3. Topology of configuration spaces

<sup>Φ</sup>ðt, <sup>m</sup>Þ ¼ <sup>φ</sup><sup>t</sup>

each level Lf into itself and carries M into a submanifold φ<sup>t</sup>

or in other words, T ¼ TðFÞ.

52 Lagrangian Mechanics

Define functions θ ¼ ðθ1, …, θnÞ by

where, in the ðt, FÞ coordinate system,

which implies

θ ¼ TðFÞ

The flow map Φ therefore induces a diffeomorphism F<sup>−</sup><sup>1</sup>

This is a smooth map which is a local diffeomorphism of <sup>R</sup><sup>n</sup> · <sup>M</sup> with <sup>F</sup><sup>−</sup><sup>1</sup>ðCÞ. Indeed, <sup>φ</sup><sup>t</sup> carries

other hand, the derivative of <sup>φ</sup><sup>t</sup> with respect to <sup>t</sup> is a surjective linear map onto <sup>T</sup>φtðmÞLf . Therefore, dΦ is surjective, so injective, hence Φ is a local diffeomorphism onto its image.

For each <sup>m</sup>∈M, let <sup>P</sup>ðmÞ⊂R<sup>n</sup> be the set of <sup>t</sup> such that <sup>Φ</sup>ðt, <sup>m</sup>Þ ¼ <sup>m</sup>. Since each level set is compact, <sup>P</sup>ðm<sup>Þ</sup> is a discrete subgroup of <sup>R</sup><sup>n</sup> isomorphic to <sup>Z</sup><sup>n</sup>. This is the "period lattice" of the action φ. If one selects a basis of PðmÞ, one obtains a map M ! GLðn; RÞ, m ! 2πTðmÞ. The implicit function theorem implies that there is a smooth map amongst these maps. Moreover, since FjM is a bijection onto its image, one can take the components of F as coordinates on M,

To complete the proof, one might show that each vector field ∂=∂θ<sup>i</sup> is Hamiltonian with Hamiltonian function Ii and that F is functionally dependent on I so that ðθ, IÞ is a canonical system of coordinates on <sup>F</sup><sup>−</sup><sup>1</sup>ðCÞ. This is performed indirectly. Define the functions Ii <sup>¼</sup> IiðF<sup>Þ</sup> by

<sup>2</sup>πIi <sup>¼</sup> <sup>∮</sup> <sup>Γ</sup>iðF<sup>Þ</sup>

where ξ ¼ P � dQ is the primitive of the symplectic form Ω and ΓiðFÞ is the cycle on LF on which θ<sup>i</sup> increases from 0 to 2π and the other angle variables are held equal to 0. To show that

> ¼ lim s!0 1 s ð

is the "cylinder" obtained by sweeping out the cycles ΓjðF þ vejÞ as the j-th component of

<sup>¼</sup> <sup>∂</sup>Fk ∂Fj

Fincreases from Fj to Fj þ s, and Ti is the i-th column of the period matrix T. Since

<sup>Ω</sup> <sup>∂</sup> ∂Fj , ∂ ∂tk � � Cj

ð

<sup>ð</sup><sup>F</sup>,<sup>s</sup><sup>Þ</sup>

, F þ υejÞju∈½0, 2π�, υ∈½0,s�}

<sup>ð</sup>θ,I<sup>Þ</sup> is a system of coordinates on <sup>F</sup><sup>−</sup><sup>1</sup>ðCÞ, one computes the Jacobian <sup>½</sup>∂Ii=∂Fj�:

2π ∂Ii ∂Fj

CjðF, sÞ ¼ {ðuTðF þ υejÞ<sup>i</sup>

Compactness and connectedness of the levels Lf imply that the image of <sup>Φ</sup> is <sup>F</sup><sup>−</sup><sup>1</sup>ðCÞ.

<sup>ð</sup>mÞ, <sup>t</sup>∈R<sup>n</sup>, <sup>m</sup>∈M: (27)

<sup>ð</sup>M<sup>Þ</sup> transverse to Lf at <sup>φ</sup><sup>t</sup>

<sup>ð</sup>CÞ ! <sup>T</sup><sup>n</sup> · <sup>C</sup> : <sup>x</sup> ! ðθðxÞ, <sup>F</sup>ðxÞÞ.

ξ, (29)

Ω, (30)

¼ δjk, (31)

<sup>−</sup><sup>1</sup> � <sup>t</sup> <sup>ð</sup>mod 2πÞ, <sup>θ</sup> : <sup>R</sup><sup>n</sup> ·<sup>C</sup> ! <sup>R</sup><sup>n</sup>=2πZ<sup>n</sup>: (28)

ðmÞ; on the

The central problem in the theory of completely integrable Tonelli Hamiltonian systems is to

Problem 3.1. Determine necessary conditions on the configuration space Σ for the existence of a completely integrable Tonelli Hamiltonian H.

This is a broad, overarching problem which has motivated research by many authors over an almost 40-year period, including many of the author's publications. It is helpful to pose several sub-problems which address aspects of this problem and that appear to be amenable to solution. The remainder of this section is devoted to an elaboration of this problem, along with known results. We start with two-dimensional configuration spaces.

### 3.1. Surfaces of genus more than one

As a rule, completely integrable Tonelli Hamiltonians are quite rare, as are the configuration spaces Σ which support such Hamiltonians. Indeed, in two dimensions, the compact surfaces that are known to support a completely integrable Tonelli Hamiltonian are the 2-sphere, S<sup>2</sup> , the 2-torus T<sup>2</sup> and their non-orientable counterparts. With some quite mild restrictions on the singular set–called condition ℵ−, and assuming that the Hamiltonian is Riemannian, Bialy has proven these are the only compact examples [5]. This extended an earlier result of V. V. Kozlov [6]; the author has obtained a similar result for super-integrable Tonelli Hamiltonians [7].

V. Bangert has suggested to the author that Bialy's argument should extend to prove the nonexistence of a C<sup>2</sup> integral that is independent of the Hamiltonian when Σ is a compact surface of negative Euler characteristic (c.f. [8]). The idea of such a proof would be the following (assuming that H is Riemannian): Suppose that H enjoys a C<sup>2</sup> integral F that is independent on a dense set, hence that the union of Liouville tori is dense. Let Γ⊂H<sup>−</sup><sup>1</sup> ð1 <sup>2</sup>Þ be the union of orbits which project to minimizing geodesics. It is known, due to results of Manning and Katok [9, 10], that Γ contains a hyperbolic invariant set Λ on which the flow is conjugate to a horseshoe. Let λ⊂Λ be a closed orbit of the geodesic flow of period T. Since the union of Liouville tori is dense, for each E > 0, there is a Liouville torus Lλ,<sup>E</sup> that contains an orbit of the geodesic flow that remains within a distance E of λ over the interval ½0, T�. Hence, π1ðLλ,EÞ has a homotopy class mapping onto λ. Since λ is minimizing, it has no conjugate points and so for E sufficiently small, the same is true for the orbit on Lλ,<sup>E</sup> over the time interval ½0, T�. This implies that the image of π1ðLλ,EÞ is (free) cyclic and the kernel is generated by a cycle that bounds a disc–in classical terminology, this means that Lλ,<sup>E</sup> is compressible. It follows that Lλ,<sup>E</sup> bounds a solid torus <sup>T</sup>λ≅T<sup>1</sup> · <sup>B</sup><sup>2</sup> that is invariant for the geodesic flow. The integral <sup>F</sup>jT<sup>λ</sup> induces a singular fibration of the solid torus by invariant 2-tori.

Thus, for each closed orbit λ in the hyperbolic invariant set Λ, we have produced an invariant solid torus T<sup>λ</sup> that shadows λ–at least in some rough, homotopic sense. This fact alone should suffice to achieve a contradiction.

Problem 3.2. Let Σ be a compact surface of negative Euler characteristic. Extend the above argument to prove the non-existence of a smooth Tonelli Hamiltonian H : T� <sup>Σ</sup> ! <sup>R</sup> with a second C<sup>2</sup> integral F that is independent on a dense set; or give an example of a completely integrable Tonelli Hamiltonian H : T� Σ ! R.

V. Bangert proposes similar problems in his contribution in ([8], Problems 1.1, 1.2).

There is a similar, but possibly more accessible, problem for twist maps. Recall that if we discretize time, the notion of a Tonelli Hamiltonian is replaced by that of a twist map f : T� Σ ! T� Σ which is a symplectomorphism that satisfies a condition analogous to T1. If f enjoys n independent, Poisson commuting first integrals, then the Liouville-Arnol'd theorem implies that some power of f acts a translation on the Liouville tori. We noted above that the Hamiltonian flow of a Tonelli Hamiltonian has a horseshoe on an energy level.

Problem 3.3. Let f : T� <sup>T</sup><sup>1</sup> ! <sup>T</sup>� T<sup>1</sup> be a twist map. If f has a horseshoe and a C<sup>1</sup> first integral F, is F necessarily constant on an open set?

## 3.2. The 2-torus

Let us turn now to the torus. The 2-torus T<sup>2</sup> admits a family of completely integrable Riemannian Hamiltonians which are called Liouville. These are of the form

$$H = \frac{p\_x^2 + p\_y^2}{2[f(\mathbf{x}) + g(\mathbf{y})]} \quad F = \frac{g(\mathbf{y})p\_x^2 - f(\mathbf{x})p\_y^2}{f(\mathbf{x}) + g(\mathbf{y})} \tag{33}$$

where <sup>f</sup> , <sup>g</sup> : <sup>T</sup><sup>1</sup> ! <sup>R</sup> are smooth positive functions and <sup>ð</sup>x, <sup>y</sup>, px, py<sup>Þ</sup> is a canonical system of coordinates on T� T2 . The degenerations of the Liouville family include the rotationally symmetric ðf ≡const:Þ and flat ðf , g≡const:Þ.

The Liouville family is obtained from two uncoupled mechanical oscillators with periodic potentials,

$$G = \frac{1}{2}(p\_x^2 + p\_y^2) + a(\mathbf{x}) + b(\mathbf{y}),\tag{34}$$

on an energy level E ¼ α þ β > maxa þ maxb such that f ¼ α−a, g ¼ β−b. The Maupertuis principle states that orbits of the Hamiltonian flow of G on the energy level {G ¼ E} are orbits of the Hamiltonian flow of H up to a change in time along the orbit. The complete integrability of G is explained in Sections 2.1c and 2.3.

the geodesic flow that remains within a distance E of λ over the interval ½0, T�. Hence, π1ðLλ,EÞ has a homotopy class mapping onto λ. Since λ is minimizing, it has no conjugate points and so for E sufficiently small, the same is true for the orbit on Lλ,<sup>E</sup> over the time interval ½0, T�. This implies that the image of π1ðLλ,EÞ is (free) cyclic and the kernel is generated by a cycle that bounds a disc–in classical terminology, this means that Lλ,<sup>E</sup> is compressible. It follows that Lλ,<sup>E</sup> bounds a solid torus <sup>T</sup>λ≅T<sup>1</sup> · <sup>B</sup><sup>2</sup> that is invariant for the geodesic flow. The integral <sup>F</sup>jT<sup>λ</sup>

Thus, for each closed orbit λ in the hyperbolic invariant set Λ, we have produced an invariant solid torus T<sup>λ</sup> that shadows λ–at least in some rough, homotopic sense. This fact alone should

Problem 3.2. Let Σ be a compact surface of negative Euler characteristic. Extend the above argument to

is independent on a dense set; or give an example of a completely integrable Tonelli Hamiltonian

There is a similar, but possibly more accessible, problem for twist maps. Recall that if we discretize time, the notion of a Tonelli Hamiltonian is replaced by that of a twist map

independent, Poisson commuting first integrals, then the Liouville-Arnol'd theorem implies that some power of f acts a translation on the Liouville tori. We noted above that the Hamilto-

Let us turn now to the torus. The 2-torus T<sup>2</sup> admits a family of completely integrable Riemann-

where <sup>f</sup> , <sup>g</sup> : <sup>T</sup><sup>1</sup> ! <sup>R</sup> are smooth positive functions and <sup>ð</sup>x, <sup>y</sup>, px, py<sup>Þ</sup> is a canonical system of

The Liouville family is obtained from two uncoupled mechanical oscillators with periodic

Σ which is a symplectomorphism that satisfies a condition analogous to T1. If f enjoys n

T<sup>1</sup> be a twist map. If f has a horseshoe and a C<sup>1</sup> first integral F, is F

x−fðxÞp<sup>2</sup> y

. The degenerations of the Liouville family include the rotationally sym-

<sup>f</sup>ðxÞ þ <sup>g</sup>ðy<sup>Þ</sup> (33)

<sup>y</sup>Þ þ aðxÞ þ bðyÞ, (34)

V. Bangert proposes similar problems in his contribution in ([8], Problems 1.1, 1.2).

nian flow of a Tonelli Hamiltonian has a horseshoe on an energy level.

ian Hamiltonians which are called Liouville. These are of the form

<sup>H</sup> <sup>¼</sup> <sup>p</sup><sup>2</sup>

<sup>G</sup> <sup>¼</sup> <sup>1</sup> 2 ðp2 <sup>x</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup>

<sup>x</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup> y <sup>2</sup>½fðxÞ þ <sup>g</sup>ðyÞ� <sup>F</sup> <sup>¼</sup> <sup>g</sup>ðyÞp<sup>2</sup>

<sup>Σ</sup> ! <sup>R</sup> with a second C<sup>2</sup> integral F that

induces a singular fibration of the solid torus by invariant 2-tori.

prove the non-existence of a smooth Tonelli Hamiltonian H : T�

<sup>T</sup><sup>1</sup> ! <sup>T</sup>�

suffice to achieve a contradiction.

H : T�

f : T�

Σ ! R.

54 Lagrangian Mechanics

Σ ! T�

Problem 3.3. Let f : T�

3.2. The 2-torus

coordinates on T�

potentials,

necessarily constant on an open set?

T2

metric ðf ≡const:Þ and flat ðf , g≡const:Þ.

It is a remarkable fact that the Liouville family exhausts the list of known completely integrable Riemannian Hamiltonians whose configuration space is T<sup>2</sup> . Indeed, in 1989, Fomenko conjectured that these are the only examples possible when the second integral in polynomialin-momenta [11]. Most recently, in 2012, Kozlov, Denisova and Treschëv reiterate Fomenko's conjecture ([12], p. 908).

Let us note that it is a well-known fact that, if the first integral F is real- analytic, then F ¼ ∑N≥0FN where each term FN is polynomial-in-momenta with real-analytic coefficients, homogeneous and of degree N and since {H, FN} is polynomial-in-momenta, homogeneous and of degree N þ 1, each graded piece of F is a first integral. So, there is no loss in generality in restricting attention to polynomial-in-momenta first integrals–and, indeed, a slight increase in generality because the coefficients of the polynomial-in-momenta first integral are not assumed to be real-analytic.

In [13, 14], Kozlov and Denisova prove that if, when ðx, yÞ are isothermal coordinates, and

$$H = \frac{1}{2\Lambda} (p\_x^2 + p\_y^2),\tag{35}$$

with the conformal factor Λ a trigonometric polynomial, then the existence of a second independent first integral that is polynomial-in-momenta implies that H is Liouville.

In [12], Denisova, Kozlov and Treschëv prove that, if one only assumes Λ is smooth, then H has no irreducible polynomial-in-momenta first integral F that is of degree 3 or 4 that is independent of H. Mironov separately proves the non-existence of F of degree 5, but as noted in ([12], p. 909), Λ satisfies an extra unstated hypothesis [15]. The line of attack used in these papers is pioneered in [16], where Kozlov and Treschëv introduce the notion of the spectrum S⊂2nZ<sup>2</sup> of the function Λ as the support of the Fourier transform of Λ. This spectrum is finite iff Λ is a trigonometric polynomial; Denisova and Kozlov prove that, in this case, any first integral of H is dependent on H unless the spectrum S is contained in a pair of orthogonal lines through (0, 0), in which case H is Liouville and has a second independent first integral that is quadratic-inmomenta. Without the hypothesis that S is finite, the problem becomes significantly more delicate. The bulk of [12], for example, is devoted to a study of solutions to a PDE that characterizes the first integral F by means of Fourier analysis.

An alternative approach, due to Bialy and Mironov, is to observe that the equation {H, F} ¼ 0 coupled with the hypothesis that F is polynomial-in-momenta of degree N implies that when we write F as

$$F = \sum\_{j=0}^{N} a\_j(\mathbf{x}, y) p\_x^{N-j} p\_y^j \tag{36}$$

then the coefficients a0, …, an satisfy a semi-linear PDE [17, 18]. Indeed, there is a system of coordinates <sup>ð</sup>τ, <sup>υ</sup><sup>Þ</sup> on <sup>T</sup><sup>2</sup> such that, when <sup>F</sup> is written in the adapted canonical coordinates as <sup>F</sup> <sup>¼</sup> <sup>∑</sup><sup>N</sup> <sup>j</sup>¼<sup>0</sup>ujðτ, <sup>υ</sup>Þðpυ=g<sup>Þ</sup> j p <sup>ð</sup>N−j<sup>Þ</sup> <sup>r</sup> then this equation is of the form

$$
\mu\_v + T(\mu)\mu\_r = 0 \tag{37}
$$

where u<sup>0</sup> ¼ 1, u<sup>1</sup> ¼ g, u ¼ ðu1, u2, …, uNÞ and

$$T(u)\ddot{\boldsymbol{\eta}} = \begin{cases} u\_{i+1} & \text{if } \boldsymbol{j} = \boldsymbol{i} + 1, \\ (\boldsymbol{i} + 1)u\_{i+1} - (\boldsymbol{N} - \mathbf{1} - \boldsymbol{i})u\_{i-1} & \text{if } \boldsymbol{j} = 1, \\ 0 & \text{otherwise,} \end{cases} \tag{38}$$

where we adopt the convention that u\_<sup>1</sup> ¼ uNþ<sup>1</sup>≡0.

A standard technique to solve a quasi-linear PDE like (37) is to diagonalize it, that is, to find Riemann invariants, so that it is equivalent to

$$(r\_\upsilon + \Delta(r)r\_\tau = 0 \quad \text{where} \quad \Delta(r) = \text{diag}(\delta\_1(r), \dots, \delta\_N(r)), \tag{39}$$

$$r = (r\_1, \dots, r\_N). \tag{30}$$

To find Riemann invariants, Bialy and Mironov employ the following trick: let p<sup>υ</sup> ¼ g cos ðθÞ, pr <sup>¼</sup> sin <sup>ð</sup>θ<sup>Þ</sup> parameterize cotangent fibres of <sup>H</sup><sup>−</sup><sup>1</sup> <sup>1</sup> 2 � �. The invariance condition <sup>f</sup>H, <sup>F</sup>g ¼ <sup>0</sup> translates to Fvg<sup>−</sup><sup>1</sup> cos <sup>ð</sup>θÞ þ <sup>F</sup><sup>τ</sup> sin <sup>ð</sup>θÞ ¼ 0 along the locus where <sup>F</sup><sup>θ</sup> <sup>¼</sup> 0, i.e. where d<sup>F</sup> and d<sup>H</sup> are co-linear. If one supposes that θ<sup>i</sup> ¼ θiðτ, υÞ, i ¼ 1, …, N, is a smooth parameterization of the critical-point set, then the critical values ri ¼ Fðr, υ, θiðr, υÞÞ are Riemann invariants with δ<sup>i</sup> ¼ gðτ, vÞ · tan ðθiÞ. Of course, the main problem is to determine the relationship between the Liouville foliation–the singular foliation of T� T<sup>2</sup> by the Liouville tori and their degenerations–and the system 39.

In ([18], Theorems 1 and 2), Bialy and Mironov prove that if N≤4, then in any region where a multiplier δ<sup>i</sup> is non-real, the metric is Liouville. One can view the result of Bialy and Mironov as a partial confirmation of Fomenko's conjecture and an important step toward resolving that conjecture.

The key step in Bialy and Mironov's proof is to show that, in any region where δ<sup>i</sup> is non-real, the imaginary part of the Riemann invariant ri satisfies an elliptic PDE. It appears that the properties of this PDE are key to proving stronger results.

Problem 3.4. Extend Bialy and Mironov's work to show that there are no regions where any multiplier δ<sup>i</sup> is non-real on T<sup>2</sup> , i.e. show that (39) is a hyperbolic system.

There is good reason to believe that the multipliers δ<sup>i</sup> are always real. When ∅⊈δ<sup>−</sup><sup>1</sup> <sup>i</sup> <sup>ð</sup>C\RÞ⊈T<sup>2</sup> , Bialy and Mironov prove that the Riemann invariant ri is real and constant, say ri ¼ si . This implies that the common level set F<sup>−</sup><sup>1</sup> <sup>ð</sup>siÞ∩H<sup>−</sup><sup>1</sup> ð1 <sup>2</sup>Þ, a subset of the complexified cotangent bundle T� CT<sup>2</sup> , has a tangent with the fibres of T� <sup>C</sup>T<sup>2</sup> on an open set. That picture is dramatically at odds with the real picture, where the tangency can occur along a one-cycle at most. Because of this, it seems likely that there is a geometric proof of Problem 3.4.

Hyperbolicity of Eq. (39) has additional meaning. As the previous paragraph alluded to, the points where F<sup>θ</sup> ¼ 0 are the critical points of the canonical projection map π : T� <sup>T</sup><sup>2</sup> ! <sup>T</sup><sup>2</sup> restricted to a common level <sup>F</sup><sup>−</sup><sup>1</sup>ðrÞ∩H<sup>−</sup><sup>1</sup> <sup>1</sup> 2 . Such tori necessarily bound a solid torus in H<sup>−</sup><sup>1</sup> <sup>1</sup> 2 and are not minimizing. Based on Fomenko's conjecture, it is expected that these solid tori must be quite rigid in a well-defined sense: in homology, they should generate at most two transverse subgroups of H1ðT� T2 Þ.

There is an alternative approach to Fomenko's conjecture that is based on topological entropy. In a series of papers based on Glasmachers dissertation results, Glasmachers and Knieper study Riemannian Hamiltonians on T� T<sup>2</sup> with zero topological entropy [19, 20]. They prove the closure of one of the above-mentioned solid tori is a union of one or two closed, minimizing geodesic orbits and their stable and unstable manifolds ([20], Theorem 3.7c).<sup>3</sup> The picture that emerges from their work is that there is a family of minimizing closed geodesics of the same homology class, and their stable and unstable manifolds, which bound a family of invariant solid tori. Bialy [5] describes the boundary of this set as a separatrix chain. The projection of the separatrix chain covers T<sup>2</sup> . A neighbourhood of the separatrix chain in the complement is fibred by invariant Lagrangian tori that are graphs, i. e. that are a union of minimizing orbits. The multipliers δi, or rather the angles θ<sup>i</sup> mentioned above, define sections of the unit cotangent bundle trapped within a separatrix chain.

Let us reformulate this as:

then the coefficients a0, …, an satisfy a semi-linear PDE [17, 18]. Indeed, there is a system of coordinates <sup>ð</sup>τ, <sup>υ</sup><sup>Þ</sup> on <sup>T</sup><sup>2</sup> such that, when <sup>F</sup> is written in the adapted canonical coordinates as

A standard technique to solve a quasi-linear PDE like (37) is to diagonalize it, that is, to find

r<sup>υ</sup> þ ΔðrÞrr ¼ 0 where ΔðrÞ ¼ diagðδ1ðrÞ, …, δNðrÞÞ,

To find Riemann invariants, Bialy and Mironov employ the following trick: let p<sup>υ</sup> ¼ g cos ðθÞ,

translates to Fvg<sup>−</sup><sup>1</sup> cos <sup>ð</sup>θÞ þ <sup>F</sup><sup>τ</sup> sin <sup>ð</sup>θÞ ¼ 0 along the locus where <sup>F</sup><sup>θ</sup> <sup>¼</sup> 0, i.e. where d<sup>F</sup> and d<sup>H</sup> are co-linear. If one supposes that θ<sup>i</sup> ¼ θiðτ, υÞ, i ¼ 1, …, N, is a smooth parameterization of the critical-point set, then the critical values ri ¼ Fðr, υ, θiðr, υÞÞ are Riemann invariants with δ<sup>i</sup> ¼ gðτ, vÞ · tan ðθiÞ. Of course, the main problem is to determine the relationship between

In ([18], Theorems 1 and 2), Bialy and Mironov prove that if N≤4, then in any region where a multiplier δ<sup>i</sup> is non-real, the metric is Liouville. One can view the result of Bialy and Mironov as a partial confirmation of Fomenko's conjecture and an important step toward resolving that conjecture. The key step in Bialy and Mironov's proof is to show that, in any region where δ<sup>i</sup> is non-real, the imaginary part of the Riemann invariant ri satisfies an elliptic PDE. It appears that the

Problem 3.4. Extend Bialy and Mironov's work to show that there are no regions where any multiplier

2

uiþ<sup>1</sup> if j ¼ i þ 1, ði þ 1Þuiþ<sup>1</sup>−ðN−1−iÞui<sup>−</sup><sup>1</sup> if j ¼ 1, 0 otherwise,

u<sup>υ</sup> þ TðuÞur ¼ 0 (37)

<sup>r</sup> ¼ ðr1, …,rNÞ: (39)

� �. The invariance condition <sup>f</sup>H, <sup>F</sup>g ¼ <sup>0</sup>

T<sup>2</sup> by the Liouville tori and their degenera-

<sup>2</sup>Þ, a subset of the complexified cotangent bundle

<sup>C</sup>T<sup>2</sup> on an open set. That picture is dramatically at odds

(38)

<sup>i</sup> <sup>ð</sup>C\RÞ⊈T<sup>2</sup>

,

. This

<sup>ð</sup>N−j<sup>Þ</sup> <sup>r</sup> then this equation is of the form

<sup>F</sup> <sup>¼</sup> <sup>∑</sup><sup>N</sup>

56 Lagrangian Mechanics

<sup>j</sup>¼<sup>0</sup>ujðτ, <sup>υ</sup>Þðpυ=g<sup>Þ</sup>

j p

where u<sup>0</sup> ¼ 1, u<sup>1</sup> ¼ g, u ¼ ðu1, u2, …, uNÞ and

TðuÞij ¼

where we adopt the convention that u\_<sup>1</sup> ¼ uNþ<sup>1</sup>≡0.

pr <sup>¼</sup> sin <sup>ð</sup>θ<sup>Þ</sup> parameterize cotangent fibres of <sup>H</sup><sup>−</sup><sup>1</sup> <sup>1</sup>

the Liouville foliation–the singular foliation of T�

properties of this PDE are key to proving stronger results.

it seems likely that there is a geometric proof of Problem 3.4.

, i.e. show that (39) is a hyperbolic system.

There is good reason to believe that the multipliers δ<sup>i</sup> are always real. When ∅⊈δ<sup>−</sup><sup>1</sup>

<sup>ð</sup>siÞ∩H<sup>−</sup><sup>1</sup>

Bialy and Mironov prove that the Riemann invariant ri is real and constant, say ri ¼ si

ð1

with the real picture, where the tangency can occur along a one-cycle at most. Because of this,

tions–and the system 39.

δ<sup>i</sup> is non-real on T<sup>2</sup>

T� CT<sup>2</sup>

implies that the common level set F<sup>−</sup><sup>1</sup>

, has a tangent with the fibres of T�

Riemann invariants, so that it is equivalent to

8 < :

> Problem 3.5. Prove the vanishing of the topological entropy of the geodesic flow of a Riemannian Hamiltonian on T� T<sup>2</sup> that is completely integrable with a polynomial-in-momenta first integral F.

> In various special cases, such as when F is real-analytic or Morse-Bott, it is known that the topological entropy vanishes [21].

> Finally, since topological entropy is an important invariant in the study of these systems, let us state a number of problems that are directly relevant to the preceding discussion. If one assumes Fomenko's conjecture is true and that the Liouville family of Riemannian Hamiltonians equals the set of completely integrable Riemannian Hamiltonians on T<sup>2</sup> , then it should be true that

#### Problem 3.6. The topological entropy of a non-Liouville Riemannian Hamiltonian on T� T<sup>2</sup> is positive.

Glasmachers and Knieper [20, 19] have studied the structure of geodesic flows with zero topological entropy on T� T2 . The picture that emerges is the phase portrait looks remarkably like that of an integrable system. It seems likely that their results admit a strengthening: in particular, they are unable to determine the number of primitive homology classes represented by non-minimizing geodesics (for Liouville metrics, this is at most 4).

On the other hand, it is known, from results of Contreras, Contreras and Paternain and Knieper and Weiss that an open and dense set of Riemannian Hamiltonians have positive

<sup>3</sup> Although the minimizing orbits have stable and unstable manifolds, it is not suggested that they are hyperbolic.

topological entropy [22–24]. In the case of this particular problem, the natural point of departure is to look at Riemannian Hamiltonians that are close to Liouville, i.e. where the conformal factor in (35) is of the form

$$
\Lambda\_{\epsilon} = \Lambda\_0 + \epsilon \Lambda\_1 + \mathcal{O}(\epsilon^2) \tag{40}
$$

where Λ<sup>0</sup> is Liouville-and has no T<sup>1</sup> symmetry–and Λ<sup>E</sup> is not Liouville for all E≠0. Based on the study in [25, 26] of the phase portrait of such systems, it should be possible to prove that the perturbed flow develops transverse homoclinic points.
