Geodesic integrator

Figure 1.8 Four arcs belonging to a surface. From the Gauss-Bonnet theorem, the integral curvature within the region of the surface bounded by the arcs (ABCD) is determined by the vertex angles (flj) and the geodesic curvature along the arcs AB, BC, CD and DA.

Geodesics are the curves that trace the shortest distance between two points, sufficiently close to each other. An n-dimensionaJ domain has the volume given by the integral... 

A system may be integrable but have not a single closed stable trajectory (although it may have many closed trajectories). A simple example is a geodesic flow of a two-dimensional flat torus, that is, a torus with a metric gij = 6ij. It is easy to see that this geodesic flow has an additional linear integral but that all closed trajectories of the system are unstable( ). 

FVom the results of Anosov, Klingenberg, and Takens, it follows that in the set of all geodesic flows on smooth Riemannian manifolds there exists an open everywhere dense subset of flows without closed stable integral trajectories [170], 17l. This means that the property of a geodesic flow to have no stable trajectories is the property of general position. Recall once again that we mean strong stability (see Definition 2.1.2). 

Consider a geodesic flow of a flat two-dimensional torus, that is, a torus with a locally Euclidean metric. This flow is integrable in the class of Bott integrals and obviously has no closed stable trajectories. By virtue of Proposition 2.1.2, we must have rank i(Q) 2. Indeed, the nonsingular surfaces Q are diffeomorphic here to a three-dimensional torus T, for which Hi T, Z) = Z 0 Z 0 Z. 

An application of this assertion is seen, for instance, from CoroUary 2.1.1 on nonintegrability of geodesic flows of general position on a two-dimensional sphere. In the case of a geodesic flow of a flat torus T, we have Q = T, Hi T, 2) = Z, R = S (that is, the conditions of Corollary 2.1.2 are not fulfilled), and although the flow has no closed stable trajectories on Q, it is nonetheless integrable in the class of Bott integrals. 

Meshcheryakov obtained some results on exact integration of geodesic flows of metrics PabD simple Lie groups by means of special functions. The functional nature of the solutions of the equations for geodesics is as follows they are either quasi-polynomials or rational functions of the restrictions of the theta functions of compact Riemann surfaces to rectilinear windings of Jacobian tori of these surfaces. These methods rest upon the papers by Novikov and Dubrovin [45]. 

It turns out that one may present an exact integration of the equations for geodesics of the metrics (pahD on the group SL(m,C). Metrics of the type (pabD appeared for the first time in the course of construction of nonlinear differential equations integrable by the inverse scattering method. FVom the paper [38] it readily follows that the Euler equation X = [X,ipahD ) on a classical Lie algebra of series Afn-i serves as a commutativity equation for a pair of operators. 

Theorem 5.2.1 can be reformulated as follows. Let be a real-analytic two-dimensional compact closed connected manifold endowed with an arbitrary real-analytic Riemannian metric. If the genus of a manifold is higher than unity then the geodesic flow of this metric (as the Hamiltonian flow on a four-dimensional manifold T M ) is not completely Liouville integrable, that is, does not have an additional (second) integral which is independent of the energy integral and is in involution with it. 

The geodesic flow has the function H Xf ) as a first integral. Examine three-dimensional level surfaces Q of the integral fT, that is, = ((x, ) T Mf = h = const). As we already know (see 2.1), these surfaces fibre over the surface Af with a circle as a fibre provided that h is greater than zero. Fix, for simplicity, the value of h to be equal to zero and examine a three-dimensional manifold Q, ... 

Theorem 5.2.4 (Bolotin). Let be a connected Riemannian compact two-dimensional real-analytic manifold with a locally geodesically convex boundary and such that x( ) < 0. Then the geodesic Sow of the Riemannian metric on the three-dimensional manifold of constant energy h = 1 does not have an analytic Brst integral (which is independent of the energy integral and is in involution with it on T M). 

Lemma 5.2.2. If there exists an inconstant analytic Brst integral of the geodesic Sow on Qi, then the set Yl ) Snite,... 

PROOF Let / be an analytic first integral of the geodesic flow on Qf, Then on the compact Qf the integral / has a finite set critical values. We put... 

Proof By the Maupertuis principle, the function / ox on (7 is an analytic first integral of restriction of the geodesic flow... 

Then / is a well defined analytic first integral of the geodesic flow on T M. 

To complete the proof, it remains to show that the geodesic flow on T M does not have analytic first integrals. 

By Theorem 5.2.3 (see [61]), the geodesic flow on a compact Riemann surface with a negative Euler characteristic does not have analytic first integrals. The surface M is noncompact, but M contains a compact submanifold with a geodesically convex boundary and homotopy equivalent to Af. 

Topological Obstacles for Analytic Integrability of Geodesic Flows on Non-Simply-Connected Manifolds... 

We have already seen that in the two-dimensional case non-simply-connectedness of a manifold may affect integrability of a system if the fundamental group is "sufficiently large, there are no analytic integrals. In this section we generalize Theorems 5.2.2 and 5.2.3 on analytic integrability of geodesic fiows of Riemannian... 

Definition 5.3.1 A complete set of involutive first integrals of a geodesic flow /i,..., /n-i, In = Hf is called geometrically simple if ... 

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