Various Notes on Math and Physics
(If you have comments/corrections on these notes, please email me.)
In what sense is
z* independent of
z?
Well, of course it is not independent, but there are some valid tricks that appear to assume this. The tricks work because
z is composed of two independent parts, its real and imaginary components.
z and
z* are linear combinations of these.
For years I did not really understand what was meant by the term "Green's Functions", even though I had been introduced to them in several contexts (electrostatics and electrodynamics, quantum mechanics). This confusion was rather common among grad students. These notes are to help physics students understand the core meaning of "Green's function", without being too mathematical.
In this rather mathematical note I demonstrate the method of multiple time scales and use it to argue the averaging principle for the slow perturbation of an initially integrable system. The notation is that of action-angle variables in classical mechanics. The averaging principle says that under slow perturbation of parameters, the action variable vector
I evolves, but remains close to a variable
J which obeys a simpler system of equations which has half as many variables, because the angle variables are averaged (integrated) over
The averaged system describes the slow evolution of the manifold that is invariant on short time scales.
Such averaging requires one assume that the motion is ergodic. These concepts were related to several different projects I worked on as a Ph.D. student.
This is further discussion about adiabatically perturbed systems, including Hamiltonian fully chaotic systems and quantum wave systems. In particular it demonstrates that the values of the actions
I for an integrable system are indeed adiabatic invariants. Considering the same type of result exists for Hamiltonian fully chaotic systems (only constant of the motion is
H(
p,
q), I supposed the analogous results hold for systems which have 1 <
k <
N constants of motion.