PHYS 3202 Classical Mechanics II  Homework #6
Due at 12:05pm, Friday March 11, 2011 in the class
In homework, quiz, and final exam, please show intermediate steps of
your calculations. Whenever appropriate, you may draw diagram(s).
Problem 1
(10 points): Find the symplectic area
A
enclosed by a
clockwise
loop
in the 2 dimensional phase plane:
q
(
θ
) =
c
1
cos
θ,
p
(
θ
) =

c
2
sin
θ,
where
c
1
and
c
2
are positive constants, and the parameter
θ
increases from 0 to 2
π
.
If we parameterize the loop as
q
=
c
1
cos(
λ
2
)
,
p
=

c
2
sin(
λ
2
)
,
where the parameter
λ
increases from 0 to
√
2
π
, what is the symplectic area enclosed
by the loop?
If every point on the plane moves according to Hamilton’s equations, governed
by a particular Hamiltonian function
H
(
q, p, t
), the above loop will in general be
transformed into different shapes at different locations in the phase plane at later
times.
If
∂H
∂t
>
0 for all values of (
q, p, t
), how does the symplectic area of the
transformed loop depend on the time
t
? Is
H
conserved?
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 Spring '11
 Tan
 mechanics, Work, Hamiltonian mechanics, initial conditions, Symplectic manifold, symplectic, symplectic matrix, symplectic area

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