Factorization of the constants of motion

2006 ◽  
Vol 84 (8) ◽  
pp. 717-722
Author(s):  
P L Nash ◽  
L Y Chen
Keyword(s):  
Model System ◽  
First Integrals ◽  
Time Dependent ◽  
05.70 Ln ◽  
Constant Of Motion ◽  
Sum Of Products ◽  
Constants Of Motion ◽  
Independent Constant ◽  
Complete Set

A complete set of first integrals, or constants of motion, for a model system is constructed using “factorization”, as described below. The system is described by the effective Feynman Lagrangian L = [Formula: see text], with one of the simplest, nontrivial, potentials V(x) = (1/2)m ω2x2 selected for study. Four new, explicitly time-dependent, constants of the motion ci±, i = 1, 2 are defined for this system. While [Formula: see text]ci± ≠ 0, [Formula: see text]ci± = [Formula: see text]ci± + [Formula: see text]ci± + [Formula: see text]ci± + · · · = 0 along an extremal of L. The Hamiltonian H is shown to equal a sum of products of the ci±, and verifies [Formula: see text] = 0. A second, functionally independent constant of motion is also constructed as a sum of the quadratic products of ci±. It is shown that these derived constants of motion are in involution.PACS Nos.: 02.30.Jr, 02.30.Ik, 02.60.Cb, 02.30.Hq, 05.70.Ln, 02.50.–r

TEMPERATURE-DEPENDENCE OF PROTON CONDUCTIVITY IN HYDROGEN-BONDED MOLECULAR SYSTEMS

2007 ◽  
Vol 21 (19) ◽  
pp. 1239-1252 ◽  
Author(s):  
XIAO-FENG PANG ◽  
BO DENG ◽  
HUAI-WU ZHANG ◽  
YUAN-PING FENG
Keyword(s):  
Experimental Data ◽  
Electric Field ◽  
Temperature Dependence ◽  
Electric Conductivity ◽  
Proton Conductivity ◽  
Model System ◽  
Time Dependent ◽  
Molecular Systems ◽  
Damping Effect ◽  
Hydrogen Bonded

The temperature-dependence of proton electric conductivity in hydrogen-bonded molecular systems with damping effect was studied. The time-dependent velocity of proton and its mobility are determined from the Hamiltonian of a model system. The calculated mobility of (3.57–3.76) × 10-6 m 2/ Vs for uniform ice is in agreement with the experimental value of (1 - 10) × 10-2 m 2/ Vs . When the temperature and damping effects of the medium are considered, the mobility is found to depend on the temperature for various electric field values in the system, i.e. the mobility increases initially and reaches a maximum at about 191 K, but decreases subsequently to a minimum at approximately 241 K, and increases again in the range of 150–270 K. This behavior agrees with experimental data of ice.

scholarly journals Complete set of quasi-conserved quantities for spinning particles around Kerr

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Geoffrey Compère ◽  
Adrien Druart
Keyword(s):  
Linear Order ◽  
Conserved Quantities ◽  
Killing Tensor ◽  
Kerr Geometry ◽  
Spinning Particles ◽  
Kerr Spacetime ◽  
Killing Tensors ◽  
Mixed Symmetry ◽  
Constants Of Motion ◽  
Complete Set

We revisit the conserved quantities of the Mathisson-Papapetrou-Tulczyjew equations describing the motion of spinning particles on a fixed background. Assuming Ricci-flatness and the existence of a Killing-Yano tensor, we demonstrate that besides the two non-trivial quasi-conserved quantities, i.e. conserved at linear order in the spin, found by Rüdiger, non-trivial quasi-conserved quantities are in one-to-one correspondence with non-trivial mixed-symmetry Killing tensors. We prove that no such stationary and axisymmetric mixed-symmetry Killing tensor exists on the Kerr geometry. We discuss the implications for the motion of spinning particles on Kerr spacetime where the quasi-constants of motion are shown not to be in complete involution.

scholarly journals Exact time-dependent decoherence factor and its adiabatic classical limit

2003 ◽  
Vol 81 (10) ◽  
pp. 1185-1191
Author(s):  
J -Q Shen ◽  
P Chen ◽  
H Mao
Keyword(s):  
Invariant Theory ◽  
Geometric Phase ◽  
Classical Limit ◽  
Unitary Transformation ◽  
Time Dependent ◽  
Quantum Decoherence ◽  
Dynamical Models ◽  
Complete Set ◽  
General Time ◽  
03.65 Ge

The present paper finds the complete set of exact solutions of the general time-dependent dynamical models for quantum decoherence, by making use of the Lewis–Riesenfeld invariant theory and the invariant-related unitary transformation formulation. Based on this, the general explicit expression for the decoherence factor is then obtained and the adiabatic classical limit of an illustrative example is discussed. The result (i.e., the adiabatic classical limit) obtained in this paper is consistent with what is obtained by other authors, and furthermore we obtain more general results concerning time-dependent nonadiabatic quantum decoherence. It is shown that the invariant theory is appropriate for treating both the time-dependent quantum decoherence and the geometric phase factor. PACS Nos.: 03.65.Ge, 03.65.Bz

scholarly journals PERIODIC FIRST INTEGRALS FOR HAMILTONIAN SYSTEMS OF LIE TYPE

2011 ◽  
Vol 08 (06) ◽  
pp. 1169-1177 ◽  
Author(s):  
RUBEN FLORES ESPINOZA
Keyword(s):  
Hamiltonian Systems ◽  
Nonlinear Oscillator ◽  
Original System ◽  
First Integrals ◽  
Time Dependent ◽  
Existence Problem ◽  
Adjoint Group ◽  
Killing Form ◽  
Modern Physics ◽  
Existence Of Periodic Solutions

In this paper, we study the existence problem of periodic first integrals for periodic Hamiltonian systems of Lie type. From a natural ansatz for time-dependent first integrals, we refer their existence to the existence of periodic solutions for a periodic Euler equation on the Lie algebra associated to the original system. Under different criteria based on properties for the Killing form or on exponential properties for the adjoint group, we prove the existence of Poisson algebras of periodic first integrals for the class of Hamiltonian systems considered. We include an application for a nonlinear oscillator having relevance in some modern physics applications.

THE QUADRATIC TIME-DEPENDENT HAMILTONIAN: EVOLUTION OPERATOR, SQUEEZING REGIONS IN PHASE SPACE AND TRAJECTORIES

1994 ◽  
Vol 08 (11n12) ◽  
pp. 1563-1576 ◽  
Author(s):  
S.S. MIZRAHI ◽  
M.H.Y. MOUSSA ◽  
B. BASEIA
Keyword(s):  
Phase Space ◽  
Unitary Transformation ◽  
Uniform Magnetic Field ◽  
Evolution Operator ◽  
Single Mode ◽  
Time Dependent ◽  
Special Cases ◽  
Complete Set ◽  
Hamiltonian Evolution ◽  
General Time

We consider the most general Time-Dependent (TD) quadratic Hamiltonian written in terms of the bosonic operators a and a+, which may represent either a charged particle subjected to a harmonic motion, immersed in a TD uniform magnetic field, or a single mode photon field going through a squeezing medium. We solve the TD Schrödinger equation by a method that uses, sequentially, a TD unitary transformation and the diagonalization of a TD invariant, and we verify that the exact solution is a complete set of TD states. We also obtain the evolution operator which is essential to express operators in the Heisenberg picture. The variances of the quadratures are calculated and a phase space of parameters introduced, in which we identify squeezing regions. The results for some special cases are presented and as an illustrative example the parametric oscillator is revisited and the trajectories in phase space drawn.

Some vorticity theorems and conservation laws for non-barotropic fluids

1981 ◽  
Vol 108 ◽  
pp. 475-483 ◽  
Author(s):  
S. D. Mobbs
Keyword(s):  
Thermodynamic Properties ◽  
Conservation Laws ◽  
Potential Vorticity ◽  
Time Dependent ◽  
Dissipative Effects ◽  
Perfect Fluids ◽  
Barotropic Fluids ◽  
Complete Set ◽  
Circulation Theorem

Some theorems concerning the vorticity in barotropic flows of perfect fluids are generalized for non-barotropic flows. The generalization involves replacing the velocity in certain parts of the equations by a time-dependent quantity which is a function of the velocity and thermodynamic properties of the fluid. Results which are generalized include Kelvin's circulation theorem and conservation laws for potential vorticity and helicity. It is shown how the results can be further generalized to include dissipative effects. The possibility of using some of the results in deriving a complete set of Lagrangian conservation laws for perfect fluids is discussed.

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