Author details

Leo T. Butler

3.6. Three-dimensional configuration spaces

^

mental group π1ð∑

62 Lagrangian Mechanics

spaces are

few.

3.7. The 3-sphere

quadratic-in-momenta first integrals.

acting as the linear isometry group of R<sup>3</sup>

then we see that G acts freely on T�

action on S<sup>3</sup>

Based on the analogous problem for the two-sphere,

three-dimensional configuration space ∑ has a finite covering p^ : ∑

S3

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

Poincaré conjecture, this result implies that, up to finite covering the only such configuration

, S<sup>2</sup> · T<sup>1</sup> or T<sup>3</sup>

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

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

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

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,

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>

,r∈R, jxj

. This is enough to see that any G-invariant Hamiltonian on T�

. If one represents

<sup>2</sup> þ jr<sup>j</sup>

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

⊂R<sup>4</sup>

some key ingredients in these constructions lead to systems with singularities.

<sup>S</sup><sup>3</sup> ¼ fðx,rÞjx∈R<sup>3</sup>

S3 \T�

commutatively integrable (analogous to the same fact for S<sup>2</sup>

structure is completely integrable with first integrals of the requisite type [45, 46].

^

Þ is abelian and of rank at most 3 [41–43]. Based on the resolution of the

! ∑ such that the funda-

: (46)

.

S<sup>3</sup> is non-

� ! R is a positive-

<sup>2</sup> <sup>¼</sup> <sup>1</sup>g, (47)

<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>-

). If K : soð3Þ

Address all correspondence to: leo.butler@ndsu.edu

Department of Mathematics, North Dakota State University, Fargo, ND, USA
