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 you substitute 0 1, 1
2 and one last 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 paths are
obtained for
t /t = m /m .
2. The Lagrangian
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 ) = 2 T .
j,k
Solution for assignment 9b: Motivation of g
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
L=
m l x1 2 + y1 2 + m l x2 2 + y2 2 m g l (z1 + z2 ) +
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) = A sin(t) + B cos(t). The integration
constants are dete
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 variables gives
k
m x2
0
t
dt
0
=
k
t=
m x2
0
x
x
dx
x
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 minimum value of the eective potential:
min
Ue = Ue (r0 )
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 momentum, we have
d L
L
=
= pj
qj
dt qj
and, therefore,
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 properties of the assignment are shown in the following:
1. R
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 ) .
Let (q1 , . . . , qn ; p1 , . . . , pn ; t) be the d
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 paths are
obtained for
t /t = m /m .
2. The Lagrangian
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 and
= kx = p
p
m
x
(4) Newtons force law follows
kx =
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 = R2 2 =
2 [E M gR (1 4 cos /(9 )]
M [3/2 8 cos /(9 )]
A.
ADVANCED DYNAMICS PHY 4936
Solution 28:
x + 1 x = 0 ,
y + 1 y = 0 ,
Solution with x0 = y0 = 0;
x(t) = A1 sin(1 t)
y (t) = A2 sin(2 t)
Now, x(t1 ) = y (t1 ) = 0 for some future time t1 > 0 implies
1 t1 = n ,
2 t1 = m
(1)
with n, m integers 1. With n and m
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 determinant. For the +
frequency the ratio of the two minor
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. Write down the Hamiltonian H (p, x).
3. Write down Ha
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, the Einstein convention is used and F is an antisymmetri
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 in favor of cos 6.
Solution: Le
iii A 91
N2 and b: b2 .
Mathematical Physics PHZ 3113
Homework 7 (February 23, 2013)
1. Continuation of Midterm 1, Problem 5:
Consider 1 = 2 = and nd the angle
0 > 0, so that for 0 < < 0
T1 = T2 > F2
(1)
holds (6 points).
Solution: Let T 1 = |T 1|, T 2 = |T 2| and
F = |F |. As 1
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. Eliminate
b
b
all a in favor of cos .
Solution: Let
1
b
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 vectors.
It holds the relation
xi xj = ij .
(1)
Name the
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)
Solution (compare book p.33):
a b c
xa
bc
m ijk i j klm l m
i
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. Do
you get all 3D values this way? Which
are positive an
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 associative law of vector addition
a + b + c = a + b + c
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 triangle in the x1 x2
plane. The triangle has a side fro