Showing papers on "Canonical transformation published in 1985"

Relations between elements of transfer matrices due to the condition of symplecticity

Abstract: Relations are derived between elements of transfer matrices. These relations result from the fact that the motion of charged particles from one profile plane to another can be described as a canonical transformation. The derived four first order relations are already known. Similarly, the higher order relations are very useful to check results of numerical ion optical calculations.

Canonical transformations theory for presymplectic systems

Abstract: We develop a theory of canonical transformations for presymplectic systems, reducing this concept to that of canonical transformations for regular coisotropic canonical systems. In this way we can also link these with the usual canonical transformations for the symplectic reduced phase space. Furthermore, the concept of a generating function arises in a natural way as well as that of gauge group.

Design of a state estimator for a class of time-varying multivariable systems

Abstract: In this note, matrix operators are first introduced to develop a canonical transformation. Then it shows that a full order state estimator with any eigenvalues can be constructed for linear time-varying systems that are uniformly observable and "lexicography-fixed." A simple algorithm is provided to design the state estimator.

Hamiltonian approach to the existence of magnetic surfaces

Abstract: A method is devised to investigate the existence of magnetic surfaces and magnetohydrodynamic (MHD) plasma equilibria in 3‐D toroidal geometry. The key feature of this method is the utilization of a Hamiltonian formulation of the lines of force. Expanding the contravariant components of the magnetic field and scalar pressure in distance ρ from the magnetic axis, the 1‐D Hamiltonian for the lines of force is written out explicitly. The Hamiltonian is then transformed to action‐angle variables. It is shown that the action J corresponds to pressure in the equilibrium problem. Specifically, it is shown that if J is an invariant, then constant pressure and hence magnetic surfaces exist. A procedure of repeated canonical transformations is formulated and carried out to displace the coordinate dependence in the Hamiltonian to successively higher order in the expansion parameter, and thus make J an increasingly better adiabatic invariant. Arising in each successive canonical transformation is a series of potentia...

Chaos and nonunitary evolution in nonintegrable Hamiltonian systems

Abstract: The Hamilton-Jacobi canonical transformation theory is extended to treat nonintegrable Hamiltonian systems with a continuous Fourier spectrum. By a natural analytic continuation of the Fourier variable to the complex plane, a nonunitary operator for the canonical transformation is obtained. We apply this transformation to describe Chirikov's diffusion process near a separatrix in nonlinear systems with two degrees of freedom. Our formalism shows a clear distinction between the irreversible evolution of an ensemble with a finite measure from the reversible evolution of a trajectory in nonintegrable systems with chaotic motion. The condition for obtaining the irreversible kinetic equation in Hamiltonian systems is connected to the condition for the existence of homoclinic points around the separatrix. We also show that Prigogine's dissipativity condition in nonequilibrium statistical systems is equivalent to the nonintegrability condition for nonlinear systems with a few degrees of freedom.

Quantization of Phonons and Raman Scattering in a Continuum Model of Trans-Polyacetylene

Abstract: The method of canonical transformation is applied in quantizing phonons around a general static solution in a continuum version of a model for t-(CH) x . The transformed effective Hamiltonian in the perfectly dimerized case is used to calculate the first and second order Raman scattering amplitudes. Especially it is shown that, through the measurement of the second order Raman process, the phonon dispersion can be derived. The application of the formulation to the case with a soliton or a polaron is also discussed.

Nonlinear canonical transformations. I

Abstract: The linear canonical transformation Q=aq+bp, P=cq+dp, (ad-bc=1,(q,p)=constant) is extended to include terms of the second, third and fourth degree. Products of second-degree transformations are used to construct third- and fourth-degree transformations. The question of factorisability of fourth-degree transformations is discussed.

A variant of the Hori-Lie series method

Abstract: If the undisturbed Hamiltonian F0 is a function of the momenta only, then a variant of the Hori-Lie series perturbation method achieves a solution to any order with a single canonical transformation, without the use of the pseudo-time.

Optical switching properties of semiconductor lasers: analysis in canonical form

Abstract: The description of optical switching occurring at a Hopf bifurcation in semiconductor lasers is developed utilising a canonical transformation of the equations of laser dynamics In canonical form, the equations of motion give an immediate identification of the switching frequency It is further shown that the canonical transformation leads to a compact formalism for studying the stability of the oscillations established at the Hopf bifurcation

Statistical Mechanics of the sine-Gorden Field: Part II

Abstract: From the work of the Part I we are now in a position to address ourselves to the main problem posed in these lectures — the evaluation of Z, (1.11), for the s-G field after canonical transformation to the action-angle variables (4.27).

On the Bound Optical Polaron

Abstract: The optical polaron bound in a Coulomb potential is investigated using a simple canonical transformation. The freedom in the choice of the generator of the transformation in this method is utilised to calculate the ground state for different types of trial generators. The results for different ranges of the coupling constant α and the Coulomb impurity parameter β are critically examined and compared with the major available results due to others. In view of the overall good and selectively excellent results obtained and in view of its simplicity the method appears to be useful.