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which are called Hamilton's equations, and the sign convention given here makes the
resulting equations of motion compatible with Newton's dynamical equations for an
N
-variable dynamical web. We should emphasize that Hamilton's equations of motion
are usually constructed from much more general conditions than what we have pre-
sented here and are a formal device for producing Newton's equations of motion. For a
general Hamiltonian of the form of the kinetic plus potential energy,
N
p
k
H
(
p
,
q
)
=
2
m
k
+
V
(
q
),
(3.5)
k
=
1
Hamilton's equations reduce to the following more familiar form for particles of
mass
m
k
:
dq
k
dt
=
p
k
m
k
,
(3.6)
dt
=−
∂
(
)
dp
k
V
q
q
k
.
∂
The first equation is the definition of the canonical momentum and the second equation
is Newton's force law. Let us now restrict our analysis to the simplest of non-trivial
dynamical systems, the one-dimensional linear harmonic oscillator.
Consider the Hamiltonian for a linear web given by
H
0
in terms of the canonical
variables,
=
H
0
(
,
).
H
q
p
(3.7)
A simple one-dimensional oscillator of mass
m
and elastic spring constant
κ
has the
Hamiltonian
p
2
m
+
κ
q
2
1
2
H
0
=
.
(3.8)
Hamilton's equations are, of course, given by
dp
dt
=−
∂
H
0
∂
dq
dt
=
∂
H
0
∂
q
;
p
,
(3.9)
which, when combined, provide the equation of motion for the linear harmonic
oscillator,
d
2
q
dt
2
2
+
ω
0
q
=
0
,
(3.10)
where the natural frequency of the oscillator is
m
.
ω
0
=
(3.11)
It is well known that, according to Hooke's law, the force required to stretch a spring
by a displacement
q
is
κ
q
and consequently Newton's third law yields the restoring
force
−
κ
q
.
