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Homework assignment 2
*
Exercise 2.30.
Let
v
∈
R
3
and
v
6
= 0 and consider the linear ODE on
R
3
:
˙
x
=
v
×
x,
where
×
denotes cross product.
Show that the solutions of this ODE are rigid rotations of the initial vector around the direction
of the vector
v
.
Writing the ODE as:
˙
x
=
Sx,
show that
S
=

S
T
(that is,
S
is skewsymmetric). Show that the Fow
φ
t
(
x
) = e
tS
x
forms a
group of orthogonal transformations.
Prove that every solution is periodic and determine the period in terms of
v
.
Solution.
The main idea is to think geometrically about this problem, in particular about the
geometric interpretation of the cross product of two vectors. Since
v
×
v
= 0, it follows that
v/

v

is a unit eigenvector of the matrix
S
, corresponding to the eigenvalue 0. Choose two orthonormal
vectors
v
⊥
1
and
v
⊥
2
in the orthogonal complement of the linear space spanned by
v
, and such that
v/

v

,v
⊥
1
,v
⊥
2
(in that order) form a right hand orthonormal basis of
R
3
(just like the standard basis
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 Summer '06
 DeLeenheer

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