A phase space moment method for classical and quantal dynamics

1981 ◽  
Vol 59 (3) ◽  
pp. 457-470 ◽  
Author(s):  
R. E. Turner ◽  
R. F. Snider
Keyword(s):  
Phase Space ◽  
Cross Sections ◽  
Moment Method ◽  
Equations Of Motion ◽  
Second Order ◽  
Conservation Of Energy ◽  
Potential Applications ◽  
The Moment ◽  
Hamilton's Equations ◽  
Hamilton’S Equations

A hierarchy of translational moment equations for the position and momentum observables is considered. The resulting set of equations is truncated in a manner that includes only first and second order moments. This closure procedure is consistent with the conservation of energy and, for spherically symmetric potentials, with the conservation of angular momentum. The method is valid for both classical and quantal mechanics with the only difference being that for quantal systems, the dispersions in momentum and position must satisfy Heisenberg's uncertainty principle. With finite dispersion in the position, the average position and momentum do not satisfy Hamilton's equations of motion since the average acceleration is modulated by the variance–covariance in the position. For classical systems, the second order moments may be taken to zero in which case Hamilton's equations are obtained. For quantal systems, a comparision is made with Heller's Gaussian wave packet approach. The moment method is illustrated by applying it to the reflection of a phase space packet from a one dimensional repulsive inverse square potential. Applications to the calculation of collision cross sections are envisaged.

scholarly journals Research of crack initiation in tension test specimen made of nickel alloy EI961sh

2018 ◽  
Vol 165 ◽  
pp. 22027
Author(s):  
Kamaliddin Karimbaev ◽  
Ivan Pleshcheev ◽  
Elena Bredihina
Keyword(s):  
Cross Sections ◽  
Test Specimen ◽  
Plastic Material ◽  
Tensile Tests ◽  
Deformation Energy ◽  
Experimental Fact ◽  
Conservation Of Energy ◽  
Software Complex ◽  
The Moment ◽  
Cylindrical Test

In this paper method of numerical computations using explicit scheme, implemented in LSDyna (Ansys) software complex, is introduced and verified. Obtained solution explains experimental fact mentioned by P. Ludwik, that cracking in cylindrical test specimen made of plastic material starts in the middle of the smallest cross-section of the specimen. Introduced method allows verifying law, obtained by N.N. Davidenkov during unique experimental research in which he studied logarithmic strain in specimen’s neck by pickling of cross-sections in this zone. Additionally it is possible to estimate amount of heat, generated during rupture of specimen, using obtained solution and law of conservation of energy. For this purpose tensile tests, in which thermal camera was used for temperature measuring, were conducted. It was shown, that all deformation energy apart from elastic and shape-forming energy in volume element, calculated in the moment before rupture is transformed to heat. Also tensile tests with various rate of loading were conducted for more detailed research of rupture process.

scholarly journals Observable and Unobservable Mechanical Motion

Entropy ◽  
2020 ◽  
Vol 22 (7) ◽  
pp. 737
Author(s):  
J. Gerhard Müller
Keyword(s):  
Information Gain ◽  
Mechanical Energy ◽  
Equations Of Motion ◽  
Minimum Energy ◽  
Radiation Energy ◽  
Mechanical Motion ◽  
Radiation Emission ◽  
Physical Action ◽  
Hamilton's Equations ◽  
Hamilton’S Equations

A thermodynamic approach to mechanical motion is presented, and it is shown that dissipation of energy is the key process through which mechanical motion becomes observable. By studying charged particles moving in conservative central force fields, it is shown that the process of radiation emission can be treated as a frictional process that withdraws mechanical energy from the moving particles and that dissipates the radiation energy in the environment. When the dissipation occurs inside natural (eye) or technical photon detectors, detection events are produced which form observational images of the underlying mechanical motion. As the individual events, in which radiation is emitted and detected, represent pieces of physical action that add onto the physical action associated with the mechanical motion itself, observation appears as a physical overhead that is burdened onto the mechanical motion. We show that such overheads are minimized by particles following Hamilton’s equations of motion. In this way, trajectories with minimum curvature are selected and dissipative processes connected with their observation are minimized. The minimum action principles which lie at the heart of Hamilton’s equations of motion thereby appear as principles of minimum energy dissipation and/or minimum information gain. Whereas these principles dominate the motion of single macroscopic particles, these principles become challenged in microscopic and intensely interacting multi-particle systems such as molecules moving inside macroscopic volumes of gas.

Second Order Poincaré Map to Study Dynamical Behavior of Nonsymmetrical Gyrostat Satellite

2000 ◽  
Author(s):  
K. H. Shirazi ◽  
M. H. Ghaffari-saadat
Keyword(s):  
Phase Space ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Dynamical Behavior ◽  
Two Dimensions ◽  
Second Order ◽  
Poincare Map ◽  
Runge Kutta Method ◽  
Final Space

Abstract The second order poincare’ map is described and used for investigation of the dynamical behavior of a gyrostat satellite. The normalized attitudinal equations of motion for a typical non-symmetric gyrostat satellite are considered. For different sets of initial conditions the equations simulated by Runge-Kutta method. The poincare’ section is used to dimension reduction of system phase space. By this map the dimension reduced from six to five. Using secondary map the dimension of phase space can be reduced to four and considering symmetry of phase space the final space has two dimensions that is presentable at the plane. Bifurcation in the attitudinal behavior can be demonstrated easily by the derived map.

Vibrational/Rotational Energy Levels

1996 ◽  
Author(s):  
Tomas Baer ◽  
William L. Hase
Keyword(s):  
Rotational Motion ◽  
Energy Levels ◽  
Equations Of Motion ◽  
Rotational Energy ◽  
Unimolecular Reaction ◽  
Coordinate Systems ◽  
Electronic Motion ◽  
Hamilton's Equations ◽  
Hamilton’S Equations

The first step in a unimolecular reaction is the excitation of the reactant molecule’s energy levels. Thus, a complete description of the unimolecular reaction requires an understanding of such levels. In this chapter molecular vibrational/rotational levels are considered. The chapter begins with a discussion of the Born-Oppenheimer principle (Eyring, Walter, and Kimball, 1944), which separates electronic motion from vibrational/ rotational motion. This is followed by a discussion of classical molecular Hamiltonians, Hamilton’s equations of motion, and coordinate systems. Hamiltonians for vibrational, rotational, and vibrational/rotational motion are then discussed. The chapter ends with analyses of energy levels for vibrational/rotational motion. The Born-Oppenheimer principle assumes separation of nuclear and electronic motions in a molecule. The justification in this approximation is that motion of the light electrons is much faster than that of the heavier nuclei, so that electronic and nuclear motions are separable.

Hamilton’s Equations & Routhian Reduction

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Angular Momentum ◽  
Phase Space ◽  
Poisson Bracket ◽  
First Integrals ◽  
Late Nineteenth Century ◽  
Algebra Structure ◽  
French Mathematician ◽  
Late Nineteenth ◽  
Hamilton's Equations ◽  
Hamilton’S Equations

In this chapter, the Poisson bracket and angular momentum are investigated and first integrals are used to develop conservation laws as a canonical Noether’s theorem. The Poisson bracket was developed by the French mathematician Poisson in the late nineteenth century and it is a reformulation, or at least a tidying up, of Hamilton’s equations into one neat package. The Poisson bracket of a quantity with the Hamiltonian describes the time evolution of that quantity as one moves along a curve in phase space. The Lie algebra structure of symmetries in mechanics is highlighted using this formulation. The classical propagator is derived using the Poisson bracket.

Sign in / Sign up

Export Citation Format

Share Document