Liouville's Theorem

A long time ago, in a 2n-dimensional symplectic manifold M, with form \omega_{\mu\nu} (and dual form \omega^{\mu\nu}), far, far away...

Prologue: Our hero, a young Hamiltonian H: M\to\R, defines a "hamiltonian flow" via a vector field (X_H)^\nu = dH_\mu \omega^{\mu\nu}. We can understand H as, for instance, the total energy, and M as the phase space. H has a friend, G, which is preserved by the hamiltonian flow (e.g. momentum in some direction). This happens exactly when X_H[G] = (dG).(X_H) = 0 (thinking of X_H in the first line as a differential operator). But dG.X_H = (dG)_\nu (X_H)^\nu = (X_G)^\mu \omega_{\mu\nu} (X_H)^\nu = (dG)_\nu \omega^{\mu\nu} (dH)_\mu. So H's flow preserves G if and only if G's flow preserves H: being friends is a reflective relationship.

Following the classical mechanists, we say that H and G are "in involution" if indeed \omega(X_H,X_G) = 0. More generally, we can define the "Poisson Bracket" \{H,G\} = \omega(X_H,X_G) = (dH)_\mu (dG)_\nu \omega^{\mu\nu}. Then clearly \{H,G\} = -\{G,H\}, and in particular H preserves itself (energy is conserved). Indeed, \{,\} behaves as a Lie bracket out: it satisfies the Jacobi identity \{\{G,H\},K\} + \{\{H,K\},G\} + \{\{K,G\},H\} = 0, and \{,\} is \R-linear. (Thus C^\infty(M) is naturally a Lie algebra; the corresponding Lie group is the space of "symplectomorphisms", or diffeomorphisms on M that preserve \omega.) Moreover, X_{\{G,H\}} = [X_G,X_H] where [,] is the (canonical) Lie bracket on vector fields. (The Hamiltonian fields X_H are exactly the differentials of symplectomorphisms, hence the identification in the previous parenthetical.)

((Actually, it's not C^\infty(M) that's tangent to the symplecto group of M, but C^\infty(M) / \R, when M is connected. Our \{,\} depends only on the differential of Hamiltonian functions, and so ignores constant terms: there are \R possible constant terms for each connected component of M. We can, of course, equip \R with the trivial Lie algebra, and then C^\infty(M) is, as a Lie algebra, T(symplectomorphisms) \times \R. The physicists would say this by observing that energy is defined only up to a total constant; this constant cannot affect our physics because it appears only in commutators. The physicists try to use this observation to justify introducing infinite constants into their expressions.))

One day our hero H met another function F, but this one unpreserved by H's flow. How does F change? In our setup, where H and F have no explicit time dependence, and we're just flowing via \dot{x} = X_H, we have that dF/dt = X_H[F] = {F,H}.

When H and G are buddies (in involution), then each of H and G is preserved by X_H: the flow stays in the common level set H = H(0) and G = G(0). Assuming that H and G are independent, in the sense that dH and dG are linearly independent (so we're not in the G = H^2 case, for instance), this common level set is (2n-2)-dimensional.

The story: As a young and attractive Hamiltonian, our hero H was particularly popular: there were n-1 other Hamiltonians H_2,...,H_n so that, along with H_1=H, all were pairwise in involution (\{H_i,H_j\} = 0), and all independent (the set of dH_i is linearly independent at each point in M, or at least at each point in some common level set, and so in a neighborhood of that level set). This is the most friends any Hamiltonian can have: the common level set is n-dimensional, and the tangent space contains n independent vectors X_i = X_{H_i}. Because, the X_i spanned each tangent, and because \omega of any two was zero, the common level set was for all to see a Lagrangian submanifold.

Being very good friends, the H_i never got in each other's way. X_i could flow, and X_j could flow, and because of the relationship between Poisson and Lie brackets, their flows always commuted. The gang used this to great affect: by giving each friend an amount to flow, the crowd defined an \R^n action on the common level set. The friends set out to explore this countryside: Hamiltonian flow is volume-preserving, since it preserves the symplectic form (whose nth power is a volume form), and a volume preserving \R-action is onto connected components.

The friends, returning home by some element of the stabilizer subgroup, understood the landscape: the only discrete subgroups of \R^n are lattices, and so the common level set was necessarily a torus (in the compact case). Picking standard coordinates q^j for a torus, the friends observed an isotropy: at every point, X_i = a_i^j \d/\dq^j with a constant "frequency" matrix a.

Our hero's life was solved. If all the frequencies a_1^j were rational multiples of each other, what the Greeks called "commensurate", H's paths were closed. Otherwise, H's flow would be dense in some subtorus, and either way, physics was simple. Indeed, because every Lagrangian manifold has a neighborhood symplectomorphic to its tangent bundle, there were "momentum" coordinates p_i conjugate to the angular position coordinates q^i, and these p_i depended only on the H_j. Indeed, in p,q coordinates, X_i was by definition -\dH_i/\dp_j \d/\dq^j + \dH_i/\dq^j \d/\dp_j, so the H_i knew themselves: H_i = -a_i^j p_j + const.

It was in this way that our hero the Hamiltonian understood how to flow not only in the level set, but in some neighborhood. The friends lived happily ever after.

Epilogue: Sadly, not all Hamiltonians can have as nice a life as our hero, because many do not have so many friends. It has been shown that the three-body problem is not Liouville-integrable, as this property of having enough mutual friends (and hence admitting Lagrangian tori) came to be called. Much analysis has gone into studying perturbations of Liouville systems — weakly-interacting gravitating bodies, for instance — but I do not know this material, and so will not exposit on it here. In my next entry, I hope to speak more about the Poisson bracket, and how it turns classical into quantum systems.

Edit: The matrix a_i^j may depend, of course, on H, or equivalently on p. What is actually true is that, up to a constant, H_i(p) = \int_0^{p} a_i^j(p') dp'_j. It is by solving this equation that one may find the conjugate p coordinates. That H_i = -a_i^j p_j + const. is true only to first order, and to first-order we cannot know whether, for instance, entries in a_i^j remain commensurate, and so whether paths stay closed as the momentum changes. Generically, Hamiltonian flows are not closed, and instead a single path is dense in the entire torus. In the general three-body problem, the flow is dense in a space greater than the dimension of any Lagrangian submanifold.