atdotde: 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

and compute a function

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

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

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)