lecture 3
Topics:
Where are we?
Consequences of Time Translation Invariance and Linearity
Uniform circular motion
Harmonic oscillation for more degrees of freedom
The double pendulum
The damped harmonic oscillator
Where are we?
Last time, we saw how initial conditions appeared in a number of different examples of force laws.
The last of these, the harmonic oscillator, is a particularly important system because it has two
general properties, time translation invariance and linearity, that appear in many many physical
systems.
Because linearity went by pretty quickly last time, let me briefly review how it works. The
equation of motion,
F
=
ma
, for the harmonic oscillator is linear because there is a single
x
in
each term. It can be written as
m
d
2
x
dt
2
+
K x
= 0
(1)
We can think of this as a single “operator” acting on
x
.
=
ˆ
m
d
2
dt
2
+
K
!
x
= 0
(2)
In this form, it may be more clear why you the solutions form a linear space. If you have two
solutions,
x
1
(
t
)
and
x
2
(
t
)
, you can form an arbitrary linear combinations and still get solutions,
because the same operator acts on both, and multiplying by a constant doesn’t affect the validity
of the solution,
ˆ
m
d
2
dt
2
+
K
!
x
1
= 0
ˆ
m
d
2
dt
2
+
K
!
x
2
= 0
⇒
ˆ
m
d
2
dt
2
+
K
!
(
a x
1
+
b x
2
) = 0
(3)
This fact has remarkable consequences, as we will see shortly.
Consequences of Time Translation Invariance and Linearity
Time translation invariance is an example of a symmetry. The physics of the harmonic oscillator
looks the same if all clocks are reset by the same amount. When a symmetry is combined with
the property of linearity, the result is an extremely powerful tool for studying the solutions of the
system’s equation of motion. The reason is that because of linearity, the solutions of the equation
1
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of motion form what mathematicians call a linear space. You can add them together and multiply
them by constants and you still have solutions. Because of this, we can use the tools of linear
algebra to understand them. In particular, we can choose a convenient set of
basis
solutions that
behave as simply as possible under time translations. For the symmetry of time translation, it is a
mathematical fact that the basis solutions are just exponentials. We can always find solutions of
the form
1
z
(
t
) =
z
(0)
e
Ht
(4)
What is special about this form (and I am not going to discuss this in detail  I hope that you will
see this beautiful argument in more detail in Physics 15c) is that when you change the setting of
your clock by taking
t
→
t
+
a
, the exponential (4) is the only function that just changes by a
multiplicative constant,
z
(
t
+
a
) =
z
(0)
e
H
(
t
+
a
)
=
z
(0)
e
Ht
e
Ha
=
e
Ha
z
(
t
)
(5)
You can always use the linearity of the space of solutions to find particularly convenient solutions
that behave in this simple way under time translations  and then the result has to be an exponential.
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 Spring '09
 JohnRobertson
 Vector Space, Complex number, Euler's formula, Time Translation Invariance and Linearity

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