Geodesic flow

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]. 

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. 

Nonintegrability of Geodesic Flows on High-Genus Riemann Surfaces with Convex Boundary... 

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, ... 

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 Liouville theorem, F b a torus if F O Qf = 0. By virtue of convexity of dAf, there exists such a neighbourhood U of the boundary dQ in Q that each trajectory of the geodesic flow on Q coming into U leaves Qf transversally to dQj. For this reason F is a torus if and only if F fl 17 = 0. FVom this and from the implicit function theorem it follows that the set of points x eV, such that Fx is a torus, is open and closed in V, Show that this set is non-empty. 

Reduce the problem to the study of geodesic flow. For our further purposes we make use of the following properties of the potential V. 

Let A 0. The Levi-Civita regularization reduces the phase flow of the problem of n attracting centres at the energy level (H = A) to the geodesic flow on the Riemann surface. 

FVom the lemma just proved there follows the Maupertuis principle. The mapping x TTU T U carries the trajectories of the geodesic flow of the metric g, ... 

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... 

Let be a closed manifold. Consider the geodesic flow on which on the cotangent bundle T M (endowed with a natural symplectic structure) is Hamiltonian with the Hamiltonian H Xfp) = S QijViPji where x M pi are coordinates in the fibre of the cotangent bundle T M and gij(x) is the Riemannian metric on... 

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 ... 

Jf the configuration space is two-dimensional, then Theorem 5.2.2 obviously follows. Note that Theorems 5.3.1 and 5.3.2 are applicable to geodesic flows on cotangent bundles to Riemannian manifolds. It would be of interest to deduce analogous statements for an arbitrary symplectic manifold, that is, to find out what topological invariants hamper the existence of a complete set of involutive integrals. We will develop the results of Ch. 2 in this direction. 

Integrahility and Nonintegrability of Geodesic Flows on Two-Dimensional Surfaces, Spheres, and Tori... 

The Holomorphic l Form of the Integral of a Geodesic Flow Polynomial in Momenta and the Theorem on Nonintegrability of Geodesic Flows on Compact Surfaces of Genus g > 1 in the Class of Functions Analytic in Momenta... 

In this section we consider the cases of complete Liouville integrahility of geodesic flows on two-dimensional Riemann surfaces. We describe the cases of complete integrahility as well as the cases of nonintegrability, that is, when the absence of analytic integrals can be proved. 

We restrict our consideration to a special case of discovery of integrals polynomial in momenta. It turns out possible to describe those Riemannian metrics whose geodesic flows admit such integrals. 

By the geodesic flow of the Riemannian manifold M with the metric ds = Y Gijdqidqj we mean a Lagrangian system in the tangent bundle TM with the Langrange function L — Identifying TM with T M by means of the... 

Riemannian metric, we may assume that the geodesic flow acts in T M. The corresponding system appears to be Hamiltonian, the Hamiltonian H being the Legendre transformation of the Lagrange function L, that is, AT = where... 

Since an integral analytic in momenta can be expanded in a converging series of homogeneous polynomials, which are obviously invariantly defined in r Af and are also integrals of a given geodesic flow, then the theorem formulated above is, in fact, equivalent to the one that follows. 

Metrics of the form (/(x) + h(y)) additional integral quadratic in momenta is of Liouville type. Consequently, the equations for geodesics are integrated in this case by way of separation of variables. 

In the assumption that the geodesic flow has an additional integral quadratic in momenta we find out the form of the function R(z) setting up the basic equation (4) in global coordinates. It can be expounded (we leave the proof to the reader) that in the global coordinates z = x + ty there hold the following asymptotic formulae A(z) = and R z) = (c H- 0(l))z as z — oo, where a and c... 

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