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.

scholarly journals The zonal satellite problem - II: Near-escape flow

1998 ◽  
pp. 37-41
Author(s):  
V. Mioc ◽  
M. Stavinschi
Keyword(s):  
Angular Momentum ◽  
Phase Space ◽  
Rotational Symmetry ◽  
Equations Of Motion ◽  
First Integrals ◽  
Local Flow ◽  
Satellite Problem ◽  
Set Up ◽  
Momentum Integral

The study of the zonal satellite problem is continued by tackling the situation r??. New equations of motion (for which the infinite distance is a singularity) and the corresponding first integrals of energy and angular momentum are set up. The infinity singularity is blown up via McGehee-type transformations, and the infinity manifold is pasted on the phase space. The fictitious flow on this manifold is described. Then, resorting to the rotational symmetry of the problem and to the angular momentum integral, the near-escape local flow is depicted. The corresponding phase curves are interpreted as physical motions.

Particle Dynamics

2020 ◽  
pp. 32-49
Author(s):  
M. Suhail Zubairy
Keyword(s):  
Quantum Mechanics ◽  
Angular Momentum ◽  
Particle Dynamics ◽  
Classical Trajectory ◽  
Late Nineteenth Century ◽  
Newtonian Mechanics ◽  
Electric And Magnetic Fields ◽  
Late Nineteenth ◽  
Mass Position ◽  
Dynamics Of Particles

In Newtonian mechanics, a particle is described as an object that is characterized by certain properties. The most important characteristics of a particle are its mass, position, velocity, and acceleration. In this chapter, it is shown how a particle follows a well defined classical trajectory. The main characteristics of the dynamics of particles such as linear and angular momentum, force, energy, moment of inertia, and torque are presented. An understanding of these effects is essential in understanding and appreciating the laws of quantum mechanics. As an example of the Newtonian mechanics, the motion of an electron in electric and magnetic fields experiencing Lorentz force is discussed. This example explains how Thomson discovered the electron in the late nineteenth century.

scholarly journals Variational problem for Hamiltonian system on so(k, m) Lie–Poisson manifold and dynamics of semiclassical spin

2014 ◽  
Vol 29 (10) ◽  
pp. 1450048 ◽  
Author(s):  
A. A. Deriglazov
Keyword(s):  
Angular Momentum ◽  
Phase Space ◽  
Hamiltonian System ◽  
Variational Problem ◽  
Poisson Bracket ◽  
Fiber Bundle ◽  
Mathematical Formulation ◽  
Poisson Manifold ◽  
Spinning Particles ◽  
Hamiltonian Equations

We describe the procedure for obtaining Hamiltonian equations on a manifold with so (k, m) Lie–Poisson bracket from a variational problem. This implies identification of the manifold with base of a properly constructed fiber bundle embedded as a surface into the phase space with canonical Poisson bracket. Our geometric construction underlies the formalism used for construction of spinning particles in [A. A. Deriglazov, Mod. Phys. Lett. A 28, 1250234 (2013); Ann. Phys. 327, 398 (2012); Phys. Lett. A 376, 309 (2012)], and gives precise mathematical formulation of the oldest idea about spin as the "inner angular momentum".

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.

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