A Note on Integrable Mechanical Systems on Surfaces

Let S be a compact, connected surface and H in C^2(T^* S) a Tonelli Hamiltonian. This note extends V. V. Kozlov's result on the Euler characteristic of S when H is real-analytically integrable, using a definition of topologically-tame integrability called semisimplicity. Theorem: If H is 2-semisimple, then S has non-negative Euler characteristic; if H is 1-semisimple and reversible, then S has positive Euler characteristic.

1. Introduction 1.1. Kozlov's Theorem. Say that a natural mechanical system is a Hamiltonian that is a sum of kinetic and potential energies. Let S be a compact surface and H : T * S → R be an analytic natural mechanical system. Kozlov proved if H enjoys a second, independent analytic integral F , then the Euler characteristic of S is non-negative and so it is homeomorphic to S 2 , T 2 or a non-orientable quotient thereof [10]. The hypotheses of Kozlov's theorem can be relaxed as follows: (i) H need only be assumed to be fibre-wise strictly convex and super-linear (i.e. 'Tonelli'); (ii) analyticity can be reduced to the combined hypotheses that H and F are C 2 , and that there is an energy level H −1 (c) where c > min {H(x, 0) | x ∈ S}, such that the critical set of F intersects a fibre of the foot-point projection in only finitely many points [11].
The present note has two aims. First, it presents a proof of Kozlov's theorem based on the theory of semisimplicity developed in [4]; see definition 1 below. Second, it extends Kozlov's theorem to non-commutatively integrable Tonelli Hamiltonians. The latter is a non-trivial extension: in [5] there are Tonelli Hamiltonians that are constructed which are non-commutatively integrable and semisimple on the unit disk bundle, but which are not tangent to a semisimple singular Lagrangian fibration. In essence, the naïve trick of discarding extra integrals to achieve complete integrability necessarily expands the critical set, and in the above-quoted example, the critical set expands from a real-analytic set to a wild set analogous to the Fox wild arc.
1.2. Non-commutative integrability. Let Σ be a smooth n-dimensional manifold. The canonical Poisson structure on the cotangent bundle T * Σ permits one to define a Poisson algebra structure on C ∞ (T * Σ) and consequently each smooth function H : T * Σ → R induces a Hamiltonian vector field X H defined by where (x i , y i ) are canonical coordinates. A first integral of the Hamiltonian vector field X H is a smooth function F which Poisson commutes with H: Following [3], the differential dimension of A is defined to be sup p dim d p A. Let A ⊂ C ∞ (T * Σ) be a subspace of first integrals of H that contains H and let Z(A) be the subspace of A which Poisson commutes with all of A. Let k (resp. l) be the differential dimension of A (resp. Z(A)). Say that a point p ∈ T * Σ is regular for A if dim d q A = k and dim d q Z(A) = l for all q in the A-level set passing through p. We say that H (or A) is non-commutatively integrable if k + l = 2n and the set of regular points is dense.
Nehorošev [13], who generalized the Liouville-Arnol'd theorem [1], proved that if H is non-commutatively integrable and p is a regular point, then there is a neighbourhood U with coordinate chart (θ, I, x, y) : where the Poisson bracket is canonical and H = H(I). Dazord and Delzant [7] globalized this result and showed that the regular point set X ⊂ T * Σ is fibred by isotropic k-dimensional tori F = T k and the quotient of X by these fibres, P , is a Poisson manifold with a foliation ζ by symplectic leaves. When this foliation is a fibration, one has the following diagram where ι • is an inclusion map, g is the fibration map, S is a symplectic leaf of P , Q is an integral affine manifold of dimension k and π * P C ∞ (Q) is the centre of C ∞ (P ) which induces the Hamiltonian vector fields that are tangent to the isotropic fibres of g.
1.3. Geometric semisimplicity. Let us abstract the notion of complete and noncommutative integrability. A smooth flow ϕ : M ×R → M is integrable if there is an open, dense subset R ⊂ M that is covered by angle-action charts which conjugate ϕ to a translation-type flow on the tori of T k × R l . There is an open dense subset L ⊂ R fibred by ϕ-invariant tori; let f : L → B be the induced smooth quotient map and let Γ = M − L be the singular set. If Γ is a tamely-embedded polyhedron, then ϕ is said to be k-semisimple with respect to (f, L, B), or just semisimple [4]. Of most interest is when ϕ is a Hamiltonian flow on a cotangent bundle or possibly a regular iso-energy surface.
Definition 1 (c.f. [14,4]). A Hamiltonian flow is geometrically k-semisimple if it is k-semisimple with respect to (f, L, B) and f is an isotropic fibration.
In this case, because the fibres of f are isotropic, ϕ is non-commutatively integrable, so geometric semisimplicity is a topologically-tame type of non-commutative integrability. Taimanov [14] introduced a related notion of geometric simplicity, see sections 2.2-2.3 of [4] for further discussion. If ϕ is real-analytically noncommutatively integrable, then the triangulability of real-analytic sets implies that ϕ is geometrically semisimple; and B may be taken to be a disjoint union of open balls. On the other hand, geometric semisimplicity is a weaker property than realanalytic non-commutative integrability [4]. A basic question is: What are the obstructions to the existence of a geometrically semisimple (resp. semisimple, completely integrable) flow?
Here are the two main theorems of this note. In both cases, S is a compact, connected surface and H : T * S → R is a C 2 Tonelli Hamiltonian.
Remark I. In two degrees of freedom, as here, 1-or 2-semisimplicity implies the fibres of the fibration are isotropic, so 1-or 2-semisimplicity implies geometric 1or 2-semisimplicity. Remark III. Bangert has asked a series of questions in [12] concerning integrable Tonelli Hamiltonians which are integrable in a weaker sense-the additional integral need only be independent of H on an open dense set. These questions are very interesting but beyond the scope and techniques of this paper. Likewise, the paper by Bialy proves an extension of Kozlov's theorem for geodesic flows when the additional integral satisfies his condition ℵ [2]. That work relies on properties of minimizing geodesics.

Preliminaries
Let us recall a few items concerning a geometric semisimplicity. Let Σ be a compact smooth manifold and H : T * Σ → R be a C 2 Tonelli Hamiltonian that is geometrically semisimple with respect to (f, L, B). The complement Γ = T * Σ−L is a tamely embedded polyhedron, so the number of components of L ∩ H −1 ((−∞, c]) is finite for any c. [4,Lemma 15] implies that there is a component L i ⊂ L such that πι Li has a finite-index image in π 1 (Σ): Suppose that B 0 ⊂ B is a nowhere dense subset such that f −1 (B 0 ) ∪ Γ = Γ 1 is tamely embedded polyhedron whose complement L 1 = f −1 (B 1 ), B 1 = B − B 0 , is dense. One calls (f 1 = f |L 1 , L 1 , B 1 ) a refinement of (f, L, B). In [4, Lemma 18] it is proven that Let 0(Σ) ⊂ T * Σ be the zero section of the cotangent bundle of Σ. An exact Lagrangian graph Λ ⊂ T * Σ is the graph of an exact 1-form; 0(Σ) is an example.

Proofs
Proposition 2.1 allows us to prove Theorem 1. The manifold Σ in the previous section is the surface S.
Proof of Theorem 1. Suppose that H is geometrically 2-semisimple. We will deal with the case of 1-semisimplicity below. By Proposition 2.1 we can suppose that each component of the base of the fibration f is homotopy equivalent to a point or a circle. Thus, each component of L i ⊂ L is homotopy equivalent to T 2 or a T 2 -bundle over T 1 . In both cases, π 1 (L i ) is solvable, and so π 1 (S) contains a solvable subgroup of finite index. Since S is a surface, the theorem is proved.
Proof of Theorem 2. To prove Theorem 2, we must adapt the diagram in (2) to our needs. In this case, the fibre F = T 1 , the base of the fibration P (= B) is a 3-dimensional Poisson manifold with a foliation ζ by symplectic surfaces S. The foliation ζ is a fibration, in fact, because the Casimirs of P are functionally dependent on the reduction of the Hamiltonian H|X. In this case, the quotient of P by ζ, Q, is a finite union of 1-manifolds: Q = ∪ i Q i where Q i R or T 1 . Since H|X = h • G for some h ∈ C 2 (Q), the Tonelli property of H implies that no component of Q is a circle.

It follows that there is a component
has a finite index image. Since X i is homotopy equivalent to an F = T 1 -principal bundle over the symplectic surface S i , a leaf of ζ|X i , it remains to examine the possibilities.
S i is compact. In this case, G −1 (q) is compact for any q ∈ Q i , and therefore it must be a connected component of an energy level. Since, above the critical value, the energy levels are connected, G −1 (q) is an energy level. If the Euler characteristic of S is negative, then π 1 (S) contains no non-trivial normal abelian subgroups. Therefore, the inclusion F → T * S is null-homotopic; this implies that all orbits of the Tonelli Hamiltonian in a super-critical energy surface are contractible-absurd. Therefore, the Euler characteristic of S must be non-negative.
Since every orbit of the Tonelli Hamiltonian is closed on a super-critical energy level, the Euler characteristic of S must be positive.
S i is non-compact. Let c be an energy level such that π(S i ) = G(H −1 (c)) = q. Let X c = G −1 (q), g c = g|X c and P c = g(X c ). X c ⊂ H −1 (c) has a complement Γ c = Γ ∩ H −1 (c) and is fibred by F = T 1 . The Hamiltonian flow of H restricted to H −1 (c) is therefore 1-semisimple with respect to (g c , X c , P c ). Now, S c is a symplectic leaf of the foliation ζ and therefore is a connected symplectic surface. By Proposition 2.1, there is a refinement (g c , X c , P c ) such that each component of P c is homotopy equivalent to a point or T 1 . Moreover, by [4,Lemma 15], the inclusion of one of the components of X c in T * S is almost surjective on π 1 . But the components of X c are homotopy equivalent to T 1 or T 2 (principal T 1 -bundles over * and T 1 respectively).
Therefore, π 1 (S) contains a finite-index abelian subgroup. Hence the Euler characteristic of S is non-negative (this completes the proof of Theorem 1). It therefore remains to prove that the Euler characteristic of S is positive. To do so, we will prove Let X ⊂ X c be a component of X c and let F / / X / / / / P be the induced fibration of X by the closed orbits of ϕ c . Each orbit is homologous to the homology class of the fibre F p . Suppose that dim H 1 (S; Q) > 0. Let ω ∈ H 1 (S; Q) be an integral homology class. Each such class ω contains a closed geodesic, and so contains the projection of a closed orbit γ of the Hamiltonian flow of H restricted to H −1 (c). If γ ⊂ X c , then ω = [γ] ∈ Z F p for some regular fibre F p as in the preceding paragraph. If γ ⊂ X c , then, by the density of X c and continuity in initial conditions of ϕ c , each multiple of γ is approximated by a broken integral curve in X c , i.e. a curve w of the form 1 2 , 1] → H −1 (c) ∩ T * π(p) S, p ∈ X c and ϕ c T (p) = p. It follows that some multiple of the homology class ω is homologous to a closed orbit of ϕ c |X c and therefore that Qω ⊂ Q [F p ].
This proves that H 1 (S; Q) is a finite union of 1-dimensional sub-spaces and therefore it is at most 1-dimensional.
If dim H 1 (S; Q) = 1, then S is a Klein bottle and has a double coverŜ → S whereŜ is a 2-torus. The pulled-back HamiltonianĤ is 1-semisimple. The conclusion of the previous paragraph implies that dim H 1 (Ŝ; Q) ≤ 1-an absurdity. Therefore, dim H 1 (S; Q) must be 0.
Given Lemma 3.1, the proof of Theorem 2 is complete.