Euler Angles
Euler angles are three angles that provide coordinates on SO (3). Denote by R1 ( )
and R3 ( ) rotations about the x and z axes, respectively, by an angle . That is
1
0
0 cos
R 1 ( ) =
0 sin
0
sin
cos
cos
sin
R 3 ( ) =
0
sin
cos
0
A Degenerate Riemannian Manifold
Dene the manifold
M = (0, 1)
(x, red)
1x<2
(y, yellow)
1y<2
(so that M contains two distinct copies of the interval [1, 2) together with one copy of the interval (0, 1)
with an atlas consisting of the two charts (U, X ) a
The Principle of Least Action
1
We have seen in class that Newtons law, 2 mx(t) = U x(t) for the motion of a particle in a potential
well is equivalent to the stationarity of the action
t2
S[t1 ,t2 ] x(t) =
L x(t), x(t) dt
t1
1
with Lagrangian L(x, v ) =
Phase Plane Analysis of the Undamped Pendulum
Phase plane analysis is a commonly used technique for determining the qualitative
behaviour of solutions of systems of ODEs in low dimensions. To illustrate the technique,
2
we consider the equation d 2 + g si
Torus Geodesics
Let 0 < < R be constants. The surface in IR3 whose equation in cylindrical coordinates is
(r R )2 + z 2 = 2
is a torus, which we shall call M . Use as coordinates on M two angles and determined
z
by
x = (R + cos ) cos
y = (R + cos ) sin
A Summary of Rigid Body Formulae
By denition, a rigid body is a family of N particles, with the mass of particle number i denoted mi and
with the position of particle number i at time t denoted x(i) (t), together with suciently many constraints of
the for
Duhamels Formula
Theorem (Duhamel) Let Ai,j (t) 1i,j n be a matrixvalued function of t IR that is
C in the sense that each matrix element Ai,j (t) is C . Then
1
d A(t)
e
dt
esA(t) A (t)e(1s)A(t) ds
=
0
Proof: We rst use Taylors formula with remainder, app
Examples of Manifolds
Example 1 (Open Subset of IRn ) Any open subset, O , of IRn is a manifold of dimension
n. One possible atlas is A = (O , id ) , where id is the identity map. That is, id (x) = x.
Of course one possible choice of O is IRn itself.
Exam
Euler Wobble
Consider a rigid body with inertia tensor
I1
I=0
0
0
0
I3
0
I2
0
with I1 = 1, I2 = 1 , I3 =
2
1
3
Under free motion, the angular momentum M , expressed in body coordinates, obeys Eulers equations (see
A Summary of Rigid Body Formulae)
M1 = M2
First Order Initial Value Problems
A rst order initial value problem is the problem of nding a function x(t) which
satises the conditions
x = F (x, t)
x(t0 ) = 0
(1)
where the initial time, t0 , is a given real number, the initial position, 0 IRd , is a g