Global invariants for variable-mass systems

We investigate the effect of mass loss on the invariants of single particle motion in potentials having translational or rotational symmetry. These systems are non-Hamiltonian in the physical momenta, but for non-velocity-dependent potentials can be made formally Hamiltonian by treating the velocities as canonical momenta. Applying Noether's theorem to the formal Hamiltonian then yields global invariants corresponding to its symmetries. For velocity-dependent potentials, an exact invariant is constructed from the (non-Hamiltonian) vector field. The results are applied to charged particle motion in a magnetic dipole and agree well with numerical calculations.     

Curious behaviour of the diffusion coefficient and friction force for the strongly inhomogeneous HMF model

We present first elements of kinetic theory appropriate to the inhomogeneous phase of the Hamiltonian Mean Field (HMF) model. In particular, we investigate the case of strongly inhomogeneous distributions for T→0 and exhibit curious behaviour of the force auto-correlation function and friction coefficient. The temporal correlation function of the force has an oscillatory behaviour which averages to zero over a period. By contrast, the effects of friction accumulate with time and the friction coefficient does not satisfy the Einstein relation. On the contrary, it presents the peculiarity to increase linearly with time. Motivated by this result, we provide analytical solutions of a simplified kinetic equation with a time dependent friction. Analogies with self-gravitating systems and other systems with long-range interactions are also mentioned.     

From Nosé–Hoover chain to Nosé–Hoover network: design of non-Hamiltonian equations of motion for molecular-dynamics with multiple thermostats

A systematic algorithm to design multiple thermostat systems in the framework of the Nosé–Hoover type non-Hamiltonian formulation is presented. Using ‘non uniform’ time transformations in a generalised Hamiltonian equation, we develop the non-Hamiltonian equations of motion for multiple thermostat systems having an arbitrary number of thermostats and arbitrary connections between a physical system and thermostats (‘Nosé–Hoover network’). We then present the algorithm to construct the Nosé–Hoover network equations based on a simple diagram only. On the basis of this algorithm, recursively attached Nosé–Hoover thermostats are introduced as an example of the Nosé–Hoover network and its high efficiency in sampling the canonical distribution for an one-dimensional double-well system is illustrated by numerical calculations.     

Non-Hamiltonian systems separable by Hamilton–Jacobi method

We show that with every separable classical Stäckel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These systems are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate the conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems.     

Matrix Algebras in Non-Hermitian Quantum Mechanics

In principle, non-Hermitian quantum equations of motion can be formulated using as a starting point either the Heisenberg's or the Schrödinger's picture of quantum dynamics. Here it is shown in both cases how to map the algebra of commutators, defining the time evolution in terms of a non-Hermitian Hamiltonian, onto a non-Hamiltonian algebra with a Hermitian Hamiltonian. The logic behind such a derivation is reversible, so that any Hermitian Hamiltonian can be used in the formulation of non-Hermitian dynamics through a suitable algebra of generalized (non-Hamiltonian) commutators.
These results provide a general structure (a template) for non-Hermitian equations of motion to be used in the computer simulation of open quantum systems dynamics.     

Modeling and Analysis of a Three-Terminal-Memristor-Based Conservative Chaotic System

In this paper, a three-terminal memristor is constructed and studied through changing dual-port output instead of one-port. A new conservative memristor-based chaotic system is built by embedding this three-terminal memristor into a newly proposed four-dimensional (4D) Euler equation. The generalized Hamiltonian energy function has been given, and it is composed of conservative and non-conservative parts of the Hamiltonian. The Hamiltonian of the Euler equation remains constant, while the three-terminal memristor’s Hamiltonian is mutative, causing non-conservation in energy. Through proof, only centers or saddles equilibria exist, which meets the definition of the conservative system. A non-Hamiltonian conservative chaotic system is proposed. The Hamiltonian of the conservative part determines whether the system can produce chaos or not. The non-conservative part affects the dynamic of the system based on the conservative part. The chaotic and quasiperiodic orbits are generated when the system has different Hamiltonian levels. Lyapunov exponent (LE), Poincaré map, bifurcation and Hamiltonian diagrams are used to analyze the dynamical behavior of the non-Hamiltonian conservative chaotic system. The frequency and initial values of the system have an extensive variable range. Through the mechanism adjustment, instead of trial-and-error, the maximum LE of the system can even reach an incredible value of 963. An analog circuit is implemented to verify the existence of the non-Hamiltonian conservative chaotic system, which overcomes the challenge that a little bias will lead to the disappearance of conservative chaos.     

Generalized Pearson distributions for charged particles interacting with an electric and/or a magnetic field

The linear Boltzmann equation for elastic and/or inelastic scattering is applied to derive the distribution function of a spatially homogeneous system of charged particles spreading in a host medium of two-level atoms and subjected to external electric and/or magnetic fields. We construct a Fokker-Planck approximation to the kinetic equations and derive the most general class of distributions for the given problem by discussing in detail some physically meaningful cases. The equivalence with the transport theory of electrons in a phonon background is also discussed.     

Heat and fluctuations from order to chaos

The Heat theorem reveals the second law of equilibrium Thermodynamics (i.e. existence of Entropy) as a manifestation of a general property of Hamiltonian Mechanics and of the Ergodic Hypothesis, valid for 1 as well as 10²³ degrees of freedom systems, i.e. for simple as well as very complex systems, and reflecting the Hamiltonian nature of the microscopic motion. In Nonequilibrium Thermodynamics theorems of comparable generality do not seem to be available. Yet it is possible to find general, model independent, properties valid even for simple chaotic systems (i.e. the hyperbolic ones), which acquire special interest for large systems: the Chaotic Hypothesis leads to the Fluctuation Theorem which provides general properties of certain very large fluctuations and reflects the time-reversal symmetry. Implications on Fluids and Quantum systems are briefly hinted. The physical meaning of the Chaotic Hypothesis, of SRB distributions and of the Fluctuation Theorem is discussed in the context of their interpretation and relevance in terms of Coarse Grained Partitions of phase space. This review is written taking some care that each section and appendix is readable either independently of the rest or with only few cross references.     

Recursive properties of Dirac and metriplectic Dirac brackets with applications

In this article, we prove that Dirac brackets for Hamiltonian and non-Hamiltonian constrained systems can be derived recursively. We then study the applicability of that formulation in analysis of some interesting physical models. Particular attention is paid to the feasibility of implementation code for Dirac brackets in Computer Algebra System.     

Using relative entropy to find optimal approximations: An application to simple fluids

We develop a maximum relative entropy formalism to generate optimal approximations to probability distributions. The central results consist of (a) justifying the use of relative entropy as the uniquely natural criterion to select a preferred approximation from within a family of trial parameterized distributions, and (b) to obtain the optimal approximation by marginalizing over parameters using the method of maximum entropy and information geometry. As an illustration we apply our method to simple fluids. The “exact” canonical distribution is approximated by that of a fluid of hard spheres. The proposed method first determines the preferred value of the hard-sphere diameter, and then obtains an optimal hard-sphere approximation by a suitably weighed average over different hard-sphere diameters. This leads to a considerable improvement in accounting for the soft-core nature of the interatomic potential. As a numerical demonstration, the radial distribution function and the equation of state for a Lennard-Jones fluid (argon) are compared with results from molecular dynamics simulations.     

The Principle of Covariance and the Hamiltonian Formulation of General Relativity

The implications of the general covariance principle for the establishment of a Hamiltonian variational formulation of classical General Relativity are addressed. The analysis is performed in the framework of the Einstein-Hilbert variational theory. Preliminarily, customary Lagrangian variational principles are reviewed, pointing out the existence of a novel variational formulation in which the class of variations remains unconstrained. As a second step, the conditions of validity of the non-manifestly covariant ADM variational theory are questioned. The main result concerns the proof of its intrinsic non-Hamiltonian character and the failure of this approach in providing a symplectic structure of space-time. In contrast, it is demonstrated that a solution reconciling the physical requirements of covariance and manifest covariance of variational theory with the existence of a classical Hamiltonian structure for the gravitational field can be reached in the framework of synchronous variational principles. Both path-integral and volume-integral realizations of the Hamilton variational principle are explicitly determined and the corresponding physical interpretations are pointed out.     

Dynamical stability of systems with long-range interactions: application of the Nyquist method to the HMF model

We apply the Nyquist method to the Hamiltonian mean field (HMF) model in order to settle the linear dynamical stability of a spatially homogeneous distribution function with respect to the Vlasov equation. We consider the case of Maxwell (isothermal) and Tsallis (polytropic) distributions and show that the system is stable above a critical kinetic temperature T_c and unstable below it. Then, we consider a symmetric double-humped distribution, made of the superposition of two decentered Maxwellians, and show the existence of a re-entrant phase in the stability diagram. When we consider an asymmetric double-humped distribution, the re-entrant phase disappears above a critical value of the asymmetry factor Δ > 1.09. We also consider the HMF model with a repulsive interaction. In that case, single-humped distributions are always stable. For asymmetric double-humped distributions, there is a re-entrant phase for 1 ≤ Δ < 25.6, a double re-entrant phase for 25.6 < Δ < 43.9 and no re-entrant phase for Δ > 43.9. Finally, we extend our results to arbitrary potentials of interaction and mention the connexion between the HMF model, Coulombian plasmas and gravitational systems. We discuss the relation between linear dynamical stability and formal nonlinear dynamical stability and show their equivalence for spatially homogeneous distributions. We also provide a criterion of dynamical stability for spatially inhomogeneous systems.     

Obstructions to the continuation of analytic integrals of Hamiltonian systems under non-Hamiltonian perturbations

In this Letter we consider n degrees-of-freedom integrable Hamiltonian systems subjected to a non-Hamiltonian perturbation controlled by a small parameter ε. An obstruction to the analytic continuation of the integrals of motion of the unperturbed system with respect to ε is developed for sufficiently small perturbations. The theory is applied to a perturbed system of Morse oscillators.     

Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom

This paper concerns Hamiltonian and non-Hamiltonian perturbations of integrable two degree of freedom Hamiltonian systems which contain homoclinic and periodic orbits. Our main example concerns perturbations of the uncoupled system consisting of the simple pendulum and the harmonic oscillator. We show that small coupling perturbations with, possibly, the addition of positive and negative damping breaks the integrability by introducing horseshoes into the dynamics.Research partially supported by ARO Contract DAAG-29-79-C-0086 and by NSF Grants ENG 78-02891 and MCS-78-06718     

Stabilization of coherent breathers in perturbed Hamiltonian coupled oscillators

We propose and study a type of non-Hamiltonian, power-preserving perturbation of Hamiltonian systems of nonlinear coupled oscillators with a global phase invariance symmetry. Our results highlight the role of non-conservative perturbations in selecting a preferred dynamically attracting mode which can be chosen to be highly coherent. The proof of principle described above should be of interest for optical systems such as fiber laser arrays whose objective is to produce high power coherent light.     

Perturbation of Super-Sech Solitons in Dispersion-Managed Optical Fibers

The dynamics of super-sech solitons in dispersion-managed optical fibers is obtained in this paper. The dynamical system of soliton parameters is obtained for such pulses for dispersion-managed fibers, in presence of various perturbation terms. The perturbation terms studied are Hamiltonian, as well as non-Hamiltonian along with non-local types.     

Wigner Functions for Non-Hamiltonian Systems on Noncommutative Space

We propose a modified form of Wigner functions for generic non-Hamiltonian systems on noncommutative space and prove that it satisfies the corresponding ＊-genvalue equation. In addition, as an example, we derive exact energy spectra and Wigner functions for a non-Hamiltonian toy model on the noncommutative space.     

Unbounded Energy Growth in Hamiltonian Systems with a Slowly Varying Parameter

We study Hamiltonian systems which depend slowly on time. We show that if the corresponding frozen system has a uniformly hyperbolic invariant set with chaotic behaviour, then the full system has orbits with unbounded energy growth (under very mild genericity assumptions). We also provide formulas for the calculation of the rate of the fastest energy growth. We apply our general theory to non-autonomous perturbations of geodesic flows and Hamiltonian systems with billiard-like and homogeneous potentials. In these examples, we show the existence of orbits with the rates of energy growth that range, depending on the type of perturbation, from linear to exponential in time. Our theory also applies to non-Hamiltonian systems with a first integral.     

Statistical Dynamics of Solitons in Optical Fibers

The soliton perturbation theory is used to study and analyze the stochastic perturbation of optical solitons in addition to deterministic perturbations of optical solitons that are governed by the nonlinear Schro¨dinger's equation. The Langevin equations are derived and analyzed. The deterministic perturbations that are considered here are of both Hamiltonian as well as of non-Hamiltonian type.