Transcript fulton_p239D
Canonical Transformations and Liouville’s Theorem Daniel Fulton P239D - 12 April, 2011
A Trivial Solution to Hamilton’s Equations Consider: - Hamiltonian is constant of motion, no explicit time dep.
- All coordinates
q i
are cyclic Then solution is trivial…
Motivation for Canonical Transformations - Want coordinates with trivial solution (see previous).
- Question: Given a set of canonical coord.
q
,
p
,
t
and Hamiltonian
H(q
,
p
,
t)
, can we transform to some
new
canonical coord.
Q
,
P
,
t
with a transformed Hamiltonian
K(Q
,
P
,
t)
such that
Q’
s are cyclic, and
K
has no explicit time dependence?
- Look for
equations of tranformation
of the form:
Structure of Canonical Transformations Both sets of coordinates must be canonical, therefore they should both satisfy Hamilton’s principle.
Integrands must be the same within a constant scaling and an additive derivative term… Can always have intermediate transformation…
: Scale Transformations Suppose we just want to change units. The transformed Hamiltonian is then and the integrands are related by…
F
- Generating Function As long as
F
=
F(q
,
p
,
Q
,
P
,
t)
, it won’t change value of the integral, however it does give information about relation between (
q
,
p
) and (
Q
,
P
).
Example: (
F
is given) Since
q i
and
Q i
are independent, each coefficient must be zero separately. This gives 2
n
eqns relating
q
,
p
to
Q
,
P
.
Four Basic Canonical Transformations If we work from eqs of trans back to
F
, might get… QuickTime™ and a decompressor are needed to see this picture.
Note: It is possible to have mixed conditions. e.g.
Example: 1D Harmonic Oscillator (i) Hamiltonian is The Hamiltonian suggests something of the form below, but we need to determine
f(P)
such that the transformation is canonical.
Try a generating function…
Example: 1D Harmonic Oscillator (ii) Immediately, write down the solutions…
Statement of Liouville’s Theorem The state of a system is represented by a single point in phase space.
In terms of large systems, it’s not realistic or practical to predict the dynamics exactly, so instead we use statistical mechanics… … we have an ensemble of points in phase space, representing all possible states of the system, and we derive information by averaging over
all
systems in this ensemble.
Liouville’s Theorem: “The density of systems in the neighborhood of some given system in phase space remains constant in time.”
Proof of Liouville’s Theorem (Goldstein) - Consider infinitesimal volume surrounding a point, bounded by neighboring points.
- Over time, the shape of the volume is distorted as points move around in phase space but… - No point that is inside the volume can move out, because if it did it would have to intersect with another point, at which point they would always be together.
- From a time
t 1
to
t 2
, the movement of the system is simply a canonical transformation generated by the Hamiltonian. - Poincare’s integral invariant indicates that the volume element should not change.
dN
and
dV
are constant, therefore
D
=
dN
/
dV
is constant.