom degenerate Hamiltonian systems is studied using the associated stroboscopic maps. For small enough amplitudes of the perturbations, a transport barrier (formed by infinitely many invariant rotational circles) is proven to exist and to form in the degenerate annulus. These results are applied to Hamiltonian models which describe some magnetic configurations in tokamak plasmas.

Dynamics and transport in mean-field coupled, many degrees-of-freedom, area-preserving nontwist maps

Area-preserving nontwist maps, i.e., maps that violate the twist condition, arise in the study of degenerate Hamiltonian systems for which the standard version of the Kolmogorov-Arnold-Moser (KAM) theorem fails to apply. These maps have found applications in several areas including plasma physics, fluid mechanics, and condensed matter physics. Previous work has limited attention to maps in 2-dimensional phase space. Going beyond these studies, in this paper, we study nontwist maps with many-degrees-of-freedom. We propose a model in which the different degrees of freedom are coupled through a mean-field that evolves self-consistently. Based on the linear stability of period-one and period-two orbits of the coupled maps, we construct coherent states in which the degrees of freedom are synchronized and the mean-field stays nearly fixed. Nontwist systems exhibit global bifurcations in phase space known as separatrix reconnection. Here, we show that the mean-field coupling leads to dynamic, self-consistent reconnection in which transport across invariant curves can take place in the absence of chaos due to changes in the topology of the separatrices. In the context of self-consistent chaotic transport, we study two novel problems: suppression of diffusion and breakup of the shearless curve. For both problems, we construct a macroscopic effective diffusion model with time-dependent diffusivity. Self-consistent transport near criticality is also studied, and it is shown that the threshold for global transport as function of time is a fat-fractal Cantor-type set.

Periodic orbits, invariant tori, and cylinders of Hamiltonian systems near integrable ones having a return map equal to the identity

Generically the return map of an integrable Hamiltonian system with two degrees of freedom in a Hamiltonian level foliated by invariant tori is a twist map. If we perturb such integrable Hamiltonian system inside the class of Hamiltonian systems with two degrees of freedom, then the Poincaré–Birkhoff theorem allows to determine which periodic orbits of the integrable can be prolonged to the perturbed one, and the KAM theory provides sufficient conditions in order that some invariant tori persist under sufficiently small perturbations. If some power of this return map is the identity, then in general for these degenerate Hamiltonian systems we cannot study which periodic orbits of the integrable can be prolonged to the perturbed one, or if some invariant tori persist. This paper studies the perturbation of integrable Hamiltonian systems with two degrees of freedom having some power of the return map equal to the identity. We show with two different models a way to study the prolongation of periodic orbits and of invariant tori or cylinders filled with periodic orbits for such kind of Hamiltonian systems. The main tool in this study is the averaging theory.

Invariant Tori in Hamiltonian Systems with High Order Proper Degeneracy

We study the existence of quasi-periodic, invariant tori in a nearly integrable Hamiltonian system of high order proper degeneracy, i.e., the integrable part of the Hamiltonian involves several time scales and at each time scale the corresponding Hamiltonian depends on only part of the action variables. Such a Hamiltonian system arises frequently in problems of celestial mechanics, for instance, in perturbed Kepler problems like the restricted and non-restricted 3-body problems and spatial lunar problems in which several bodies with very small masses are coupled with two massive bodies and the nearly integrable Hamiltonian systems naturally involve different time scales. Using KAM method, we will show under certain higher order non-degenerate conditions of Bruno–Russmann type that the majority of quasi-periodic, invariant tori associated with the integrable part will persist after the non-integrable perturbation. This actually concludes the KAM metric stability for such a properly degenerate Hamiltonian system.

Large excursions of action within the resonance of a degenerate Hamiltonian system with two degrees of freedom

We investigate the width of the resonance zone in a degenerate Hamiltonian system with two degrees of freedom, in which the Hamiltonian lacks the quadratic term in the Taylor expansion. This leads to larger excursions of action in the phase space than the nondegenerate one, and corresponding resonance frequency widths would become narrower. However, in contrast to the nonautonomous Hamiltonian system with one and half degree of freedom, we find that the above case is not generic and only occurs at particular resonances. An example relevant to the interaction of resonances is considered. Analytic results are verified in numerical simulations.

Light-cone Yang-Mills mechanics: SU(2) vs. SU(3)

We investigate the light-cone SU(n) Yang-Mills mechanics formulated as the leading order of the long-wavelength approximation to the light-front SU(n) Yang-Mills theory. In the framework of the Dirac formalism for degenerate Hamiltonian systems, for models with the structure groups SU(2) and SU(3), we determine the complete set of constraints and classify them. We show that the light-cone mechanics has an extended invariance: in addition to the local SU(n) gauge rotations, there is a new local two-parameter Abelian transformation, not related to the isotopic group, that leaves the Lagrangian system unchanged. This extended invariance has one profound consequence. It turns out that the light-cone SU(2) Yang-Mills mechanics, in contrast to the well-known instant-time SU(2) Yang-Mills mechanics, represents a classically integrable system. For calculations, we use the technique of Grobner bases in the theory of polynomial ideals.

Degenerate lower-dimensional tori in Hamiltonian systems

We study the persistence of lower-dimensional tori in Hamiltonian systems of the form H ( x , y , z ) = 〈 ω , y 〉 + 1 2 〈 z , M ( ω ) z 〉 + ε P ( x , y , z , ω ) , where ( x , y , z ) ∈ T n × R n × R 2 m , ε is a small parameter, and M ( ω ) can be singular. We show under a weak Melnikov nonresonant condition and certain singularity-removing conditions on the perturbation that the majority of unperturbed n-tori can still survive from the small perturbation. As an application, we will consider the persistence of invariant tori on certain resonant surfaces of a nearly integrable, properly degenerate Hamiltonian system for which neither the Kolmogorov nor the g-nondegenerate condition is satisfied.

Stability of Nearly Integrable, Degenerate Hamiltonian Systems with Two Degrees of Freedom

We consider the problem of the stability of action variables in properly degenerate, nearly integrable Hamiltonian systems and prove, in particular, stability results for systems with two degrees of freedom. An application of such results to celestial mechanics is presented.

On the stability of some properly--degenerate Hamiltonian systems with two degrees of freedom

Properly degenerate nearly--integrable Hamiltonian systems with two degrees of freedom such that the intermediate system depend explicitly upon the angle--variable conjugated to the non--degenerate action--variable are considered and, in particular, model problems motivated by classical examples of Celestial Mechanics, are investigated. Under suitable convexity assumptions on the intermediate Hamiltonian, it is proved that, in every energy surface, the action variables stay forever close to their initial values. In non convex cases, stability holds to a small set where, in principle, the degenerate action--variable might (in exponentially long times) drift away from its initial value by a quantity independent of the perturbation. Proofs are based on a blow up (complex) analysis near separatrices, KAM techniques and energy conservation arguments.

The cyclicity of period annuli of degenerate quadratic Hamiltonian systems with elliptic segment loops

We study the cyclicity of period annuli (or annulus) for general degenerate quadratic Hamiltonian systems with an elliptic segment or a saddle loop, under quadratic perturbations. By using geometrical arguments and studying the respective Abelian integral based on the Picard–Fuchs equation, it is shown that the cyclicity of period annuli (or annulus) for such systems equals two. This result, together with those of Gavrilov and Iliev (2000), Iliev (1996), Zhao et al (2000) and Zhao and Zhu (2001) gives a complete solution to the infinitesimal Hilbert 16th problem in the case of degenerate quadratic Hamiltonian systems under quadratic perturbations.

PERTURBATIONS OF A KIND OF DEGENERATE QUADRATIC HAMILTONIAN SYSTEM WITH SADDLE-LOOP

The authors investigate a kind of degenerate quadratic Hamiltonian systems with saddle-loop. Under quadratic perturbations, it is proved that the perturbed system has at most two limit cycles in the finite plane. The proof relies on a careful analysis of a related Abelian integral.