HW6 - PHYS 3202 Classical Mechanics II Homework#6 Due at...

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?