1
CDS 140b Winter 2008 Week 6
Spherical Tippe Top Inversion as a
DissipationInduced Instability
Due: Thursday, Feb 21, 2008
This assignment references the preprint “DissipationInduced Heteroclinic
Orbits in Tippe Tops” which accompanies this homework.
Problem A
[Quadratic Lagrangians]
Let
M
∈
L
(
R
n
,
R
n
) be a symmetric positivedefinite matrix. Consider a me
chanical system with quadratic Lagrangian
L
:
R
2
n
→
R
:
L
(
q
,
˙
q
) = ˙
q
T
M
˙
q
+ ˙
q
T
G
q

q
T
K
q
.
Show that
the only
conservative forces derivable from such a quadratic La
grangian are gyroscopic and potential.
(Recall a gyroscopic force is a force
that can be written as
A
˙
q
where
A
is skewsymmetric and a potential force is
a force that can be written as
B
q
where
B
is symmetric.)
Problem B
[Linear Hamiltonian Systems]
Let
q
(
t
)
∈
R
2
. Consider
A, B
∈
L
(
R
2
,
R
2
) where
A
is skewsymmetric and
B
is symmetric. Verify that the following differential equations are Hamiltonian:
¨
q
+
A
˙
q
+
B
q
= 0.
(*)
Show that if
σ
is an eigenvalue of (*) then so are 1
/σ
, ¯
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 list
 Force, Hamiltonian mechanics, Noether's theorem, Tippe Tops, spherical tippe

Click to edit the document details