March 12, 2010
Cornell University, Department of Physics
P3318, Analytical Mechanics, HW#7, due: 3/19/10, 2:30PM (at section)
Question 1
: Another 2
×
2 system
Consider a particle of mass
m
moving in two dimensions such that
V
(
x
1
, x
2
) =
V
0
b
cosh(
ax
1
+
bx
2
) +
cx
2
1
B
.
(1)
1. Expand the potential assuming small oscillations. Write the Lagrangian in a matrix
notation as
L
=
mδ
ij
v
i
v
j
2
−
k
ij
x
i
x
j
2
i, j
= 1
,
2
.
(2)
and write
k
ij
explicitly.
2. Diagonalize
k
ij
and Fnd the normal modes (that is, the mixing angle) and their fre
quencies. There is no need to simplify the expressions.
3. Write the solution, that is,
x
1
(
t
) and
x
2
(
t
), as a function of
ω
1
,
2
, the mixing angle
θ
and the initial conditions which we take to be
x
1
(
t
= 0) =
A
1
,
x
2
(
t
= 0) =
A
2
,
˙
x
1
(
t
= 0) = ˙
x
2
(
t
= 0) = 0
.
(3)
4. When is the small oscillation approximation valid? Write your answer as a condition
of
A
1
,
A
2
and the Lagrangian parameters, or, equivalently, the normal frequencies.
5. Consider the case
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 Spring '08
 FLANAGAN
 Mass, Cornell University, Normal mode, Diagonalize kij, small oscillation approximation

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