12 Class exercise sheet

12
Class exercise sheet
Exercise 12.1: Hamilton formalism
Giving the following Lagrangian
2
L0 = ab
ȧ
aȧḃ aä
− 3 + 2 − ω2
2
b
b
b
!
(1)
Where ω is constant.
1. Find and equivalent Lagrangian L1 which does not contain a second derivative of a(t)
and b(t).
2. Calculate the generalized momenta and build the corresponding Hamiltonian H1 .
3. Prove that Pb is a constant of motion (what is it?) which results in the zeroing of the
Hamiltonian.
4. Solve a(t) analytically.
Exercise 12.2: Stick on a hill
A uniform solid rod, with length L and mass m, is moving with out friction on half a
sphere with radius R. In equilibrium the rod is parallel to the ground. There is a uniform
gravitational field g present.
1. Build The Lagrangian L(θ, θ̇) which describes the problem.
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2. Prove that there are small oscillations.
3. Calculate the frequency of the small oscillations.
Exercise 12.3: Coordinate transformation
For a given Hamiltonian
H(p, q, t) =
pq 3
2t
(2)
and coordinate transformation
3
1
q2
P = pq 1 + cte
a
Q = 2 + b log tpq 2
q
(3)
(4)
1. Find the parameters a, b and c which will make the transformation canonical.
2. Find the appropriate creating function.
3. Calculate the transformed Hamiltonian.
Exercise 12.4: Scattering from a lens
Particles with mass m are scattered on ”lenses” according to the laws of geometric optics.
The radius (hight) of the lens is R and its focal length is f .
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1. What is the differential cross section for scattering (assume the lens is stationary)?
2. Assume the lenses has a mass that is equal to the particles mass (the lens can move
but cannot rotate). Find the differential cross section for the particles and the lenses.
3. Find the total cross section for scattering.
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