Description
1. (20 points) The Lagrangian for a system can be written as
L = ax˙
2 + b
y˙
x
+ cx˙y˙ + fy2x˙ z˙ + gy˙ − k
√
x
2 + y
2
, (1)
where a, b, c, f, g, k are constants. What is the Hamiltonian? What quantities are conserved?
2. (20 points) For a given Lagrangian
L = ˙q
2
1 +
q˙
2
2
a + bq2
1
+ k1q
2
1 + k2q˙1q˙2, (2)
with a, b, k1, k2 being constants, find the equations of motion in the Hamiltonian formulation.
3. (35 points) A Hamiltonian of one degree of freedom has the form
H =
p
2
2a
− bqpe−αt +
ba
2
q
2
e
−αt(α + be−αt) + kq2
2
, (3)
where a, b, α, k are constants.
a) Find the Lagrangian corresponding to this Hamiltonian.
b) Find an equivalent Lagrangian that is not explicitly dependent on time.
c) What is the Hamiltonian corresponding to the second Lagrangian, and what is the relationship
between the two Hamiltonians?
4. (25 points) a) The Lagrangian for a system with one degree of freedom reads
L =
m
2
( ˙q
2
sin2 ωt + ˙qqω sin 2ωt + q
2ω
2
). (4)
What is the corresponding Hamiltonian? Is it conserved?
b) Introduce a new coordinate defined by Q = q sin ωt. Find the Lagrangian in terms of the new
coordinate and the corresponding Hamiltonian. Is H conserved?
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