Lie Transformation Method Research Articles

Effective and Efficient Resonant Transitions in Periodically Modulated Quantum Systems

We analyze periodically modulated quantum systems with SU(2) and SU(1,1) symmetries. Transforming the Hamiltonian into the Floquet representation we apply the Lie transformation method, which allows us to classify all effective resonant transitions emerging in time-dependent systems. In the case of a single periodically perturbed system, we propose an explicit iterative procedure for the determination of the effective interaction constants corresponding to every resonance both for weak and strong modulation. For coupled quantum systems we determine the efficient resonant transitions appearing as a result of time modulation and intrinsic non-linearities.

Lie symmetry analysis of a stochastic gene evolution in double-chain deoxyribonucleic acid system

ABSTRACT In this work, we study the double chain Deoxyribonucleic acid (DNA) model, the study will be carried out by using two analytical methods, namely, Lie transformation method, then combined with the singular manifold method (SMM). Six cases are studied for different vector combinations. Two successive symmetry reductions transform the equation to an ordinary differential equation (ODE). We next apply the singular manifold method to the obtained ordinary differential equations. This will result in a Schwarzian derivative whose solution leads to soliton solutions for longitudinal and transversal motions. The solutions contain different types of wave solutions, N-solitons, kink and periodic wave solutions. Some of the resulting solutions are graphically illustrated.

A study of the moderate altitude frozen orbits around the Moon

Lunar frozen orbits, characterized by fixed eccentricity and argument of perigee on average, have been previously studied using different dynamical models. In this work, frozen orbits about the Moon are investigated on the basis of an averaged Hamiltonian. The gravitational field of the Moon is considered up to the seven zonal harmonic plus the third body perturbation (Earth). The third body is assumed to move in an elliptic inclined orbit. The averaging procedure is performed through Lie transformation method. We used the eccentricity-inclination diagrams to obtain a deep insight about the evolution of frozen orbits. For the reduced system, we found two frozen solutions which correspond to argument of perigee ω=π2,3π2. Moreover, we studied the evolution of the eccentricity and inclination as a function of time. The results showed that, for moderate altitude orbits, the eccentricity oscillates with small amplitudes around its initial value, while the inclination almost remains constant. In higher altitudes case, we observed that variations in eccentricity and inclination are larger than the moderate ones. The present dynamical model gives acceptable results for low initial eccentricity and inclination.

The dynamics of an ion acting on two monochromatic obliquely propagating Alfvén waves

The interaction between a magnetized ion and two monochromatic shear Alfvén waves propagating obliquely to an ambient magnetic field is investigated both analytically and numerically. According to the Hamiltonian of this system, the invariant of motion at the lowest order and the half-island widths at the corresponding resonances are obtained analytically using the Lie transformation method. It is shown that these theoretical results agree with the numerical ones from the Poincaré surface of section. The regular motions from the invariant and the transition to stochasticity due to resonance overlapping are demonstrated. Compared to the case of a single wave, there may be a lower stochastic threshold in the multiple-wave problem.

Extension of the Lie Transform Theory Depending on a Small Parameter for Multi-parametric Dynamical Systems

Lie transform method derived by Deprit and Hori in the 1960s allowed researchers to solve the perturbation problems depending on a small parameter. But actually the real systems are very complicated. In the astrodynamics, one usually encounters problems involving several perturbations which in turn yields dynamical system with several small parameters, e.g., oblateness of the massive objects, radiation pressure, mass loss, relativistic effects, drag perturbations, etc. To involve as many perturbations as the system requires, the theory of canonical Lie transform depending on a small parameter is extended to N-small parameters. Lie transform based on one small parameter is briefly surveyed. Some useful lemmas are proved. Then, the generalized Lie transformation method is developed.

Lie transformation method on quantum state evolution of a general time-dependent driven and damped parametric oscillator

A variety of dynamics in nature and society can be approximately treated as a driven and damped parametric oscillator. An intensive investigation of this time-dependent model from an algebraic point of view provides a consistent method to resolve the classical dynamics and the quantum evolution in order to understand the time-dependent phenomena that occur not only in the macroscopic classical scale for the synchronized behaviors but also in the microscopic quantum scale for a coherent state evolution. By using a Floquet U-transformation on a general time-dependent quadratic Hamiltonian, we exactly solve the dynamic behaviors of a driven and damped parametric oscillator to obtain the optimal solutions by means of invariant parameters of Ks to combine with Lewis–Riesenfeld invariant method. This approach can discriminate the external dynamics from the internal evolution of a wave packet by producing independent parametric equations that dramatically facilitate the parametric control on the quantum state evolution in a dissipative system. In order to show the advantages of this method, several time-dependent models proposed in the quantum control field are analyzed in detail.

The dynamics of ion with electrostatic waves in a sheared magnetic field

The interaction between an ion and multiple electrostatic waves propagating perpendicularly to an ambient magnetic field with shear is investigated. Based on the Lie transformation method, with the wave amplitude and the magnetic shear both as the perturbation parameters, the analytical formulas for the reduced Hamiltonian is derived and results are compared with numerical calculations of the complete equations of motion for the case of two on-resonance waves and the case of two off-resonance waves, respectively. It is found that the effect of magnetic shear drastically prevents the acceleration of an ion in both cases. This result will help us to understand the behaviors of ions in a magnetic sheared device, such as tokamak.

Second order guiding-center Vlasov–Maxwell equations

Second order gyrogauge invariant guiding-center coordinates with strong E×B-flow are derived using the Lie transformation method. The corresponding Poisson bracket structure and equations of motion are obtained. From a variational principle the explicit Vlasov–Maxwell equations are derived including second order terms. The second order contributions contain the lowest order finite-Larmor-radius corrections to the electromagnetic field. Therefore, the model is capable of describing situations where strong E×B-flows and finite-Larmor-radius effects are mutually important.

Lie transformation method for dynamical systems having chaotic behavior

The paper persents recent developments in a singular perturbation method, known as the “Lie transformation method” for the analysis of nonlinear dynamical systems having chaotic behavior. A general approximate solution for a system of first-order differential equations having algebraic nonlinearities is introduced. Past applications to simple dynamical nonlinear models have shown that this method yields highly accurate solutions of the systems. In the present paper the capability of this method is extended to the analysis of dynamical systems having chaotic behavior: indeed, the presence of “small divisors” in the general expression of the solution suggests a modification of the method that is necessary in order to analyze nonlinear systems having chaotic behavior (indeed, even non-simple-harmonic behavior). For the case of Hamiltonian systems this is consistent with the KAM (Kolmogorov-Arnold-Moser) theory, which gives the limits of integrability for such systems; in contrast to the KAM theory, the present formulation is not limited to conservative systems. Applications to a classic aeroelastic problem (panel flutter) are also included.

Improved linear representation of surface waves. Part 2. Slowly varying bottoms and currents

We extend the results of a previous paper to fluids of finite depth. We consider the Hamiltonian theory of waves on the free surface of an incompressible fluid, and derive the canonical transformation that eliminates the leading order of nonlinearity for finite depth. As in the previous paper we propose using the Lie transformation method since it seems to include a nearly correct implementation of short waves interacting with long waves. We show how to use the Eikonal method for slowly varying currents and/or depths in combination with the nonlinear transformation. We note that nonlinear effects are more important in water of finite depth. We note that a nonlinear action conservation law can be derived.

Satellite vibration-rotation motions studied via canonical transformations

Coupled vibration-rotation motion of a satellite is considered using a perturbation theory based on the Lie transformation method. Short-period oscillating terms are removed from the Hamiltonian function. The transformed damping forces directly affect rotational variables which were not directly influenced in the original variables. Motions and stability are more easily studied in the new variables. A dual-spin spacecraft model is used as an example; results for the usual nonresonant case are identical with the energy-sink method. Resonance cases produce a wealth of new dynamical phenomena. This canonical method extends and unifies various approximation methods in attitude dynamics.