Noncanonical Transformation Research Articles

Chaos of two particles in the ding-a-ling model

The smallest ding-a-ling system consists of two particles of equal masses. They move along a ring, and interact via elastic collisions. One of them is bound by a harmonic potential. Here, the system is described by a set of coupled maps, which drive it from one collision of the particles to another. This kind of Poincaré section is equivalent to a non-canonical transformation. We investigate a series of times t(n) between subsequent collisions. The method applied is the numerical calculation of the box counting fractal dimension of the phase portrait t(n+1) vs. t(n). We show that the phase transition from quasiperiodic to chaotic behaviour depends on the value of the ratio ξ of initial velocities of the particles. For large energy of the system we find chaotic behaviour for positive ξ, and quasiperiodic for negative ξ. We conclude that in the latter case the kinetic energy of the particles can be treated as an adiabatic invariant of motion.

Diffraction in time with dissipation

The more usual problems discussed in classical mechanics in elementary textbooks are the non-dissipative ones in which there is a Hamiltonian representing the energy as a constant of motion. The translation of this type of problem to quantum mechanics is very well known. Conversely, there are very simple classical mechanical problems that involve dissipation, but whose translation to quantum mechanics on the basis of a corresponding Hamiltonian is sometimes misinterpreted, since the underlying classical formalism involves non-canonical transformations that lead to non-unitary transformations of the quantum mechanical wavefunctions. In this paper we shall discuss the problem of a beam of particles of given momentum incident from the left on a shutter that is opened at time t = 0. The solution has the well known properties of diffraction in time, but we will analyse it quantum mechanically both when dissipation is and is not present. The objective is to get a better understanding of the effect of dissipation in the quantum mechanical picture and to estimate the influence of the above mentioned non-unitary transformations on this particular problem.

Nonlinear Gauge Transformation for a Quantum System Obeying an Exclusion-Inclusion Principle

We introduce a nonlinear and noncanonical gauge transformation which allows the reduction of a complex nonlinearity, contained in a Schrödinger equation, into a real one. This Schrödinger equation describes a canonical system, whose kinetics is governed by a generalized Exclusion-Inclusion Principle. The transformation can be easily generalized and used in order to reduce complex nonlinearities into real ones for a wide class of nonlinear Schrödinger equations. We show also that, for one dimensional system and in the case of solitary waves, the above transformation coincides with the one already adopted to study the Doebner–Goldin equation.

Nonlinear Gauge Transformation for a Quantum System Obeying an Exclusion-Inclusion Principle

We introduce a nonlinear and noncanonical gauge transformation which allows the re- duction of a complex nonlinearity, contained in a Schrodinger equation, into a real one. This Schrodinger equation describes a canonical system, whose kinetics is governed by a generalized Exclusion-Inclusion Principle. The transformation can be easily general- ized and used in order to reduce complex nonlinearities into real ones for a wide class of nonlinear Schrodinger equations. We show also that, for one dimensional system and in the case of solitary waves, the above transformation coincides with the one already adopted to study the Doebner-Goldin equation. Let us consider the kinetics of N particles in a one-dimensional discrete space, which is an one-dimensional Markovian chain. The generic site is labeled by the index i (i =0 , ±1, ±2 ,... ); the position at the ith site is xi = i∆x, where ∆x is a constant. We call ρi(t) the occupational probability of the ith site. Let us assume that only transitions to the nearest neighbors are allowed and define the transition probability π ± i (t) from the site i to the site i ± 1 in the following way: π ± i (t )= α ± (t)

A reversible problem in non-equilibrium thermodynamics: Hamiltonian evolution equations for non-equilibrium molecular dynamics simulations

The reversible contribution to contemporary theories of non-equilibrium thermodynamics is reviewed as a methodology for attacking difficult, conservative problems in complex fluid dynamics. Several examples of past successes are discussed, and a new application is addressed: non-equilibrium molecular dynamics (NEMD) simulations. NEMD simulations of fluids are generally based on either a DOLLS or SLLOD tensor algorithm. The former is always considered to be a Hamiltonian system, but not particularly useful in high strain rate flow simulations, while the latter is considered not to be a Hamiltonian system, but much more practical and accurate in flow simulations. We demonstrate herein using non-canonical transformations of the particle momenta of the system that the SLLOD equations, when written in terms of appropriate non-canonical variables, are completely Hamiltonian, whereas the DOLLS equations are not so. A modified set of DOLLS equations in terms of the non-canonical variables which again is completely Hamiltonian is also derived. Both algorithms then lead to a phase space distribution function which is canonical in both the coordinates and momenta.

The Lax pairs for the Holt system

By using non-canonical transformation between the Holt system and the Henon-Heiles system the Lax pairs for all the integrable cases of the Holt system are constructed from the known Lax representations for the Henon-Heiles system.

Duality between integrable Stäckel systems

For the Stackel family of the integrable systems a non-canonical transformation of the time variable is considered. This transformation may be associated to the ambiguity of the Abel map on the corresponding hyperelliptic curve. For some Stackel's systems with two degrees of freedom the 2x2 Lax representations and the dynamical r-matrix algebras are constructed. As an examples the Henon-Heiles systems, integrable Holt potentials and the integrable deformations of the Kepler problem are discussed in detail.

Noncanonical time transformations relating finite-dimensional integrable systems

We consider dual Stackel schemes related to each other by a noncanonical transformation of the time variable. We prove that this duality of different integrable systems arises from the multivaluedness of the Abel mapping. We construct the Lax matrices and the r-matrix algebras for some integrable systems on a plane. The integrable deformations of the Kepler problem and the Holt-type systems are considered in detail.

Effective description of the dissipative interaction between simple model-systems and their environment

Chemical reactions are generally connected with energy transfer between reactands and their environment. There are different ways of taking this environment into account. The traditional system-plus-reservoir (S+R) approach couples the relevant system to a large number of environmental degrees of freedom, represented, e.g., by harmonic oscillators. However, this has the disadvantage—from a computational point of view—that a system of many coupled differential equations has to be solved due to the large number of degrees of freedom of the reservoir. The number of degrees of freedom can be drastically reduced if an effective description of the interaction between system and reservoir is applied. On the quantum mechanical level, the effect of the environment can be included, e.g., by the use of explicitly time-dependent Hamiltonians or nonlinear additions to the Schrodinger equation (SE). An advantage of the former method is the preservation of the canonical formalism and the linearity of the theory as well as the fact that its operators can be derived directly from the S+R approach. Therefore, it seems to be even more of a paradox that this method apparently violates the uncertainty principle. This violation does not occur in the nonlinear approaches but the nonlinearity seems, at first sight, to be problematic. However, it can be shown that a particular logarithmic nonlinear SE is equivalent to the approach using the time-dependent Hamiltonian and can be transformed into this. The fact that this transformation is nonunitary provides the key for the solution of the paradox and also allows one to show the equivalence of all three of the above-mentioned approaches. In investigating the classical counterpart, a noncanonical transformation is used between the dissipative physical level and the canonical formal level, which makes it even possible to transform the physical interacting system into a formal isolated system that can be treated in the conventional way. As a result, the effect of the interaction with the environment can be taken into account classically by noncanonical transformations and quantum mechanically by nonunitary transformations. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 537–547, 1999

Noncanonical Hamiltonian methods for particle motion in magnetospheric hydromagnetic waves

Hamiltonian equations of motion for a nonrelativistic charged particle in magnetospheric hydromagnetic perturbations are derived. The equations are gyroaveraged, allowing much larger time steps in numerical solutions of the equations of motion compared to integrating the full Lorentz equations of motion, but they contain finite‐gyroradius effects to all orders in k⊥ρ, where k⊥. is the perpendicular wave number and ρ is the particle gyroradius. The finite‐gyroradius effects are essential for the important class of particles which undergo magnetic drift‐bounce resonances with the waves. The equations are derived by finding a Lie transform of the perturbed guiding center phase‐space Lagrangian to a new Lagrangian which is independent of the gyrophase angle. The resulting Euler‐Lagrange equations contain nonlinear terms which automatically preserve the Hamiltonian properties of the original Lorentz system, such as conservation of energy (for static systems) and conservation of phase‐space volume. The Hamiltonian conservation properties are useful for checking the accuracy of numerical integration schemes and they are essential for the use of Poincaré surface‐of‐section plots. Compared to more traditional canonical Hamiltonian methods, the phase‐space Lagrangian Lie transform methods allow general, noncanonical phase‐space coordinates and transformations. This results in more power and flexibility in finding convenient forms for the final equations of motion. The results are given in coordinate‐free form and in terms of magnetic field coordinates. Applications of these results to calculations of hydromagnetic wave‐induced particle motion in the inner magnetosphere are discussed.

Studies of Melnikov method and transversal homoclinic orbits in the circular planar restricted three-body problem

Non-Hamiltonian systems containing degenerate fixed points obtained from two degrees of freedom near-integrable Hamiltonian systems through non-canonical transformations are dealt with in this paper. Two criteria for determining the existence of transversal hontoclinic and heteroclinic orbits are presented. By exploiting these criteria the existence of the transversal homoclinic orbits and so, of the transversal homoclinic tangle phenomenon in the near-integrable circular planar restricted three-body problem with sufficiently small mass ratio of the two primaries is proven. Under some assumptions, the existence of the transversal heteroclinic orbits is proven. The global qualitative phase diagram is also illustrated.

An Introduction to The Lie-Santilli Isotopic Theory

Lie's theory in its current formulation is linear, local and canonical. As such, it is not applicable to a growing number of non-linear, non-local and non-canonical systems which have recently emerged in particle physics, superconductivity, astrophysics and other fields. In this paper, which is written by a physicist for mathematicians, we review and develop a generalization of Lie's theory proposed by the Italian–American physicist R. M. Santilli back in 1978 when at the Department of Mathematics of Harvard University and today called Lie–Santilli isotheory. The latter theory is based on the so-called isotopies which are non-linear, non-local and non-canonical maps of any given linear, local and canonical theory capable of reconstructing linearity, locality and canonicity in certain generalized spaces and fields. The emerging Lie–Santilli isotheory is remarkable because it preserves the abstract axioms of Lie's theory while being applicable to non-linear, non-local and non-canonical systems. After reviewing the foundations of the Lie–Santilli isoalgebras and isogroups, and introducing seemingly novel advances in their interconnections, we show that the Lie–Santilli isotheory provides the invariance of all infinitely possible (well-behaved), non-linear, non-local and non-canonical deformations of conventional Euclidean, Minkowskian or Riemannian invariants. We also show that the non-linear, non-local and non-canonical symmetry transformations of deformed invariants are easily computable from the linear, local and canonical symmetry transforms of the original invariants and the given deformation. We then briefly indicate a number of applications of the isotheory in various fields. Numerous rather fundamental and intriguing, open mathematical and physical problems are indicated during the course of our analysis.

Synchronization of perturbed non-linear Hamiltonian oscillators

Abstract We develop a new transformation method for simplifying perturbed non-linear Hamiltonians with one degree of freedom in “generalized” action-angle variables. The method uses non-canonical Lie transformations. The first transformation is chosen to eliminate all perturbation terms from the transformed Hamiltonian. It completely determines the shape of the perturbed orbits; however, a separate time equation must be solved to relate the motion of the perturbed system with the transformed. Thus, we view the transformation as a synchronization of the perturbed motion with the motion along an unperturbed orbit. The second transformation inverts the time equation to produce solutions which are explicit, rather than implicit, in time. These solutions are shown to be equivalent to solutions found using a traditional normalization procedure. Our approach disencumbers the shape of a perturbed orbit from the phase-time relation on the orbit. Furthermore, it avoids the difficulties found in the averaging procedure involving partitioning and integration of periodic functions for as long as possible. This is important for non-linear oscillators since partitioning and integration can be much more involved than for harmonic oscillators described by trigonometric functions. Finally, we demonstrate the method on a perturbed Duffing oscillator in which the periodic functions are elliptic functions.

Quantum and classical mechanics of q-deformed systems

Quantum and classical mechanics of a system of q-deformed bosonic oscillators are considered. The q-deformed Heisenberg-Weyl algebra of creation and destruction operators is realized by differential operators in a space of functions of real commutative variables. The corresponding Hilbert space is constructed. In this approach, the deformation parameter turns out to be a function of the Planck constant, an oscillator frequency and a parameter with dimension of length. The Hamiltonian path integral is derived and its semiclassical approximation is investigated to obtain the corresponding q-deformed classical theory. The phase space spanned by the usual commutative coordinates is shown to be a cylinder in classical theory. The dimensional parameter introduced determines its radius. It is argued that the q-deformation can be associated with a special non-canonical transformation. The principle of least action for the classical q-deformed system is formulated. A representation of Uq(m) in a space of functions on a phase space spanned by commutative coordinates is constructed.

Synchronization of perturbed non-linear Hamiltonians

We propose a new method based on Lie transformations for simplifying perturbed Hamil- tonians in one degree of freedom. The method is most useful when the unperturbed part has solutions in non-elementary functions. A non-canonical Lie transformation is used to eliminate terms from the perturbation that are not of the same form as those in the main part. The system is thus transformed into a modified version of the principal part. In conjunction with a time transformation, the procedure synchronizes the motions of the perturbed system onto those of the unperturbed part. A specific algorithm is given for systems whose principal part consists of a kinetic energy plus an arbitrary potential which is polynomial in the coordinate; the perturbation applied to the principal part is a polynomial in the coordinate and possibly the momentum. We demonstrate the strategy by applying it in detail to a perturbed Duffing system. Our procedure allows us to avoid treating the system as a perturbed harmonic oscillator. In contrast to a canonical simplification, our method involves only polynomial manipulations in two variables. Only after the change of time do we start manipulating elliptic functions in an exhaustive discussion of the flows.

The Poisson bracket for q-deformed systems

It is shown that the quantum system of q-deformed oscillators with the Uq(n) group symmetry can be obtained by the canonical quantization of a classical system with a modified Poisson bracket. The modification of the Poisson bracket is connected with a non-canonical transformation of phase space variables. In this approach, the deformation parameter turns out to be a function of the Planck constant and some dimensional parameters characterizing the classical system.

Non-Hamiltonian Perturbations of Integrable Systems and Resonance Trapping

This paper studies general, non-Hamiltonian perturbations of integrable systems with two degrees of freedom and derives conditions for temporary and permanent resonance trapping. The analysis involves a noncanonical transformation of variables near the resonant manifold and averaging with respect to the fast phase to investigate oscillatory behavior on the intermediate timescale. The resulting reduced system is Hamiltonian to leading order and permits, after averaging on the intermediate, or libration, timescale, a canonical transformation to action-angle variables in the oscillation zone. The final system so obtained reveals the possible existence of two- and three-dimensional invariant tori in the vicinity of the resonant manifold. An explicit divergence condition for general perturbations to be dissipative on the slow timescale follows from the analysis. An application of this approach to the problem of resonant trapping and escape is outlined for the restricted problem of three bodies subject to dissipative perturbations with a radial symmetry.

A new method for treating the correlations in HTSC

A further study of a new method for treating the correlations proposed by Ovchinnikov and Krivnov is presented. It consists in noncanonical transformation of the real operators of hole creation to new Fermi operators acting in auxiliary space with exact normalization of the wave function. The simplest version of this method is applied to electronic models of the CuO 2 plane of HTSC to elucidate the influence of correlations on the coupling constants of the possible pairing interactions, in particular the possible kinematic origin of the correlated hopping interaction.

Role of the noncanonical transformation in anomalous gauge theories.

The method for calculating the commutator anomalies in anomalous Abeliangauge theories proposed by Yeung and Yu is reanalyzed based on the theory ofthe noncanonical transformation. Their method is easily extended to thenon-Abelian theory, but in general it does not lead to the correct Schwingerterm.

A note on the application of a generalized canonical approach to non-linear Hamiltonian systems

In the author's treatment of the ideal resonance problem (1988), a non-canonical transformation was employed to bring the original Hamiltonian to a form amenable to the use of standard action-angle variables. Though the strictly Hamiltonian form of equations of motion was thus compromised, their general form was maintained, allowing transformation of the system to arbitrary order and forestalling the introduction of elliptic functions until a final explicit integration required in this approach. The general theory of such transformations is presented, and some points regarding their application are discussed, leading to the conclusion that the approach is practically limited to systems with a single degree of freedom only.