Successive Canonical Transformations Research Articles

Tautochronic centrifugal pendulum vibration absorbers

Abstract Since the 1930s, centrifugal pendulum vibration absorbers have been used in rotating and reciprocating machinery for the attenuation of torsional vibrations. A large variety of absorber types were suggested and the design was done by linearization theory until the introduction of the tautochronic bifilar pendulum absorbers. Since then, the performance and dynamic stability of this specific absorber type have been considered in analytical and numerical investigations. Different perturbations, e.g. nonlinear mistuning, were considered in order to optimize the system performance, but the characteristic bifilar design remained unchanged. In this paper, a general approach for the design of tautochronic pendulum vibration absorbers is proposed. As a result, it is possible to deal with a large variety of non-bifilar centrifugal vibration absorber designs which provide application-related optimal performance and resolve some of the existing design limitations. Established analytic predictions that show a satisfactory agreement with numerical as well as experimental investigations for bifilar absorbers are not applicable for the comparison of different tautochronic absorbers. Therefore, the second part of this work shows how to analyze this class of centrifugal vibration absorbers using a Hamiltonian formulation. Successive canonical transformations lead to nonlinear equations in action-angle variables, which are then approximated to first order and analyzed by using the method of averaging. These results provide a basis for the design and analysis of tautochronic bifilar and non-bifilar vibration absorbers.

Properties of a polaron in a quantum dot: a squeezed-state variational approach

The ground-state energy and the average number of virtual phonons around the electron in a parabolic quantum dot for the entire range of the electron—phonon coupling constant are obtained using the single-mode squeezed-state variational approach. The variational approach we applied is based on two successive canonical transformations and using a displaced-oscillator type unitary transformation to deal with the bilinear terms which are usually neglected. In order to study the relationship between the ground-state energy and the average number of virtual phonons around the electron of a polaron in a parabolic quantum dot with the electron—LO-phonon coupling constant and the confinement length, numerical calculations are carried out in the electron—LO-phonon strong- and weak-coupling regions.

Optimization of Low-Thrust Limited-Power Trajectories in a Noncentral Gravity Field—Transfers between Orbits with Small Eccentricities

Numerical and first-order analytical results are presented for optimal low-thrust limited-power trajectories in a gravity field that includes the second zonal harmonicJ2in the gravitational potential. Only transfers between orbits with small eccentricities are considered. The optimization problem is formulated as a Mayer problem of optimal control with Cartesian elements—position and velocity vectors—as state variables. After applying the Pontryagin Maximum Principle, successive canonical transformations are performed and a suitable set of orbital elements is introduced. Hori method—a perturbation technique based on Lie series—is applied in solving the canonical system of differential equations that governs the optimal trajectories. First-order analytical solutions are presented for transfers between close orbits, and a numerical solution is obtained for transfers between arbitrary orbits by solving the two-point boundary value problem described by averaged maximum Hamiltonian, expressed in nonsingular elements, through a shooting method. A comparison between analytical and numerical results is presented for some maneuvers.

Energy spectrum of the organic quasi-1D conductors with NNN and correlated hopping

A model for organic quasi-one-dimensional conductors (TMTTF)2X and (TMTSF)2X is considered. The anisotropic character of these compounds is modelled by two different hopping parameters: t between nearest neighbors (NN) in a chain of tetramethyl-tetrathiafulvalene (TMTTF) or tetramethyl-tetraselenfulvalene (TMTSF) molecules and t between the chains (NNN between next nearest neighbors). Taking into account the correlated hopping of electrons allows us to describe the effect of site occupancy on hopping processes. In a regime of strong intraatomic correlation, high energy processes are cut off by applying two successive canonical transformations. An effective model is obtained for concentration of electrons n < 1 which contains kinetic exchange terms of antiferromagnetic (AF) nature. Oppositely, NNN hopping and correlated hopping disfavor the AF order. The energy spectrum of the effective model is calculated. Application of the obtained results to quasi-one-dimensional conductors is discussed.

Impurity magnetopolaron in a parabolic quantum dot: the squeezed-state variationalapproach

We present a calculation of the ground-state binding energy of an impurity magnetopolaronconfined in a three-dimensional (3D) parabolic quantum dot potential, in the framework ofa variational approach based on two successive canonical transformations. First, we apply adisplaced-oscillator type unitary transformation to diagonalize the relevant FröhlichHamiltonian. Second, a single-mode squeezed-state transformation is introducedto deal with bilinear terms arising from the first transformation. Finally, theparameters of these transformations together with the parameters included in theelectronic trial wavefunction are determined variationally to obtain the ground-statebinding energy of an impurity magnetopolaron confined in a 3D parabolic quantumdot potential. Our approach has two advantages: first, the displaced-oscillatortransformation allows one to obtain results valid for whole range of electron–phononcoupling strength since it is a special combination of Lee–Low–Pines and Huybrechts(LLP–H) canonical transformations, and second, the later transformation improvesall-coupling results. It has been shown that the effects of quadratic terms arising fromthe all-coupling approach are very important and should be taken into accountin studying the size-dependent physical properties of nanostructured materials.

The post-Newtonian effects in the critical inclination problem in artificial satellite theory

The critical inclination problem in artificial satellite theory and the post-Newtonian (PN) effects on it are discussed. The Hamiltonian is constructed and is expressed in terms of the Delaunay variables containing the zonal harmonics up to J 4 and the PN contribution up to order J 2/ c l 2. The Hamiltonian is ordered out such that the PN terms are of order J 2. Two successive canonical transformations based on Lie series and transform are carried out to remove short and long period terms from the Hamiltonian, respectively. Terms are retained up to the third order in the secular perturbations and to the second order in the periodic perturbations, where J 2 is considered of first order.

Canonical solution of the critical inclination problem taking into account PN-corrections

The critical inclination problem in artificial satellite theory and the post-Newtonian (PN) effects on it are discussed. The Hamiltonian is constructed and is expressed in terms of the Delaunay variables containing the zonal harmonics up to J 4 and the PN contributions up to order J 2/ c l 2. The Hamiltonian is ordered out such that the PN terms are of order J 2. Two successive canonical transformations based on Lie series and transform are carried out to remove short- and long-period terms from the Hamiltonian, respectively. Terms are retained up to third order in the secular and to the second order in the periodic terms, where J 2 is considered of first order.

The Spin-Orbit Resonant Rotation of Mercury: A Two Degree of Freedom Hamiltonian Model

The paper develops a hamiltonian formulation describing the coupled orbital and spin motions of a rigid Mercury rotation about its axis of maximum moment of inertia in the frame of a 3:2 spin orbit resonance; the (ecliptic) obliquity is not constant, the gravitational potential of mercury is developed up to the second degree terms (the only ones for which an approximate numerical value can be given) and is reduced to a two degree of freedom model in the absence of planetary perturbations. Four equilibria can be calculated, corresponding to four different values of the (ecliptic) obliquity. The present situation of Mercury corresponds to one of them, which is proved to be stable. We introduce action-angle variables in the neighborhood of this stable equilibrium, by several successive canonical transformations, so to get two constant frequencies, the first one for the free spin-orbit libration, the other one for the 1:1 resonant precession of both nodes (orbital and rotational) on the ecliptic plane. The numerical values obtained by this simplified model are in perfect agreement with those obtained by Rambaux and Bois [Astron. Astrophys. 413, 381–393].

Generalized adiabatic product expansion: A nonperturbative method of solving the time-dependent Schrödinger equation

We outline a method based on successive canonical transformations which yields a product expansion for the evolution operator of a general (possibly non-Hermitian) Hamiltonian. For a class of such Hamiltonians this expansion involves a finite number of terms, and our method gives the exact solution of the corresponding time-dependent Schrödinger equation. We apply this method to study the dynamics of a general nondegenerate two-level quantum system, a time-dependent classical harmonic oscillator, and a degenerate system consisting of a spin 1 particle interacting with a time-dependent electric field E⃗(t) through the Stark Hamiltonian H=λ(J⃗⋅E⃗)2.

Classical canonical transformation theory as a tool to describe multidimensional tunnelling in reactive scattering. Hopping method revisited and collinear H+H2 exchange reaction near the classical threshold

Classical canonical perturbation theory is applied in the vicinity of the saddle point for a chemical reaction. This is done by applying successive canonical transformations in the scope of the Gustavson–Birkhoff approach. It is shown that the calculated approximate classical integrals of motion can be used to describe classically forbidden tunnelling processes. They are also organically embedded into a hopping method to incorporate tunnelling effects into classical trajectory simulations of chemical reactions. The applicability of the proposed scheme is demonstrated for the collinear H+H2 exchange reaction using the double many-body expansion potential energy surface.

Analytical Effects of Gravitational Waves on the Motion of an Artificial Satellite

AbstractThe motion of an artificial satellite in the Earth’s gravitational field is discussed in the post-Newtonian framework including the effect of weak gravitation waves using the perturbation technique of the canonical Lie-transformations. Two successive canonical transformations are used to derive analytical expressions for the short-period, long-period and secular perturbations of orbital elements. The solution is expressed in terms of the Delaunay variables.

Application of complex canonical point transformations to linear second-order differential equations

We present a canonical method to solve one-dimensional linear differential equations making use of pseudodifferential calculus. We apply two successive canonical point transformations on the cartesian momentum and position spaces to obtain a nonlinear complex-valued canonical transformation which maps a very simple linear differential equation into the desired differential equation. This method yields a closed contour integral representation for the exact solution in terms of arbitrary functions, which, may be determined from the mapping equations in a similar way to that followed in classical mechanics. This method does not require the completeness condition on the intermediary states and avoids calculation of the kernel of the generator. We explicitly develop the case of second-order differential equations and give some standard examples to show how this method works.

Nonlinear Asymmetric Excitation of Edge Waves

The resonant excitation of two weakly nonlinear oppositely travelling edge waves on a gently sloping beach by a perfectly reflected obliquely incident gravity wave is calculated. The four first-order nonlinear differential equations that govern the slowly varying amplitudes and phases of the edge waves are associated with a Hamiltonian supplemented by a dissipation function that provides for both viscous and radiation damping. This system is reduced, through successive canonical transformations in action-angle variables, to two evolution equations that are isomorphic to those for the symmetric problem of normal incidence. Asymmetric edge waves of finite amplitude, like their symmetric counterparts, occur in a narrow band of frequency if and only if the dissipation is inferior to a certain critical value, in which case they appear as a slowly modulated (in both space and time) standing wave and induce a second-order mean flow along the shore. The effects of imperfect reflexion are described by reference to the symmetric problem

Adiabatic Invariance and Passage Through Resonance for Nearly Periodic Hamiltonian Systems

This is a continuation of previous work on passage through resonance to nearly periodic Hamiltonian systems. We review the classical technique for calculating adiabatic invariants and exhibit the occurrence of zero divisors in the results as a certain critical term evolves slowly through a resonance condition. We then isolate the coordinate associated with the singularity and remove the remaining coordinate from the Hamiltonian to any desired order by successive canonical transformations. This is a variant of the von Zeipel procedure used extensively in celestial mechanics. The momentum conjugate to the cyclic coordinate is an adiabatic invariant, and the reduced Hamiltonian is then solved by constructing and matching three multiple variable expansions which describe the solution before, during, and after resonance passage.

Perturbation theory for the general classical x2p+2 non-linear damped oscillator

The first order perturbation solution for the frequency of the general classical x 2p+2 non-linear damped oscillator is obtained by generating successive canonical transformations in the framework of the Hamilton-Jacobi theory.

Small Oscillation Theory of the Interaction of a Particle and Scalar Field

We study the interaction of a nonrelativistic particle with a scalar field, with particular application to the theory of polarons. The approach is based on a general classical method for the integration of equations of motion. The Hamiltonian is transformed by successive canonical transformations, the first corresponding to describing the motion relative to special solutions of the equations of motion. This stage as applied to suitably ordered Heisenberg equations of motion is identical with intermediate coupling theory. The second transformation treats the coupled small oscillations of particle and field oscillators about the chosen special solution. This affords a natural extension of intermediate coupling theory for this problem. Differences between the classical and quantum theories arise in the ordering of operators; the differences play a crucial role in determining the effective cutoff in wave vector space.