A classical mechanics aperitif

From: Larne Pekowsky (larne_at_yahoo.com)
Date: 03/22/05 

Date: Tue, 22 Mar 2005 08:23:38 +0000 (UTC)

Following a conversation with one of my professors I've been looking
into the formalism of classical mechanics a bit. I can just about
follow the treatment in Arnold, but my classes don't allow me the time
needed to really understand the material at that level.

So, I've picked up bits and pieces from other sources and assembled
them into a sort of overview. In this post (and followups if there's
interest) I'd like to take an example-based look at what I think is
going on and invite people to correct what's wrong, clarify what's
muddled, or add their own insights.

I'll start with a simple one-dimensional problem and the good old p,q
plane, where we set up our first thatched-roof cottages back in
first-year physics. Now we'll spruce up the place by adding a
symplectic form:

        w = dp/\dq

It's tempting to identify this as the volume form, but the definition
of the volume form requires a metric and until someone tells me
otherwise I'm going to assume we don't have one.

Next we need a hamiltonian, and we'll start off with the simplest
possible case, a particle in free motion (and hopefully this won't be
too simple to be interesting)

      H = p^2/2m

Next we take the exterior derivative to get a 1-form:

      dH = p/m dp

Then the symplectic from can be used to turn this into a vector field,
much as a metric provides a map between vectors and 1-forms. Here we
want a vector h such that for any other vector, um, let's call it z:

      w(h,z) = (dH)z

Or, in components

      h = h^q \partial_q + h^p \partial_p
      z = z^q \partial_q + z^p \partial_p

      w(h,z) = (h^q)(z^p)-(h^p)(z^q)
      dH(z) = (p/m)(z^p)

equating these tells us that h^p=0, h^q=p/m or in other words

      h=(p/m) \partial_q

So now we've got a vector field, and we want to find the integral
curves (the flow). Let gamma be a curve parameterized by t, so
gamma(t) = (p,q). Then at any point the time rate of change of gamma
is equal to the value of the vector field at that point

      d gamma/d t = (p-dot, q-dot) = (0,p/m)

And the nifty thing here is that this was just a roundabout way of
getting to Hamilton's equations!

      q-dot = dH/dp = {q,H}
      p-dot = -dH/dq = {p,H}

The result of integrating these is, unsurprisingly,

      gamma=(p_0, q_0+(p_0/m) t)

Now looking at this as a family of transformations of the plane
parameterized by t, there's an identity element (t=0), and that the
composition of two transformations (t=t1 and t=t2) is another
transformation (t=t1+t2), that this is associative, and that each
transformation has an inverse (-t). Looks like a group to me. At
this point I'll be sloppy and just assume it's also a manifold, and
hence it's a Lie group.

That means there should also be a Lie algebra lurking around, and I've
heard that "the hamiltonian is the generator of flows" so let's take a
look at h. It's certainly a one-dimensional vector space indexed by
p. My first question (not counting the implied "is any of this even
vaguely right") is: what is the appropriate operation on this space to
make it a Lie algebra? But skipping over that for the moment and
considering

      e^{h t} =
      e^{t (p/m) \partial_q} =
      1+(t p/m)\partial_q + (1/2)(t p/m)^2\partial^2_q + ...

When applied to some function f(q) this gives a taylor series in (t
p/m) at q, which is f(q+(t p/m)). Which is the same translation we
got by finding the flow! So, yup, the hamiltonian is the Lie algebra
that generates the Lie group of flows.

I have a few more questions at this point. First, the connection
between exponentiation and integration here seems very deep and more
than a little mysterious. Anyone have any insights on this? Second,
while it's cool that we have a Lie group and algebra, what does that
tell us about what's actually going on? Is this situation too simple
to see the utility? I gather that there are other relevant Lie
algebras/groups, notably the algebra given by the Poisson brackets on
functions. I assume there's some relationship between this algebra
and the one on the hamiltonian, and that this relationship has
something to do with h=dH?

Finally, I've heard it said that flows "preserve the symplectic
structure." Which is true in this case; if the map is written as
(p.q) -> (p,q+tp) then the Jacobian is

      1 t
      0 1

      dp(p,q+tp) = (1,0), which multiplied by the Jacobian
                 = (1,0)
                 = dp

      dq(p,q+tp) = (0,1), which multiplied by the Jacobian
                 = (t,1)
                 = t dp + dq

and

      dp/\(t dp+dq) = dp /\ dq

I've also heard that flows should preserve volumes, which follows from
the previous bit plus thinking of dp /\ dq as a volume form. Or it
can be seen directly from the fact that the flow stretches out
parallelograms but leaves the base and height (and hence the area)
unchanged. 

--
This account is a spamtrap.  To send me real mail, use
larne@caneprince.com but change a prince into a toad.

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