Mathematical Physics PHZ 3113
Levi-Cevita Tensor 1
(January 11, 2013)
Group #
Participating students (print):
1. Use binary numbers 0, 1 and write down
the numbers 0 to 3. Add one more column in which
Solution for assignment 14: Problems 1 and 2, Landau-Lifshitz p.24.
1. The Lagrangians are
1
m
L=
2
dr
dt
1
=
m
2
2
dr
dt
L
U,
2
U
After substituting t = t m /m in L the Lagrangians agree and the same
Solution for assignment 9a:
For a bilinear kinetic Energy we nd by dierentiation and carrying the sum over i
out
qi
i
T
=
qi
qi
i
=
qi
ajk qj qk =
j,k
qi (ij qk + qj ik )
i,j,k
ajk (qj qk + qj qk )
ADVANCED DYNAMICS PHY 4241/5227
HOME AND CLASS WORK SET 2
Solution for assignment 8: Double Pendulum.
Derivation of the Lagrangian:
x1
y1
x2
y2
=
=
=
=
l sin
l cos
l sin + l sin
l cos + l cos
1
1
HOME AND CLASS WORK SET 1
Solution for assignment 5.
1. The Lagrangian is
12 12
x x.
2
2
Thus, the Euler-Lagrange equation becomes
L=
0=
L d L
= x x
x dt x
and the general solution is given by x(t) =
Solution for assignment 15:
The total Energy the is
E=
1 2 1k
x+
.
2
2 x2
(1)
1k
.
2 x2
0
(2)
1
1
2
2
x0 x
(3)
The initial values imply
E=
Therefore,
x2 =
k
m
x=
x2 x2
0
.
x
k
m x2
0
(4)
Separation of
Solution for assignment 19: Eective Potential.
Ue (r) =
M2
+
r
2mr2
with M angular momentum and m reduced mass.
(A) Solve now for r = r0 :
Ue (r) = +
2M 2
M2
M2
r0 =
.
= 0 r0 =
r
2mr3
m
m
(B) The mi
Solution for assignment 36:
With
H=
qj
j
L
L
qj
we nd
dH =
d
j
L
qj
qj +
L
L
L
dqj
dqj
dqj
qj
qj
qj
.
The two central terms cancel out. Using Euler-Lagrange and the denition of the
generalized
Solution for assignment 37
Poisson Brackets
(21a) Let us consider functions g = g (qk , pk , t) and h = h(qk , pk , t). The Poisson
bracket is dened by
[g, h]
def
=
k
g h
h g
qk pk qk pk
.
The proper
Solution for assignment 38
Liouvilles Theorem
We consider motion of point particles with n degrees of freedom in phase space,
which is described by a Hamiltonian
H (q1 , . . . , qn ; p1 , . . . , pn )
Solution for assignment 14: Problems 1 and 2, Landau-Lifshitz p.24.
1. The Lagrangians are
1
m
L=
2
dr
dt
1
=
m
2
2
dr
dt
L
U,
2
U
After substituting t = t m /m in L the Lagrangians agree and the same
SOLUTIONS FINAL PHY 4936 Fall 2011
PROBLEM 1
(1) The momentum is
L
= mx
x
(2) Eliminating x in favor of p we obtain the Hamiltonian
p=
H =T +V =
p2
1
+ kx2
2m 2
(3) Hamiltons equations are
H
p
H
=
= x
2
A
B
1.5
1
v [m/s]
0.5
0
-0.5
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
[radiant]
PHY 4936 Solution 32
The energy is
E=
1
3 8 cos + M gR 1 4 cos .
M R2 2
2
2 9
9
Solving for v 2 = R2 2 gives
v 2 = R
ADVANCED DYNAMICS PHY 4241/5227
HOME AND CLASS WORK SET 6
Solution for assignment 26:
Double pendulum solution and plot (continuation of 25).
Let us take minors with respect to the rst row of the dete
FINAL PHY 4936 (December 15, 2011)
PROBLEM 1 (25 points)
The Lagrangian of the 1D harmonic oscillator is
L=
1
1
m x2 k x 2 .
2
2
1. Use the denition of the generalized momentum to nd the momentum p.
2
Mathematical Physics PHZ 3113
Curl; Vector Integration Homework
(February 7, 2013)
The dual of the Euclidean electromagnetic
eld tensor is dened by
1
F
(1)
F
=
2
where the indices run from 1 to 4, t
Mathematical Physics * PHZ 3113
Einstein Convention Homework 1
1. Use the 3D identity
Eijkeilm I jz5km 5jm5kz (1)
L_l
to calculate
(ma-(aw) (2)
Where a and [3 are unit vectors. Eliminate
all 51, - b i
Mathematical Physics PHZ 3113
Levi-Cevita Tensor Homework 1
(January 23, 2013)
1. Use the 3D identity
3
i=1
ijk ilm = jl km jmkl
(1)
to calculate
( (
a b) a b)
(2)
where a and are unit vectors. Elimi
Mathematical Physics PHZ 3113
Vectors 1 (Classwork January 7, 2013)
Group #
Participating students (print):
In the following i = 1, . . . , n, j = 1, . . . , n.
1. Let xi and xj be Cartesian unit vect
Mathematical Physics PHZ 3113
Levi-Cevita Tensor Homework 2
(January 25, 2013)
1. Use the identity
3
ijk ilm = jl km jmkl
i=1
(1)
to eliminate the vector products from the
expression
a b c
(2)
Solutio
Mathematical Physics PHZ 3113
Levi-Cevita Tensor 2 Applications
(January 14, 2013)
Group #
Participating students (print):
1. Write down the values of the cyclic permutations of 123 and then of 213. D
Mathematical Physics PHZ 3113
Vectors 2 (Classwork January 9, 2013)
Group #
Participating students (print):
1. Write down the commutative law of
vector addition
a + b = b + a.
(1)
2. Write down the as
Mathematical Physics PHZ 3113
Vector Integration Homework
(February 20, 2013)
Integrate (explicitly!) the force
F = x2 x1 + x1 x2
(1)
in anti-clockwise direction along the boundary of an equilateral t