
110 Classical Mechanics
© 2010 Taylor & Francis Group, LLC
Solution:
The problem possesses cylindrical symmetry, so we choose ρ, ϕ, and z as the generalized coordi-
nates, and we let the axis of the paraboloid correspond to the z-axis and the vertex of the parabo-
loid be located at the origin (Figure 4.11). The Lagrangian of the system is
LTVm
gz=−=++−
1
2222
ρρϕ
where the reference level for the potential energy is set at the vertex of the paraboloid.
As ϕ is a cyclic coordinate, ∂L/∂ϕ = 0. Then, Lagrange’s equation for coordinate ϕ reduces to
d
d
d
dt
L
t
m
∂
∂
ϕ
ρϕ
2
0
from which we obtain
mρϕ
2
= constant
. (4.45)
Note that
ρϕ
=
is just the angu ...