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lecture 2
Topics:
Where are we?
Forces of the form
F
(
v
)
Example:
F
(
v
) =

m
Γ
v
Another example:
F
(
v
) =

mβ v
2
Forces of the form
F
(
x
)
Review of the harmonic oscillator
Linearity and Time Translation Invariance
Back to
F
(
v
) =

m
Γ
v
Where are we?
Last time, we discussed Newton’s second law —
F
=
ma
. I tried to convince you that this is
essentially equivalent to the statement that the motion of any given classical mechanical system
is determined by a set of
initial conditions
, the values of the coordinates which specify its con
ﬁguration, and their ﬁrst derivatives at any given time. I also suggested that this is a very deep
and interesting fact about the world, and promised that we would come back to it at the end of the
course and give at least a provisional explanation of it.
We also discussed how to solve for the motion of a system numerically by keeping track of
q
and
˙
q
as functions of time, and using the Taylor expansion and
F
=
ma
to calculate approximately
how they change in a small time step
Δ
t
. By putting together many small time steps, we can trace
out the trajectory of the system. This procedure works for any any number of degrees of freedom,
and it should convince you that in principle, giving a second order differential equation for the
conﬁguration of a classical does just what expect  it determines the trajectory in terms a set of two
initial conditions per degree of freedom.
In a sense, our numerical analysis completely solves the problem  at least least your computer
can construct the solution to any problem. But for us people, it is nice to have analytic solutions that
we can use to develop our intuition. So we also talked about systems with one degree of freedom
in which the force depends only on time. In this VERY special case, we found that we could write
down the formal solution simply by integration. Then if the integral can be done analytically, we
get a completely analytic solution.
Today, we will give some more examples of very special systems in which we can do more
than just solving numerically. In some sense, I will just be showing you a collection of dirty tricks,
because it is only in very special cases that they work. But more generally, today’s lecture should
be regarded as a bunch of examples of the different ways in which initial conditions can enter into
the solutions of classical mechanics problems. We always need two initial conditions per degree
of freedom. But they appear in the actual trajectories in many different ways. In fact, I have
something else in mind as well. At the end, when we come to discuss the harmonic oscillator, we
will see that there are some very important general principles at work. These will be very useful,
and we will come back to them many times.
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 Spring '09
 JohnRobertson

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