Today, over lunch, togther with Christian Römmelsberger, we tried to understand Hamilton-Jacobi theory from a more geometric point of view.
The way this is usually presented (in the very end of a course on classical mechanics) is in terms of generating functions for canonical transformations such that in the new coordinates the Hamiltonian vanishes. Here I will rewrite this in the laguage of symplectic geometry.
As always, let us start with a 2N dimensional symplectic space M with symplectic form
. In addition, pick N functions
such that the submanifolds
=\{m\in M|q^i(m)=x^i\})
are Lagrangian, that is
)
is a Lagrangian subspace of
(meaning that the symplectic form of any two tangent vectors of
)
vanishes). If this holds, the
can be regarded as position coordinates.
Starting from these Lagrangian submanifolds, we can locally find a family of 1-forms in the normal bundle
)
such that
. You should think that
for appropriate momentum coordinates
on the Lagrangian leaves of constant
. But here, these are just coefficient funtions to make
a potential for
.
Now we repeat this for another set of position coordinates
which we assume to be "sufficiently independent" of the
meaning that
\oplus TL(Q^i))
. This implies that locally
)
are coordinates on M. With the
comes another 1-form
and since
are locally related by a "gauge transformation". We have
for a function F.
Let's look at
a little bit closer. A general normal 1-form would look like
dq^q)
. But since we started from Lagrangian leaves, there is no
in
and thus
dq^i)
. But expressing this in coordinates yields
dq^i=\theta=\Theta + dF= P_idQ^i + \frac{\partial F}{\partial q^i}dq^i + \frac{\partial F}{\partial Q^i}dQ^i)
Comparing coefficients we find
and
. You will recognize the expressions for momenta in terms of a "generating function".
What we have done was to take two Lagrangian foliations given in terms of
and
and compute a function
from them. The trick is now to turn this procedure around: Given only the
and a function
)
of these
and some N other variables
, one can compute the
as functions on M: Take a point
\in M)
and define
)
by inverting
}{\partial q^i})
. For this remember that
was defined implicitly above: It is the coefficient of
in
.
Up to here, we have only played symplectic games independent of any dynamics. Now specify this in addition in terms of a Hamilton function h. Then the Hamilton-Jacobi equations are nothing but the requirement to find a generating function
such that the
are constants of motion.
Even better, by making everything (that is h and Q and F) explicitly time dependent, by the requirement that the action 1-form is invariant:
giving
we get a transforming Hamilonian and we can require this to vanish:
If we think of the Hamiltonian h given in terms of the coordinates
)
this is now a PDE for F which has to hold for all
. That is, writing F as a function of
and
it has to hold for all (fixed)
as a PDE in the
and t.
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1 comment:
Nice, makes me want to take a look at my old course notes on theoretical mechanics again.
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