## 1. Introduction

Lagrangian mechanics employs the least-action principle to derive Newton’s equations from a scalar function, the action function

Due to the advantages of the Hamiltonian perspective, this chapter studies Lagrangian systems from this dual point of view. The organization of the chapter is this: Section 2 recalls the classic construction of angle-action variables in

## 2. Integrability in Hamiltonian mechanics

### 2.1. Integrability in 1 degree of freedom

One of the central problems in classical mechanics is the *integrability* of the equations of motion. The classical notion of integrability is loosely related to exact solvability, and roughly corresponds to the ability to solve a system of differential equations by means of a finite number of integration steps.

2.1a. *Example: Harmonic oscillator* Let us take the simple harmonic oscillator, or an idealized Hookean spring-mass system, with mass

The change of variables

where

The differential equations in (3) are trivial to integrate since the right-hand sides are constants. Let us explain the sequence of transformations. The change of coordinates

Therefore, the change of coordinates

Suppose that for some reason one did not know to introduce “polar” coordinates. One might still determine the change of coordinates using only that the transformation

where

If

On the other hand, since

Equating (5) and (6) shows that

These calculations show that one may determine

for

so

Let it be observed that if, in Eq. (4), one had chosen the anti-derivative to be

2.1b. *Example: the planar pendulum*. Let us take the idealized planar pendulum with a mass-less rigid rod of length **Figure 1**). The total energy is

To simplify the exposition, assume that the mass

If one tries to solve for a generating function

where

If

where **Figure 2** shows the geometric meaning of

which determines

**Figure 3** graphs

The first, calculus-based, proof is this: as

The second, topological, proof is this: each level set *c.f*. **Figure 2**). If the generating function v were differentiable in _{,} and so the level sets of

To derive the change of coordinates

where

where

2.1c. *Example: a mechanical system*. Let

If one attempts to find the generating function

up to a function depending only on

and, upon solving (18) for

to obtain

### 2.2. The generating function

The above three examples use a generating function

2.2a. *Question: why do the angle-action variables exist?* In order to understand the generating function, it is necessary to clarify the existence of the coordinates *A* is connected and

Since the tangent space at

Let

The function _{,} the Hamiltonian vector field

This proves the existence of an area-preserving diffeomorphism

*2.2b. Question: what kind of “function” is* *? In the first instance*, *is not single-valued. Indeed, one postulates the area-preserving change of coordinates* *to deduce that*

so that *locally* there is a function

But since

The way to resolve these ambiguities or difficulties is simple: the domain of the change of coordinates *c.f*. 21). In this case, the lift of a closed contour

So to answer the question that started the section, the generating function

### 2.3. Integrability in 2 or more degrees of freedom and Tonelli Hamiltonians

Integrability in 2 or more degrees of freedom is substantially more involved than the case of 1 degrees of freedom. Of course, a sum of *n* distinct, non-interacting 1-degree-of-freedom Hamiltonians is a simple case; and upon reflection, a not-so- simple case, because this condition is not coordinate independent. Indeed, a necessary and sufficient condition is that the Hamiltonian vector field be Hamiltonian with respect to two distinct non-degenerate Poisson brackets

Let us turn now to a definition which generalizes mechanical Hamiltonians.

**Definition 2.1** (Tonelli Hamiltonian). Let*be a smooth n-manifold and* *its cotangent bundle. A smooth function* *which satisfies* *is strictly convex for each* *and* *uniformly as* *is called a Tonelli Hamiltonian*.

As noted, Tonelli Hamiltonians are natural generalizations of mechanical systems. For this reason,

If

for all

A fundamental result in Hamiltonian mechanics is the Liouville-Arnol’d theorem, which provides a semi-local description of a completely integrable Hamiltonian and the Poisson bracket.

**Theorem 2.1** (Liouville-Arnol’d). Let*be a smooth Hamiltonian. Assume there exists n functionally independent, Poisson commuting conserved quantities* *is a compact component of a regular level set, then there is a neighbourhood* *and a diffeomorphism* *such that*

*that maps *.

In such a situation, it is said that *Liouville torus*, the neighbourhood *toroidal ball* and the conserved quantities are *first integrals*. Systems with *k* first integrals, of which *non-commutatively integrable*; when *super-integrable c.f*. [3, 4].

There are several proofs of the Liouville-Arnol’d theorem in the literature. The basic ideas are already captured in the one-dimensional case discussed in Section 2.2.

It can be assumed, without loss, that

Because the functions

for all

This is a smooth map which is a local diffeomorphism of

For each *t* such that

Define functions

The flow map

To complete the proof, one might show that each vector field

where

where, in the

is the “cylinder” obtained by sweeping out the cycles

which implies

Since the period matrix

Finally, the functions

The remainder of the theorem follows from the fact that the angle-action coordinates

## 3. Topology of configuration spaces

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

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

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 *known* to support a completely integrable Tonelli Hamiltonian are the 2-sphere,

V. Bangert has suggested to the author that Bialy’s argument should extend to prove the non-existence of a *c.f*. [8]). The idea of such a proof would be the following (assuming that *T*. Since the union of Liouville tori is dense, for each *compressible*. It follows that

Thus, for each closed orbit

**Problem 3.2.** Let*be a compact surface of negative Euler characteristic. Extend the above argument to prove the non-existence of a smooth Tonelli Hamiltonian* *with a second* *integral* *that is independent on a dense set; or give an example of a completely integrable Tonelli Hamiltonian*

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 * which is a symplectomorphism that satisfies a condition analogous to

**Problem 3.3.** Let*be a twist map. If f has a horseshoe and a* *first integral* *is* *necessarily constant on an open set?*

### 3.2. The 2-torus

Let us turn now to the torus. The 2-torus *Liouville*. These are of the form

where

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

on an energy level

It is a remarkable fact that the Liouville family exhausts the list of known completely integrable Riemannian Hamiltonians whose configuration space is *only* examples possible when the second integral in polynomial-in-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

In [13, 14], Kozlov and Denisova prove that if, when

with the conformal factor

In [12], Denisova, Kozlov and Treschëv prove that, if one only assumes *spectrum * of the function

An alternative approach, due to Bialy and Mironov, is to observe that the equation

then the coefficients *semi-linear* PDE [17, 18]. Indeed, there is a system of coordinates

where

where we adopt the convention that

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

To find Riemann invariants, Bialy and Mironov employ the following trick: let

In ([18], Theorems 1 and 2), Bialy and Mironov prove that if

The key step in Bialy and Mironov’s proof is to show that, in any region where

**Problem 3.4.** Extend Bialy and Mironov’s work to show that there are no regions where any multiplier*is non-real on* *i.e. show that (39) is a hyperbolic system.*

There is good reason to believe that the multipliers

Hyperbolicity of Eq. (39) has additional meaning. As the previous paragraph alluded to, the points where

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

Let us reformulate this as:

**Problem 3.5.** Prove the vanishing of the topological entropy of the geodesic flow of a Riemannian Hamiltonian on *that is completely integrable with a polynomial-in-momenta first integral*

In various special cases, such as when

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

**Problem 3.6.** The topological entropy of a non-Liouville Riemannian Hamiltonian on*is positive.*

Glasmachers and Knieper [20, 19] have studied the structure of geodesic flows with zero topological entropy on

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 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

where

### 3.3. The 2-sphere

The unit two-dimensional sphere

The fundamental problem is to describe the moduli space of completely integrable Hamiltonians on *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

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

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-momenta. In conformal coordinates *c*.

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 existence of an isometry group containing *H* and polynomial-in-momenta integral *F* can be written in the adapted coordinates as

where

where

In case

**Problem 3.7.** Solve the*case of the differential system (42).*

It appears to the author that this differential system may be soluble via hypergeometric functions. A successful resolution to the

**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 *–*super-integrable or not [35, 38].

Let us explain Kiyohara’s construction in some detail. Let

with simple branch points at **Figure 4**) satisfies the second-order PDE

where

Kiyohara writes a function

Then, by means of this perturbed function *given* values of

Condition (45b) ensures that

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

In particular, the obtained metrics are unlikely to have a

Second, Kiyohara’s construction produces a polynomial-in-momenta first integral

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

It ought to be fruitful to ask three related questions. The reducibility of

* 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

**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

### 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

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 *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

Based on the analogous problem for the two-sphere,

**Problem 3.13.** Describe the structure of the super-integrable Riemannian Hamiltonians on

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

then we see that

for some function

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

where *ansatz* suggests that the coefficients

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