Theory Of Hamiltonian Systems Research Articles

Equivalent Differential Equations in Problems of Control Theory and the Theory of Hamiltonian Systems

Equivalent Differential Equations in Problems of Control Theory and the Theory of Hamiltonian Systems

On the linear stability of simple and semi‐simple periodic waves for a system of cubic Klein–Gordon equations

AbstractWe study the linear stability of traveling wave solutions for the nonlinear wave equation and coupled nonlinear wave equations. It is shown that periodic waves of the dnoidal type are spectrally unstable with respect to co‐periodic perturbations. Our arguments rely on a careful spectral analysis of various self‐adjoint operators, both scalar and matrix and on instability index count theory for Hamiltonian systems.

Generalized focal points and local Sturmian theory for linear Hamiltonian systems

Generalized focal points and local Sturmian theory for linear Hamiltonian systems

Enhanced Synchronization Stability of Grid-Forming Inverters With Passivity-Based Virtual Oscillator Control

In this article, a passivity-based virtual oscillator control strategy with enhanced synchronization stability for grid-forming inverters (GFMs) is proposed. By adopting the port-controlled Hamiltonian system theory for orbital stabilization problems, an energy pumping-and-damping block is proposed to render GFMs globally asymptotically stable with respect to the prespecified solutions of the power-flow equations from any initial condition. This allows for stable integrations of GFMs to any other globally asymptotically stable systems without their explicit knowledge, e.g., helping maintain synchronism with the bulk power system in a wide range of short-circuit-ratio conditions or under large disturbances and keeping synchronism among multiple GFMs in power systems. Both simulations and experiments are presented to demonstrate the proposed control approach.

Hamiltonian theory of the crossing of the 2Qx−2Qy=0 nonlinear coupling resonance

In a recent paper, the adiabatic theory of Hamiltonian systems was successfully applied to study the crossing of the linear coupling resonance, ${Q}_{x}\ensuremath{-}{Q}_{y}=0$. A detailed explanation of the well-known phenomena that occur during the resonance-crossing process, such as emittance exchange and its dependence on the adiabaticity of the process, was obtained. In this paper, we consider the crossing of the resonance of nonlinear coupling $2{Q}_{x}\ensuremath{-}2{Q}_{y}=0$ using the same theoretical framework. We perform the analysis using a Hamiltonian model in which the nonlinear coupling resonance is excited and the corresponding dynamics is studied in detail, in particular looking at the phase-space topology and its evolution, in view of characterizing the emittance exchange phenomena. The theoretical results are then tested using a symplectic map. Thanks to this approach, scaling laws of general interest for applications are derived.

Planar Hamiltonian systems: Index theory and applications to the existence of subharmonics

We consider a planar Hamiltonian system of the type Jz′=∇zH(t,z), where H:R×R2→R is a function periodic in the time variable, such that ∇zH(t,0)≡0 and ∇zH(t,z) is asymptotically linear for |z|→+∞. After revisiting the index theory for linear planar Hamiltonian systems, by using the Poincaré–Birkhoff fixed point theorem we prove that the above nonlinear system has subharmonic solutions of any order k large enough, whenever the rotation numbers (or, equivalently, the mean Conley–Zehnder indices) of the linearizations of the system at zero and at infinity are different. Applications are given to the case of planar Hamiltonian systems coming from second order scalar ODEs.

On the Eigenvalues of the Fermionic Angular Eigenfunctions in the Kerr Metric.

In view of a result recently published in the context of the deformation theory of linear Hamiltonian systems, we reconsider the eigenvalue problem associated with the angular equation arising after the separation of the Dirac equation in the Kerr metric, and we show how a quasilinear first order PDE for the angular eigenvalues can be derived efficiently. We also prove that it is not possible to obtain an ordinary differential equation for the eigenvalues when the role of the independent variable is played by the particle energy or the black hole mass. Finally, we construct new perturbative expansions for the eigenvalues in the Kerr case and obtain an asymptotic formula for the eigenvalues in the case of a Kerr naked singularity.

Renormalized oscillation theory for singular linear Hamiltonian systems

Working with a general class of linear Hamiltonian systems on intervals with at least one singular endpoint which can be limit-point, limit-circle, or limit-intermediate, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of C2n. In the first part of the analysis we associate our linear Hamiltonian systems with families of well-defined self-adjoint operators, and in the latter part we employ the renormalized oscillation approach to count the number of eigenvalues these operators have on fixed intervals (λ1,λ2) whose closures do not intersect the essential spectrum of the operators. We conclude the analysis with two illustrative examples, indicating how the theory can be implemented in practice. This extends previous work by the authors for regular linear Hamiltonian systems.

The linear Vlasov-Poisson-Ampère equation from the viewpoint of abstract scattering theory

We review some recent results on the scattering structure of the linearized Vlasov-Poisson equation in d=1 space dimension. It started with [9] where the linearized Vlasov-Poisson equation is rewritten as a linear Vlasov-Ampère set of equations which makes the L 2 structure more visible. A consequence is that the linear Landau damping becomes an application of the scattering theory for Hamiltonian systems. Then we review the extension, firstly of the linearization around non homogeneous profiles which is treated with the theory of trace-class operators, secondly of the case with a forcing magnetic field which has the ability to eliminate the possibility of a linear Landau damping effect. Finally, we evoke some possibility for extension to space dimension x∈ℝ d with d>1.

Geometric Hamilton–Jacobi theory for systems with external forces

In this paper, we develop a Hamilton–Jacobi theory for forced Hamiltonian and Lagrangian systems. We study the complete solutions, particularize for Rayleigh systems, and present some examples. Additionally, we present a method for the reduction and reconstruction of the Hamilton–Jacobi problem for forced Hamiltonian systems with symmetry. Furthermore, we consider the reduction of the Hamilton–Jacobi problem for a Čaplygin system to the Hamilton–Jacobi problem for a forced Lagrangian system.

Renormalized Oscillation Theory for Linear Hamiltonian Systems on [0, 1] Via the Maslov Index

Working with a general class of regular linear Hamiltonian systems, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of \({\mathbb {C}}^{2n}\). We verify that our applicability class includes Dirac and Sturm–Liouville systems, as well as a system arising from differential-algebraic equations for which the spectral parameter appears nonlinearly.

Linear instability of Vlasov-Maxwell systems revisited-A Hamiltonian approach

<p style='text-indent:20px;'>We consider linear stability of steady states of 1<inline-formula><tex-math id="M1">\begin{document}$ \frac{1}{2} $\end{document}</tex-math></inline-formula> and 3DVlasov-Maxwell systems for collisionless plasmas. The linearized systems canbe written as separable Hamiltonian systems with constraints. By using ageneral theory for separable Hamiltonian systems, we recover the sharp linearstability criteria obtained previously by different approaches. Moreover, weobtain the exponential trichotomy estimates for the linearized Vlasov-Maxwellsystems in both relativistic and nonrelativistic cases.</p>

Hamiltonian control of magnetic field lines: Computer assisted results proving the existence of KAM barriers

<p style='text-indent:20px;'>We reconsider a control theory for Hamiltonian systems, that was introduced on the basis of KAM theory and applied to a model of magnetic field in previous articles. By a combination of Frequency Analysis and of a rigorous (Computer Assisted) KAM algorithm we prove that in the phase space of the magnetic field, due to the control term, a set of invariant tori appear, and it acts as a transport barrier. Our analysis, which is common (but often also limited) to Celestial Mechanics, is based on a normal form approach; it is also quite general and can be applied to quasi-integrable Hamiltonian systems satisfying a few additional mild assumptions. As a novelty with respect to the works that in the last two decades applied Computer Assisted Proofs into the framework of KAM theory, we provide all the codes allowing to produce our results. They are collected in a software package that is publicly available from the <i>Mendeley Data</i> repository. All these codes are designed in such a way to be easy-to-use, also for what concerns eventual adaptations for applications to similar problems.</p>

Lidskii angles and Sturmian theory for linear Hamiltonian systems on compact interval

In this paper we investigate the Sturmian theory for general (possibly uncontrollable) linear Hamiltonian systems by means of the Lidskii angles, which are associated with a symplectic fundamental matrix of the system. In particular, under the Legendre condition we derive formulas for the multiplicities of the left and right proper focal points of a conjoined basis of the system, as well as the Sturmian separation theorems for two conjoined bases of the system, in terms of the Lidskii angles. The results are new even in the completely controllable case. As the main tool we use the limit theorem for monotone matrix-valued functions by Kratz (1993). The methods allow to present a new proof of the known monotonicity property of the Lidskii angles. The results and methods can also be potentially applied in the singular Sturmian theory on unbounded intervals, in the oscillation theory of linear Hamiltonian systems without the Legendre condition, in the comparative index theory, or in linear algebra in the theory of matrices.

Classical multiseparable Hamiltonian systems, superintegrability and Haantjes geometry

We show that the theory of classical Hamiltonian systems admitting separating variables can be formulated in the context of (ω,H) structures. They are symplectic manifolds endowed with a compatible Haantjes algebra H, namely an algebra of (1,1)tensor fields with vanishing Haantjes torsion. A special class of coordinates, called Darboux-Haantjes coordinates, will be constructed from the Haantjes algebras associated with a separable system. These coordinates enable the additive separation of variables of the corresponding Hamilton-Jacobi equation.We shall prove that a multiseparable system admits as many ωH structures as separation coordinate systems. In particular, we will show that a large class of multiseparable, superintegrable systems, including the Smorodinsky-Winternitz systems and some physically relevant systems with three degrees of freedom, possesses multiple Haantjes structures.

The Hamilton–Jacobi Theory for Contact Hamiltonian Systems

The aim of this paper is to develop a Hamilton–Jacobi theory for contact Hamiltonian systems. We find several forms for a suitable Hamilton–Jacobi equation accordingly to the Hamiltonian and the evolution vector fields for a given Hamiltonian function. We also analyze the corresponding formulation on the symplectification of the contact Hamiltonian system, and establish the relations between these two approaches. In the last section, some examples are discussed.

Quantum Zeno Jumps in a Resonantly Driven Qubit under Frequent Measurements

In Nature, 570, 200 (2019), Minev and co-authors’ experiment shows how to deterministically “catch and reverse a quantum jump mid-flight” in a continuously-observed Rabi-stimulated qubit. Its interpretation is in debate (La Recherche, 555, 40, (2020)). We show that the quantum Zeno effect (QZE) of continuous measurement —by use of photon emission from a 3rd high-rate monitored ancilla level— can be described by an action-angle canonical transformation of the original Hamiltonian dynamical system (HDS) theory of QZE. Then energy whose mean value yields the well-known resonant Rabi harmonic dynamics is actually defined by large-amplitude high-frequency oscillations of the internal as well as of the overall phase of the two-level system. By making use of their standard deviation, we show that the separatrix crossing of the HDS trajectory yields the quantized action nh where n = 1, 2, 3 .... Therefore, the jump dynamics observed in Minev et al. experiment belongs to a series of discrete quantum jumps: it corresponds in this experiment to n = 3.

Analysis of Novel Oscillations of Quantized Mechanical Energy in Mass-Accreting Nano-Oscillator Systems

Quantum characteristics of a mass-accreting oscillator are investigated using the invariant operator theory, which is a rigorous mathematical tool for unfolding quantum theory for time-dependent Hamiltonian systems. In particular, the quantum energy of the system is analyzed in detail and compared to the classical one. We focus on two particular cases; one is a linearly mass-accreting oscillator and the other is an exponentially mass-accreting one. It is confirmed that the quantum energy is in agreement with the classical one in the limit ℏ→0. We showed that not only the classical but also the quantum energy oscillates with time. It is carefully analyzed why the energy oscillates with time, and a reasonable explanation for that outcome is given.

Haantjes algebras of classical integrable systems

A tensorial approach to the theory of classical Hamiltonian integrable systems is proposed, based on the geometry of Haantjes tensors. We introduce the class of symplectic-Haantjes manifolds (or $$\omega {\mathscr {H}}$$ manifolds), as a natural setting where the notion of integrability can be formulated. We prove that the existence of suitable Haantjes algebras of (1,1) tensor fields with vanishing Haantjes torsion is a necessary and sufficient condition for a Hamiltonian system to be integrable in the Liouville–Arnold sense. We also show that new integrable models arise from the Haantjes geometry. Finally, we present an application of our approach to the study of the Post–Winternitz system and of a stationary flow of the KdV hierarchy.

Canonical analysis of Brans-Dicke theory addresses Hamiltonian inequivalence between the Jordan and Einstein frames

Jordan and Einstein frame are studied under the light of Hamiltonian formalism. Dirac's constraint theory for Hamiltonian systems is applied to Brans-Dicke theory in the Jordan Frame. In both Jordan and Einstein frame, Brans-Dicke theory has four secondary first class constraints and their constraint algebra is closed. We show, contrary to what is generally believed, the Weyl (conformal) transformation, between the two frames, is not a canonical transformation, in the sense of Hamiltonian formalism. This addresses quantum mechanical inequivalence as well. A canonical transformation is shown.