Problem 3.7. Solve the N ¼ 3 case of the differential system (42).

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

Λ<sup>E</sup> ¼ Λ<sup>0</sup> þ EΛ<sup>1</sup> þ OðE

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

the geodesic flow of an ellipsoid is also completely integrable with the second integral of

The fundamental problem is to describe the moduli space of completely integrable Hamilto-

mechanical) Hamiltonians H has received wide-spread attention. When H is Riemannian, the most common approach is to assume the second integral F is polynomial-in-momenta, and without loss of generality, homogeneous. If the degree of F is fixed, then the problem of determining H&F is reducible to a non-linear PDE in the coefficients of F. When the degree is 1, the first integral F is a momentum map of a T<sup>1</sup> isometry group (see below). When the degree is 2, then the Hamiltonian is Liouville, a classical result due to Darboux c.f. [27]. In degree 3, there is the well-known case due to Goryachev-Chaplygin, and more recent cases due to Selivanova, Dullin and Matveev and Dullin, Matveev and Topalov and Valent [28–33]. In degree 4, Selivanova and Hadeler & Selivanova have produced a family of examples using the results of Kolokol'tsov [34, 27]. Beyond degree 4, Kiyohara has provided a construction of a smooth Riemannian metric H with an independent first integral F of degree k for any k≥1. In this construction, the metric H depends on a functional modulus, and so for each k, the set is

Let us review the work of Matveev and Shevchishin in more detail [36]. These authors impose an additional formal constraint that the metric possess one first integral that is linear-in-

From a geometric perspective, it is more natural to introduce coordinates adapted to the isometry group. That is, the existence of a linear-in-momenta first integral is equivalent to the

1 structure. The fixed set of the <sup>T</sup><sup>1</sup> action is a set of points {p−, <sup>p</sup>þ} which are equidistant along

<sup>γ</sup> : <sup>½</sup>−T, <sup>T</sup>� ! <sup>S</sup><sup>2</sup> is a minimal geodesic such that <sup>γ</sup>ð�TÞ ¼ <sup>p</sup>�, then we can let <sup>ð</sup>r, <sup>θ</sup><sup>Þ</sup> be 'polar'

2cðxÞðp<sup>2</sup>

<sup>x</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup>

. The action of T<sup>1</sup> on S<sup>1</sup> induces a cohomogeneity-


<sup>y</sup>Þ, the existence of a cubic

. The sub-problem of describing the integrable Riemannian (resp. natural or

2

⊂R<sup>3</sup> admits a completely integrable geodesic flow. Indeed,

Þ (40)

factor in (35) is of the form

58 Lagrangian Mechanics

3.3. The 2-sphere

nians on T�

S2

infinite dimensional [35].

The unit two-dimensional sphere S<sup>2</sup>

perturbed flow develops transverse homoclinic points.

motion that is, in general, a quadratic form in the momenta.

3.4. Super-integrable systems with a linear-in-momenta first integral

momenta. In conformal coordinates ðx, yÞ where H ¼ <sup>1</sup>

integral is reduced to a second-order ODE involving c.

existence of an isometry group containing T<sup>1</sup>

any minimal geodesic; and the principal T<sup>1</sup>

It appears to the author that this differential system may be soluble via hypergeometric functions. A successful resolution to the N ¼ 3 case will naturally lead to the higher degree cases, which appear to be somewhat more involved.

Problem 3.8. Solve the higher degree cases of the differential system (42).

#### 3.5. Super-integrable systems with a higher degree first integral

The author believes that the differential system 42 provides the key to understanding the subspace of super-integrable Riemannian Hamiltonians which admit a cohomogeneity-1 structure. Super-integrability alone does not imply the existence of such a cohomogeneity-1 structure. Without this additional hypothesis, there is very little known. Indeed, the extremely valuable construction of Kiyohara is the only construction that provides a smooth Riemannian Hamiltonian with a polynomial-in-momenta first integral of degree N > 3–super-integrable or not [35, 38].

Let us explain Kiyohara's construction in some detail. Let H<sup>0</sup> be the Riemannian Hamiltonian of the standard unit sphere in R<sup>3</sup> . Let F0, F<sup>1</sup> be linear-in-momenta first integrals of H<sup>0</sup> that are linearly independent and let l≥k≥1 be integers such that N ¼ k þ l≥3. Define a polynomial-inmomenta first integral <sup>G</sup><sup>0</sup> <sup>¼</sup> <sup>F</sup><sup>k</sup> 0Fl <sup>1</sup>. For almost all q∈S<sup>2</sup> , the functions G0, H0jT� <sup>q</sup>S<sup>2</sup> are dependent along two distinct lines through 0; this defines a pair of mutually transverse line bundles L� � over S<sup>2</sup> \fp� <sup>0</sup> , p� <sup>1</sup> g. The excluded, singular set consists of the anti-podal points p� <sup>j</sup> where Fj vanishes identically on the fibre (equivalently, the corresponding Killing field vanishes). This pair of line bundles provides a branched double covering

$$\Phi: \mathbf{T}^2 = \mathbf{R}^2 / 2\pi \mathbf{Z}^2 \to \mathbf{S}^2 \tag{43}$$

with simple branch points at fp� <sup>0</sup> , p� <sup>1</sup> g ¼ <sup>Φ</sup>ðπZ<sup>2</sup> Þ. The line bundles L� � pullback to the line bundles R dxj on <sup>T</sup><sup>2</sup> <sup>¼</sup> {ðx1, <sup>x</sup>2<sup>Þ</sup> mod2πZ}. Kiyohara shows that in these coordinates, the pullback of the function r which measures the time along the unique geodesic γ through fp� <sup>0</sup> , p� <sup>1</sup> g (see Figure 4) satisfies the second-order PDE

$$
\frac{\partial^2 r}{\partial \mathbf{x}\_1 \partial \mathbf{x}\_2} + \frac{1}{B\_1 + B\_2} \frac{\partial B\_1}{\partial \mathbf{x}\_2} \frac{\partial r}{\partial \mathbf{x}\_1} + \frac{1}{B\_1 + B\_2} \frac{\partial B\_2}{\partial \mathbf{x}\_1} \frac{\partial r}{\partial \mathbf{x}\_2} = 0 \tag{44}
$$

where B<sup>1</sup> and B<sup>2</sup> are functions that describe the line bundles L� in terms of the basis {dr, sin ðrÞ dθ}.

Kiyohara writes a function R ¼ r<sup>0</sup> þ r where r<sup>0</sup> is the solution to (44) given by Φ� r and r is a solution of (44) with C<sup>2</sup> small boundary conditions satisfying

$$r(\mathbf{s}, \mathbf{0}) = \mu\_1(\mathbf{s}), r(\mathbf{0}, \mathbf{s}) = \mu\_2(\mathbf{s}), \tag{45a}$$

$$\text{where}\quad\mu\_i(\mathbf{s}) = \mu\_i(\neg\mathbf{s}) = \mu\_i(\pi\neg\mathbf{s}), \quad \text{for all } i, \mathbf{s}, \text{ and} \tag{45b}$$

$$
\mu\_i([\!-\epsilon,\epsilon\!])=\mathbf{0}.\tag{45c}
$$

Then, by means of this perturbed function R, Kiyohara writes down an explicit formula for the perturbed Riemannian Hamiltonian H and polynomial-in-momenta first integral F. The condition for the Poisson bracket {H, F} to vanish is shown to reduce to the satisfaction of Eq. (44) by R for the given values of B<sup>1</sup> and B<sup>2</sup> (this legerdemain is the real trick that makes the construction work).

Condition (45b) ensures that R factors through Φ to a function on S<sup>2</sup> , while the condition (45c) ensures that <sup>R</sup> is <sup>C</sup><sup>∞</sup> on <sup>S</sup><sup>2</sup> and coincides with <sup>r</sup> on a neighbourhood of the branch set <sup>f</sup>p� <sup>0</sup> , p� <sup>1</sup> g (hence that H and F coincide with H<sup>0</sup> and F0, respectively, on a neighbourhood of the cotangent fibres of the branch set).

Let us now state several problems related to Kiyohara's construction. First, Kiyohara's vanishing condition on the boundary values (45c) is used to deduce the Riemannian Hamiltonians are not real-analytic. Since all the remaining constructions involve real-analytic data, this serves to show his examples are genuinely different.

Problem 3.9. Does Kiyohara's construction extend to real-analytic boundary conditions u1, u<sup>2</sup> that satisfy (45b) ? Do these real-analytic metrics include other known cases?

Figure 4. Kiyohara's construction. The zero set of the pair of Killing fields determines the equatorial geodesic γ. A choice of zeros {p<sup>þ</sup> <sup>0</sup> , p<sup>þ</sup> <sup>1</sup> } determines the polar coordinate system ðr, θÞ.

In particular, the obtained metrics are unlikely to have a T<sup>1</sup> isometry group, so the question is really whether the known examples in degree 3 and 4 are obtainable via this construction [12– 14, 28–30, 34, 39, 40].

Second, Kiyohara's construction produces a polynomial-in-momenta first integral F factors as A<sup>l</sup> 0Am <sup>1</sup> where Ai are linear-in-momenta functions. It is clear that the reducibility of the first integral F is forced by the desire to use a very simple branched covering.

## Problem 3.10. Is reducibility of the first integral F necessary?

linearly independent and let l≥k≥1 be integers such that N ¼ k þ l≥3. Define a polynomial-in-

along two distinct lines through 0; this defines a pair of mutually transverse line bundles L�

vanishes identically on the fibre (equivalently, the corresponding Killing field vanishes). This

bundles R dxj on <sup>T</sup><sup>2</sup> <sup>¼</sup> {ðx1, <sup>x</sup>2<sup>Þ</sup> mod2πZ}. Kiyohara shows that in these coordinates, the pullback of the function r which measures the time along the unique geodesic γ through fp�

> ∂r ∂x<sup>1</sup> þ

where B<sup>1</sup> and B<sup>2</sup> are functions that describe the line bundles L� in terms of the basis

Then, by means of this perturbed function R, Kiyohara writes down an explicit formula for the perturbed Riemannian Hamiltonian H and polynomial-in-momenta first integral F. The condition for the Poisson bracket {H, F} to vanish is shown to reduce to the satisfaction of Eq. (44) by R for the given values of B<sup>1</sup> and B<sup>2</sup> (this legerdemain is the real trick that makes the

ensures that <sup>R</sup> is <sup>C</sup><sup>∞</sup> on <sup>S</sup><sup>2</sup> and coincides with <sup>r</sup> on a neighbourhood of the branch set <sup>f</sup>p�

(hence that H and F coincide with H<sup>0</sup> and F0, respectively, on a neighbourhood of the cotan-

Let us now state several problems related to Kiyohara's construction. First, Kiyohara's vanishing condition on the boundary values (45c) is used to deduce the Riemannian Hamiltonians are not real-analytic. Since all the remaining constructions involve real-analytic data, this

Problem 3.9. Does Kiyohara's construction extend to real-analytic boundary conditions u1, u<sup>2</sup> that

1 B<sup>1</sup> þ B<sup>2</sup>

<sup>1</sup> g. The excluded, singular set consists of the anti-podal points p�

, the functions G0, H0jT�

Þ. The line bundles L�

∂B<sup>2</sup> ∂x<sup>1</sup>

∂r ∂x<sup>2</sup>

rðs, 0Þ ¼ u1ðsÞ,rð0,sÞ ¼ u2ðsÞ, (45a)

uið½−E,E�Þ ¼ 0: (45c)

where uiðsÞ ¼ uið−sÞ ¼ uiðπ−sÞ, forall i,s, and (45b)

<sup>=</sup>2πZ<sup>2</sup> ! <sup>S</sup><sup>2</sup> (43)

<sup>q</sup>S<sup>2</sup> are dependent

� pullback to the line

¼ 0 (44)

, while the condition (45c)

<sup>0</sup> , p� <sup>1</sup> g

<sup>j</sup> where Fj

<sup>0</sup> , p� <sup>1</sup> g

r and r is a

�

<sup>1</sup>. For almost all q∈S<sup>2</sup>

<sup>Φ</sup> : <sup>T</sup><sup>2</sup> <sup>¼</sup> <sup>R</sup><sup>2</sup>

<sup>1</sup> g ¼ <sup>Φ</sup>ðπZ<sup>2</sup>

∂B<sup>1</sup> ∂x<sup>2</sup>

Kiyohara writes a function R ¼ r<sup>0</sup> þ r where r<sup>0</sup> is the solution to (44) given by Φ�

<sup>0</sup> , p�

1 B<sup>1</sup> þ B<sup>2</sup>

momenta first integral <sup>G</sup><sup>0</sup> <sup>¼</sup> <sup>F</sup><sup>k</sup>

with simple branch points at fp�

(see Figure 4) satisfies the second-order PDE

∂<sup>2</sup>r ∂x1∂x<sup>2</sup> þ

solution of (44) with C<sup>2</sup> small boundary conditions satisfying

Condition (45b) ensures that R factors through Φ to a function on S<sup>2</sup>

satisfy (45b) ? Do these real-analytic metrics include other known cases?

serves to show his examples are genuinely different.

over S<sup>2</sup>

60 Lagrangian Mechanics

\fp� <sup>0</sup> , p�

{dr, sin ðrÞ dθ}.

construction work).

gent fibres of the branch set).

0Fl

pair of line bundles provides a branched double covering

It ought to be fruitful to ask three related questions. The reducibility of F is very special, with just two distinct factors.

Problem 3.11. Is it possible to extend Kiyohara's construction so that the polynomial-in-momenta first integral F has more than 2 distinct linear factors?

It would be natural to try to extend the construction to the case where the zeros all lie on the same geodesic γ. More generally, one might attempt to mirror Kiyohara's construction but in a more abstract way: start with a simple ramified covering <sup>Φ</sup> : <sup>∑</sup> ! <sup>S</sup><sup>2</sup> with a branch set Y⊂S<sup>2</sup> . Let F<sup>0</sup> be a product of linear first integrals of H<sup>0</sup> that vanishes identically on T� <sup>Y</sup>S<sup>2</sup> and not elsewhere. The stumbling block is that we need to clarify the intrinsic geometric meaning of the PDE that governs the perturbed systems (44).

Problem 3.12. Describe in explicit terms the third, independent first integral of H that is of least degree.

Kiyohara proves in his paper that H is super-integrable (he proves the geodesic flow is 2πperiodic, in fact), but that proof does not proceed by finding this third first integral.

#### 3.6. Three-dimensional configuration spaces

In comparison to the wealth of results and examples for surfaces that were surveyed above, comparatively little is known about the three-dimensional analogues. Tăĭmanov tells us that if the Tonelli Hamiltonian is completely integrable with real-analytic first integrals, then the

three-dimensional configuration space ∑ has a finite covering p^ : ∑ ^ ! ∑ such that the fundamental group π1ð∑ ^ Þ is abelian and of rank at most 3 [41–43]. Based on the resolution of the Poincaré conjecture, this result implies that, up to finite covering the only such configuration spaces are

$$\mathbf{S}^3, \qquad \mathbf{S}^2 \times \mathbf{T}^1 \qquad \text{or} \qquad \mathbf{T}^3. \tag{46}$$

The author generalized Kozlov's result on surfaces to three-manifolds. In this result, if the Tonelli Hamiltonian is completely integrable and the singular set is topologically tame, then Tăĭamanov's list extends to include those three-manifolds ∑ such that π1ð∑Þ is almost solvable (equivalently, due to the resolution of the geometrization conjecture, ∑ admits either a Nil or Sol geometry) [44]. Both results are sharp, like Kozlov's, in the sense that all such admissible configuration spaces admit a geometric structure and the Riemannian Hamiltonian of such a structure is completely integrable with first integrals of the requisite type [45, 46].

There are a large number of questions that this strand of research has opened. Let us sketch a few.

#### 3.7. The 3-sphere

The case of S<sup>3</sup> is perhaps best understood. It has been known since Jacobi proved the complete integrability of the geodesic flow of an ellipsoid via separation of variables, that the Liouville family of metrics on S<sup>3</sup> is completely integrable. These systems possess three independent quadratic-in-momenta first integrals.

Based on the analogous problem for the two-sphere,

#### Problem 3.13. Describe the structure of the super-integrable Riemannian Hamiltonians on S<sup>3</sup> .

Researchers who specialize in super-integrable classical and quantum systems have developed tools for constructing and classifying super-integrable systems c.f. [47–49]. Unfortunately, some key ingredients in these constructions lead to systems with singularities.

The first method is based on the cohomogeneity-1 structure of <sup>S</sup><sup>3</sup> with the group <sup>G</sup> <sup>¼</sup> SOð3<sup>Þ</sup> acting as the linear isometry group of R<sup>3</sup> ⊂R<sup>4</sup> . If one represents

$$\mathbf{S}^3 = \{ (\mathbf{x}, r) \vert \mathbf{x} \in \mathbf{R}^3, r \in \mathbf{R}, \vert \mathbf{x} \vert^2 + \vert r \vert^2 = 1 \}, \tag{47}$$

then we see that G acts freely on T� S3 \T� <sup>F</sup>S<sup>3</sup> where <sup>F</sup> <sup>¼</sup> {ð0, � <sup>1</sup>Þ} is the fixed-point set of the <sup>G</sup>action on S<sup>3</sup> . This is enough to see that any G-invariant Hamiltonian on T� S<sup>3</sup> is noncommutatively integrable (analogous to the same fact for S<sup>2</sup> ). If K : soð3Þ � ! R is a positivedefinite quadratic form, and Ψ : T� <sup>S</sup><sup>3</sup> ! soð3<sup>Þ</sup> � is the momentum map of the SO(3)-action, then an invariant Riemannian Hamiltonian can be written as

$$H = \frac{1}{2}p\_r^2 + \frac{1}{2}s(r)\Psi^\*\mathbf{K},\tag{48}$$

for some function s > 0 such that s · ð1 � rÞ <sup>2</sup> ! const:≠0 as <sup>r</sup> ! <sup>∓</sup>1.

If one employs the ansatz of Matveev & Shevchishin (c.f. Section 3.3), one would like to find first integrals that are polynomial-in-momenta of the form

$$F = \sum\_{j=0}^{N} b\_j(\mathbf{x}, r) \ p\_r^j \,\Psi^\* \eta\_{N-j} \tag{49}$$

where ηN−<sup>j</sup> : soð3Þ � ! R is a homogeneous polynomial of degree N−j. In (41), the pre-factor exp ðivθÞ appears to ensure that the coefficients of the first integral F are common eigenfunctions of the Casimir ΔS<sup>1</sup> ¼ <sup>∂</sup><sup>2</sup> <sup>∂</sup>θ<sup>2</sup> parameterized by r. In the current case, the ansatz suggests that the coefficients bj should factor as φλðθÞajðrÞ where φλ is an eigenfunction of the Casimir ΔS<sup>2</sup> with eigenvalue λ and θ ¼ x=jxj.

Problem 3.14. Extend the construction sketched, above to higher dimensional spheres.
