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Unformatted text preview: Physics
601
Homework
4­­­Due
Friday
October
1
 
 
 Goldstein:
9.7,
9.25
 
 In
addition:
 
 1. This
problem
uses
the
result
in
problem
2
of
homework
3
in
the
context
of
a
6‐ dimension
phase
space
associated
with
a
single
particle
in
3
dimensions.



The
 canonical
variables
are

 x, y, z, px , py , pz 
.


In
this
problem
various
transformations
 with
clear
physical
meanings
are
proposed.


State
the
physical
meaning,
show
that
 they
are
canonical
and
find
the
generator
of
the
transformation

(as
defined
in
in
 ∂ξ (η;ε) € = [ξ , g]PB given
 problem
2
of
homework
3)
and
show
it
satisfies
the
relation
 ∂ε 
 in
problem
2
of
homework
3.
In
all
of
these
ε
is
a
constant.
 
 X = x +ε X = x cos(ε) + y sin(ε) X=x X = (1 + ε) x € Y=y Y = − x sin(ε) + y cos(ε) Y=y Y = (1 + ε) y Z=z Px = px Py = py Pz = pz Z=z Px = px cos(ε) + py sin(ε) Py = − px sin(ε) + py cos(ε) Pz = pz Z=z Px = px + ε Py = py Pz = pz Z = (1 + ε) z Px = (1 + ε)−1 px 
 Py = (1 + ε)−1 py Pz = (1 + ε)−1 pz a. b. c. d 
 € 
 
 
 € € € € 2. Show
that
if
a
one‐parameter
family
of
time‐independent
canonical
 transformations
leaves
the
Hamiltonian
invariant
and
its
generator
has
no
explicit
 time
dependence
then
its
generator
is
conserved.

That
is
given
a
canonical
 transformation
 ξ (η;ε) 
with
 H (ξ (η;ε)) = H (η) 
and
with
g
as
a
generator‐‐‐
 ∂ξ (η;ε) = [ξ , g]PB ‐‐‐‐then
g
is
conserved.
 ∂ε 
 € € 3. In
problem
1
of
HW
2
we
showed
an
explicit
example
of
a
conserved
quantity
 which
was
not
associated
with
a
point
transformation.


Use
the
result
shown
above
 in
problem
2
to
show
that
the
conserved
quantity
Δ
is
associated
with
a
family
of

 canonical
transformations
by
explicitly
constructing
the
family
of
transformations.
 4. Consider
the
canonical
transformation
for
a
system
with
one
degree
of
freedom
 P 2t generated
by:
 F 2 (q, P, t ) = (q + 1 gt 2 )( P " m g t ) " .
 2 2m a. Find
the
explicit
form
for
the
canonical
transform.
 b. Verify
that
it
satisfies
the
canonical
Poisson
bracket
relations.
 
 ! For
the
remainder
of
this
problem
consider
particle
moving
in
one
dimension
in
a
 ˙ 2˙ constant
gravitational
field
 L(q, q) = 1 mq 2 − mgq .
 dQ dA ∂A = 0 
and
 = [ A, H ]PB + c. Use
the
fact
that
for
any
A,
 to
show
that

 dt ∂t dt dP = 0 .


 € dt d. Part
c.
implies
that
K
the
Hamiltonian
associated
with
Q
,
P
must
be
zero
up
 € ! to
a
possibly
time
dependent
constant
independent
of
Q
,
P.

Show
from
the
 "F 2 (q, P, t ) explicit
form
of
 
that
this
is
in
fact
the
case.
 ! "t e. What
is
the
physical
interpretation
of
Q
,
P.
 f. 
Show
that
 F 2 (q, P, t ) 
satisfies
a
Hamilton‐Jacobi
equation:
 ∂F 2 ∂F 2 H q, ! + = 0 .
 ∂q ∂t g. Introduce
the
function

 f (Q, P, t ) = F 2 (q(Q, P, t ), P, t ) .

Show
that
 € ∂f (Q, P, t ) ˙ = L(q(Q, P, t ) , q(Q, P, t )) .
 ∂t € 
 € ∂S (q, P ) = 0 where
 5. In
class
we
derived
the
Hamilton
Jacobi
equation
 H q, ∇S (q, P ) + ∂t € 
 P 
is
a
constant
of
the
motion.


Derive
an

analogus
expression
 Q ˜ ˜ (Q, p), p) + ∂S (Q, p) = 0 where

 Q 
is
a
constant
of
the
motion
and
the
tilde
is
 H −∇ p S ∂t € ˜ 
to
distinguish
it
from
S.
 
 on
 S € 
 € ( ( € € € ) ) ...
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This note was uploaded on 12/29/2011 for the course PHYSICS 601 taught by Professor Hassam during the Fall '11 term at Maryland.

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