Department of Chemical Engineering
University of California, Santa Barbara
ChE 210C
Spring 2008
Instructor: Glenn Fredrickson
Homework #1
Homework Due: Friday, April 18, 2008
1. Show that the total linear momentum is conserved for a system of
N
classical
particles with an interaction potential
U
that only depends on the relative
distances between particles.
2. Show that the Liouville operator as deﬁned in class for a single particle conﬁned
to move in onedimension has an adjoint operator
L
†
=
L
. Argue that if we
redeﬁne the operator according to
i
L
g
≡ {
g,H
}
, where
i
≡
√

1, then
L
is
Hermitian. Generalize the proof to
N
particles in threedimensions.
3. Derive Hamilton’s equations for a particle moving in twodimensions under a
central potential
U
(
r
). [Use polar coordinates (
r,θ
)]. Which equation illustrates
the law of conservation of angular momentum? Is angular momentum conserved
if the potential also depends on
θ
?
4. By making the assumption that the potential energy of an
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 Fall '09
 Ceweb
 Chemical Engineering, Angular Momentum, Force, Fundamental physics concepts, total linear momentum, Glenn Fredrickson, microscopic momentum density

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