6
Parametric
oscillator
6.1
Mathieu
equation
We
now
study
a
different
kind
of
forced
pendulum.
Specifically,
imagine
subjecting
the
pivot
of
a
simple
frictionless
pendulum
to
an
alternating
vertical
motion:
rigid rod
This
is
called
a
“parametric
pendulum,”
because
the
motion
depends
on
a
timedependent
parameter.
Consider
the
parametric
forcing
to
be
a
timedependent
gravitational
field:
g
(
t
) =
g
0
+
λ
(
t
)
The
linearized
equation
of
motion
is
then
(in
the
undamped
case)
d
2
β
g
(
t
)
+
β
= 0
.
d
t
2
l
The
timedependence
of
g
(
t
)
makes
the
equation
hard
to
solve.
We
know,
however,
that
the
rest
state
β
=
β
˙
= 0
is
a
solution.
But
is
the
rest
state
stable?
We
investigate
the
stability
of
the
rest
state
for
a
special
case:
g
(
t
)
periodic
and
sinusoidal:
g
(
t
) =
g
0
+
g
1
cos(2
γt
)
Substituting
into
the
equation
of
motion
then
gives
d
2
β
+
γ
0
2
[1
+
h
cos(2
γt
)]
β
=
0
(14)
d
t
2
38
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where
γ
2
=
g
0
/l
and
h
=
g
1
/g
0
�
0
.
0
Equation
(14)
is
called
the
Mathieu
equation
.
The
excitation
(forcing)
term
has
amplitude
h
and
period
2
α
α
T
exc
=
=
2
γ
γ
On
the
other
hand,
the
natural,
unexcited
period
of
the
pendulum
is
2
α
T
nat
=
γ
0
We
wish
to
characterize
the
stability
of
the
rest
state.
Our
previous
methods
are
unapplicable,
however,
because
of
the
timedependent
parametric
forcing.
We
pause,
therefore,
to
consider
the
theory
of
linear
ODE’s
with
periodic
coeﬃcients,
known
as
Floquet
theory
.
6.2
Elements
of
Floquet
Theory
Reference:
Bender
and
Orszag,
p.
560
.
We
consider
the
general
case
of
a
secondorder
linear
ODE
with
periodic
coeﬃcients.
We
seek
to
determine
the
conditions
for
stability.
We
begin
with
two
observations:
1.
If
the
coeﬃcients
are
periodic
with
period
T
,
then
if
β
(
t
)
is
a
solution,
so
is
β
(
t
+
T
).
2.
Any
solution
β
(
t
)
is
a
linear
combination
of
two
linearly
independent
solutions
β
1
(
t
)
and
β
2
(
t
):
β
(
t
) =
Aβ
1
(
t
) +
Bβ
2
(
t
)
(15)
where
A
and
B
come
from
initial
conditions.
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 Fall '06
 DanielRothman
 Cos, Mathieu, rest state, Mathieu Equation, Floquet Theory

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