Thursday, February 07, 2008

Geometric Hamilton Jacobi

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 \omega. In addition, pick N functions q^i such that the submanifolds L(x^i)=\{m\in M|q^i(m)=x^i\} are Lagrangian, that is T_mL(x^i) is a Lagrangian subspace of T_mM (meaning that the symplectic form of any two tangent vectors of L(x^i) vanishes). If this holds, the q^i can be regarded as position coordinates.

Starting from these Lagrangian submanifolds, we can locally find a family of 1-forms in the normal bundle \theta\in N^*L(x^i) such that \omega=d\theta. You should think that \theta=p_idq^i for appropriate momentum coordinates p_i on the Lagrangian leaves of constant q^i. But here, these are just coefficient funtions to make \theta a potential for \omega.

Now we repeat this for another set of position coordinates Q^i which we assume to be "sufficiently independent" of the q^i meaning that TM=TL(q^i)\oplus TL(Q^i). This implies that locally (q^i, Q^j) are coordinates on M. With the Q^i comes another 1-form \Theta and since d\Theta=\omega=d\theta are locally related by a "gauge transformation". We have \theta = \Theta + dF for a function F.

Let's look at \theta a little bit closer. A general normal 1-form would look like f_i(q^j,Q^k)dq^q. But since we started from Lagrangian leaves, there is no dq^i\wedge dq^j in \omega and thus \theta=p_i(Q^j)dq^i. But expressing this in coordinates yields

p_i(Q^j)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 p_i = \frac{\partial F}{\partial q^i} and P_i = -\frac{\partial F}{\partial Q^i}. 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 q^i and Q^i and compute a function F from them. The trick is now to turn this procedure around: Given only the q^i and a function F(q^i,X^j) of these q^i and some N other variables X^j, one can compute the Q^i as functions on M: Take a point m=(q,p)\in M and define Q^i(m) by inverting p_i = \frac{\partial F(q,X=Q)}{\partial q^i}. For this remember that p_i was defined implicitly above: It is the coefficient of dq^i in \theta.

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 F such that the Q^i 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: \theta- hdt = \Theta - Hdt giving H = h +\frac Ft we get a transforming Hamilonian and we can require this to vanish:

0= H = h +\frac Ft

If we think of the Hamiltonian h given in terms of the coordinates (q^i,p_i) this is now a PDE for F which has to hold for all m\in m. That is, writing F as a function of q^i and X^i it has to hold for all (fixed) X^i as a PDE in the q^i and t.

1 comment:

Thomas said...

Nice, makes me want to take a look at my old course notes on theoretical mechanics again.