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Unformatted text preview: Physics
601
Homework
1­­­Due
Friday,
September
10
 
 Hint:

For
some
of
these
problems

it
will
be
helpful
to
use
Mathematica
or
some
 other
symbolic
manipulation
program.

If
you
make
use
of
such
a
program
please
 include
the
output
with
your
homework
solutions.
 
 Goldstein

2.2,

2.4,

2.12
 
 In
addition:
 
 1. Consider
a
particle
that
is
constrained
to
move
in
1
dimension
and
is
 confined
to
a
1‐dimensional
box
of
length
L.

The
particle
bounces
elastically
 back
and
forth
between
the
walls
of
the
box.


Thus
at
the
wall
there
is
an
 ˙ ˙ impulsive
force
which
flips
the
velocity
( x → − x 

)
conserving
energy.

This
 problem
explores
what
happens
if
at
time
t=0
the
particle
has
velocity
 v 0 
and
 one
of
the
walls
is
moved
slowly
(either
inward
or
outward)
so
that
size
of
 the
box
is
now
a
function
of
time
L(t).

Since
the
wall
is
moving
it
can
add
or
 € remove
energy
from
the
system.

The
goal
is
to
find
an
adiabatic
invariant
 € relating
the
energy
to
L.
 ˙ ˙ ˙ a. 

Show
that
when
the
particle
hits
the
moving
wall
 x → − x + 2 L .

(Hint,
 what
does
the
process
look
like
from
wall’s
point
of
view.)
 b. Quantify
what
is
meant
by
“slowly”
in
terms
of
the
parameters
of
the
 problem.
 € c. Show
that
 L2 E 
is
an
adiabatic
invariant.
 d. From
your
knowledge
of
the
1‐dimenisonal
particle
in
the
box
in
 quantum
mechanics,
explain
why
this
result
is
expected.
 


 € 2. Consider
the
Lagrangian
for
a
simple
1‐dimesnional
harmonic
oscillator:
 2 ˙ ˙ L( x, x ) = 1 m x 2 − 1 mω 0 x 2 where
 ω 0 
is
a
parameter.


 2 2 
 a. Find
the
equation
of
motion
from
Lagrange’s
equations.
 
 € € Consider
a
change
of
variables
to
the
generalized
coordinate
 q = sinh−1 ( x ) 
 
 b. 
Using
the
result
of
a.
,

find
the
equation
of
motion
for
q.
 ˙ c. Find
the
Lagrangian
for
q
(i.e.
find
 L(q, q) ).
 € ˙ d. Find
the
Lagrange’s
equation
of
motion
for
 L(q, q) .
 e. How
do
the
results
of
b.
and
d.
compare?


Why
is
this
expected?
 
 € 
 € 3. Again
consider
a
simple
1‐dimensional
harmonic
oscillator
with
 2 ˙ ˙ L( x, x ) = 1 m x 2 − 1 mω 0 x 2 .



Consider
a
family
of
trajectories
with
 x ( t ) 
subject
 2 2 to
the
boundary
conditions
 x (0) = 0, x (T ) = l .


Suppose
further
that
the
 family
of
trajectories
includes
the
solution
of
the
exact
equations
of
motion.


 € € € An
example
of
such
a
family
is
the
set
of
function
 x ( t,ω ) = l sin(ω t ) 
where
 ω 
 sin(ω T ) is
a
parameter.


 
 a. Verify
that
this
family
satisfies
the
boundary
conditions.
 € € b. Show
that
for

 ω = ω 0 
this

trajectory
correspond
to
the
solution
of
 the
full
equations
of
motion.
 c. Calculate
the
action
as
a
function
of
 ω :
i.e.
find
 S (ω ).
 € d. Show
explicitly
by
calsulationg
the
action
that
 dS(ω ) = 0 
at
 ω = ω 0 .

 dω Explain
why
this
is
expected
 € € 
 4. Problem
2
considered
a
family
of
trajectories
containing
the
exact
solution
of
 € € the
equations
of
motion.

Suppose
one
has
a
when
a
family
of
trajectories
 which
does
not
contain
the
exact
solution.

The
variational
principal
canstill
 be
of
use
in
finding
approximate
solutions.

Consider
the
harmonic
oscillator

 of

problem
2
with
the
same
boundary
conditions.

Consider
the
family
of
 t3 t t trajectories
given
by
 x ( t, c ) = l + c l 3 − 
where
c
is
a
parameter.


 T T T 
 a. Verify
that
this
family
satisfies
the
boundary
conditions.


 b. Calculate
the
action
as
a
function
of
c:
i.e.
find
 S (c ).
 € c. Minimize
 S (c ) 
to
find
the
“best”
approximation
to
the
full
solution
of
 the
form
considered.
 d. Plot
the
exact
solution
and
the
approximate
solutions
for
three
cases:
 € i) Tω 0 = 1 ii) Tω 0 = 2 iii) Tω 0 = 3 .

Qualitatively
discuss
the
difference
 € in
these
cases
and
why
these
differences
make
sense.


 
 
 
 
 € ...
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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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