Since this result [i.e., equation (22) above] is non-zero, it follows from the dynamical evolution theorem [i.e., equation (16)] of Quantum Mechanics (QM) that none of the angular momentum components [L.sub.i] are--for the Dirac particle-going to be constants of motion
. This result obviously bothered the great and agile mind of Paul Dirac.
A way to perturb a three-dimensional Hamilton-Poisson systems consists in alteration of its constants of motion. This method leads to integrable deformations of the initial system.
In , observing that the constants of motion of the Euler top determine its equations, integrable deformations of the Euler top were given.
Paramount among them are the so called constants of motion
, they say, which are physical quantities that are conserved along the solution trajectories of a large set of differential systems named conservation problems.
In the second step, these effective constants of motion
are used to predict positions of particles swept across the interference field.
For atomic and molecular systems, his calculation is based on the fact that the wave function and geometric elements of the wave described by the Schrodinger equation are mathematical objects that describe the same physical system and depend on its constants of motion
. The method is as accurate as the Hartee-Fock method, he says.
[P.sub.t] and [P.sub.[phi]] are constants of motion
. In fact, they correspond to the test particle's energy per unit mass E and the angular momentum parallel to the symmetry axis per unit mass L, respectively.
The first four chapters contain extended reviews of the main results on dynamical symmetries and corresponding constants of motion
in non-relativistic and relativistic classical mechanics and non-relativistic quantum mechanics.
Besides, we show that the Lax integrability is preserved in the peakon case; some constants of motion
for the peakon dynamical systems of N-order CH equations are obtained.
The constants of motion
E and L are found from the boundary conditions of the system, i.e.
In Section 2, we construct integrable deformations of an integrable version of the Rikitake system by modifying its constants of motion
. In Section 3, we obtain two Hamilton-Poisson realizations of the new system.
In other words, the issue is searching for some general method by which selecting such constants of motion
related to the emergence of classical trajectories without arbitrarily choosing regions of the phase-space where momenta are conserved.