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Transcript
Action-Angle Variables
based on FW-36
Hamilton-Jacobi theory can be used to calculate frequencies of
various motions without completely solving the problem if the
motion of the system is both separable and periodic.
libration
e.g. harmonic oscillator
rotation
e.g. pendulum going over the top
192
Let’s define the action variables of the system:
libration
se space
a
h
p
n
i
od
area
one peri
r
e
v
o
n
take
rotation
they are constants of the motion:
we assume the relations are invertible
we already know one constant:
193
we can use action variables as the integration constants 's in S:
just a name
This generates following transformation:
S satisfies Hamilton-Jacobi equation:
194
Let’s define the angle variables of the system:
“frequency”, a constant of the motion
the angle variables increase linearly with time
We will need the differential of the angle variables:
depends on coordinates
and constants
definition
separability
195
If the motion of the system is periodic, then in one period of the entire system
all the degrees of freedom execute some integral number of individual periods.
The corresponding change in the angle variables:
but also
indeed it is frequency
fundamental frequencies of the system:
follow directly from the Hamiltonian
expressed as a function of the action variables
196
Example (harmonic oscillator in two dimensions with different
spring constants):
total energy
Hamilton-Jacobi equation for Hamilton’s characteristic function
separability
197
Example (continued):
The action variables are then:
can be inverted
and we get the hamiltonian as a
function of the action variables:
and the fundamental frequencies of the system:
198
Poisson brackets
based on FW-37
Let’s define the Poisson bracket of two functions, F and G:
it is obviously antisymmetric
Poisson bracket of a function F and the Hamiltonian:
Hamilton’s equations:
Poisson bracket formulation of classical mechanics!
199
PB is equivalent formulations of classical mechanics:
Hamilton’s equations:
applied to q and p
PB of coordinates and canonical momenta:
Canonical transformation to a new set of
coordinates and momenta, Q and P, preserves
Hamilton equations and thus Poisson brackets.
A canonical transformation can be defined as one that
preserves Poisson-bracket description of mechanics.
200
Transition to quantum mechanics:
Define the commutator of two quantities:
Canonical quantization prescription:
non-commuting operators acting on a Hilbert space
canonical commutation relations
Equation of motion:
Heisenberg operator equation of motion
201