MTH-696A: TOPICS IN GEOMETRIC MECHANICS ASSIGNMENT 3
DR. LEO BUTLER
A.
Let M be the conguration space of the planar double pendulum. See gure 1.
(a) The conguration space of the planar double pendulum is the set of pairs of angles (x, y ) mod 2 .
What is
MTH-696A: TOPICS IN GEOMETRIC MECHANICS ASSIGNMENT 2
DR. LEO BUTLER
A.
Let M be the conguration space of the spherical pendulum.
(a) Let c : [0, 1] M be a smooth curve. Show directly that the work done by the sti rod along
this curve is zero.
Solution.
Th
MTH-696A: TOPICS IN GEOMETRIC MECHANICS ASSIGNMENT 1
DR. LEO BUTLER
A. Let
: R3 R3 R3
be the vector (=cross) product. Dene the following operator
x (u, v ) := x, uv ,
xR
for
3
3
3
(1)
u, v Tx R R . Let e1 , e2 , e3 be the standard basis of R
x = x1 e1 +
MTH-696A: TOPICS IN GEOMETRIC MECHANICS ASSIGNMENT 7
DR. LEO BUTLER
A.
Let
M
be a smooth manifold and
L : TM R
be a smooth, strictly convex Lagrangian. Show that
(a) the energy
E (x, v ) = v ,
L
L(x, v )
v
is constant along solutions to the Euler-Lagrang
MTH-696A: TOPICS IN GEOMETRIC MECHANICS ASSIGNMENT 6
DR. LEO BUTLER
A.
Let L : T T2 R be the kinetic energy of the double pendulum (see assignment 3).
(a) Compute the Euler-Lagrange equations
L
d L
=
,
dt xi
xi
x1 = x, x2 = y,
for this Lagrangian.
(b) Int
MTH-696A: TOPICS IN GEOMETRIC MECHANICS ASSIGNMENT 5
DR. LEO BUTLER
A.
Let N be the Lie group
1
N = g = 0
0
x
1
0
z
y s.t. x, y, z R
1
with the standard matrix multiplication. Let n = T1 N be the Lie algebra of N .
(a) Compute the exponential map exp : T1
MTH-696A: TOPICS IN GEOMETRIC MECHANICS ASSIGNMENT 4
DR. LEO BUTLER
A.
Let G = O(n; R) and let g = o(n; R) be the Lie algebra of G.
(a) Show that g can be identied with the skew-symmetric real n n matrices with Lie bracket
[x, y ] = xy yx.
(b) Show that t
MTH-696A: TOPICS IN GEOMETRIC MECHANICS ASSIGNMENT 3
DR. LEO BUTLER
A.
Let M be the conguration space of the planar double pendulum. See gure 1.
(a) The conguration space of the planar double pendulum is the set of pairs of angles (x, y ) mod 2 .
What is
MTH-696A: TOPICS IN GEOMETRIC MECHANICS ASSIGNMENT 2
DR. LEO BUTLER
A.
Let M be the conguration space of the spherical pendulum.
(a) Let c : [0, 1] M be a smooth curve. Show directly that the work done by the sti rod along
this curve is zero.
(b) Determin
MTH-696A: TOPICS IN GEOMETRIC MECHANICS ASSIGNMENT 1
DR. LEO BUTLER
A.
Let : R3 R3 R3 be the vector (=cross) product. Dene the following operator
(1)
for x R and u, v Tx R R . Let e1 , e2 , e3 be the standard basis of R and let xi be coordinates
induced