ISSN 1560-3547, Regular and Chaotic Dynamics, 2009, Vol. 14, Nos. 4–5, pp. 455–465. c  Pleiades Publishing, Ltd., 2009. ARTICLES Isomorphisms of Geodesic Flows on Quadrics A. V. Borisov * and I. S. Mamaev ** Institute of Computer Science, Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia Received June 06, 2008; accepted March 2, 2009 Abstract—We consider several well-known isomorphisms between Jacobi’s geodesic problem and some integrable cases from rigid body dynamics (the cases of Clebsch and Brun). A relationship between these isomorphisms is indicated. The problem of compactification for geodesic flows on noncompact surfaces is stated. This problem is hypothesized to be intimately connected with the property of integrability. MSC2000 numbers: 53C22, 37Kxx DOI: 10.1134/S1560354709040030 Key words: quadric, geodesic flows, integrability, compactification, regularization, isomorphism In a series of papers [1–7] different analogies between the classical integrable Jacobi problem on the geodesic motion on an ellipsoid and the Clebsch case in the Kirchhoff equations (identical to the Brun case and the Neumann system), which govern the motion of a solid body in an unbounded ideal liquid, were noticed. Primarily our paper is of a methodical character. We tried to systematize the above mentioned results and thereby explain recently found isomorphisms for the two integrable problems: 1) the nonholonomic Chaplygin ball and 2) the Clebsch case. It is interesting to note that in some cases the transformation of one system into another can solve the problem of compactification of motion, which is useful for further qualitative and numerical explorations. However such a compactification is not always possible. We have formulated a conjecture connecting the possibility of compactification with the existence of an additional analytic integral. Let us write the equations of various classical integrable systems in variables most convenient for establishing sought-for isomorphisms. 1. JACOBI PROBLEM ON GEODESICS Consider a particle moving in a potential U (x) on the two-dimensional ellipsoid in R 3 = {x}. The ellipsoid is defined by the equation (x, Ix)= ζ, (1.1) where I = diag(I 1 ,I 2 ,I 3 ), ζ = const and (·, ·) is the conventional scalar product. Using the Lagrange function of the system L = 1 2 (˙ x, ˙ x) − U (x), one gets the equations of motion with an undetermined multiplier ¨ x = − ∂U ∂ x + λIx, λ = ( Ix, ∂U ∂x ) − (˙ x, I ˙ x) (Ix, Ix) , (1.2) where λ can be found from constraint (1.1). * E-mail: borisov@ics.org.ru ** E-mail: mamaev@ics.org.ru 455