Vladimir Matveev

Gottfried-Claren-Strasse 26, 53225 Bonn
Max-Planck-Institute f. Mathematik


Dullin H. R., Matveev V. S., Topalov P. I.
On Integrals of the Third Degree in Momenta
1999, vol. 4, no. 3, pp.  35-44
Consider a Riemannian metric on a surface, and let the geodesic flow of the metric have a second integral that is a third degree polynomial in momenta. Then we can naturally construct a vector field on the surface. We show that the vector field preserves the volume of the surface, and therefore is a Hamiltonian vector field. As examples we treat the Goryachev–Chaplygin top, the Toda lattice and the Calogero–Moser system, and construct their global Hamiltonians. We show that the simpliest choice of Hamiltonian leads to the Toda lattice.
Citation: Dullin H. R., Matveev V. S., Topalov P. I.,  On Integrals of the Third Degree in Momenta, Regular and Chaotic Dynamics, 1999, vol. 4, no. 3, pp. 35-44
Matveev V. S., Topalov P. I.
We suggest a simple approach for obtaining integrals of Hamiltonian systems if there is known a trajectorian map of two Hamiltonian systems. An explicite formila is given. As an example, it is proved that if on a manifold are given two Riemannian metrics which are geodesically equivalent then there is a big family of integrals. Our theorem is a generalization of the well-known Painleve–Liouville theorems.
Citation: Matveev V. S., Topalov P. I.,  Geodesical equivalence and the Liouville integration of the geodesic flows, Regular and Chaotic Dynamics, 1998, vol. 3, no. 2, pp. 30-45
Matveev V. S.
In the present paper we construct and topologically describe a series of examples of metrics on the Klein bottle such that for each metric
  • the corresponding geodesic flow has an integral, which is a polynom of degree four in momenta
  • the corresponding geodesic flow has no integral, which is a polynom of degree less than four in momenta.
Citation: Matveev V. S.,  Geodesic Flows on the Klein Bottle, Integrable by Polynomials in Momenta of Degree Four, Regular and Chaotic Dynamics, 1997, vol. 2, no. 2, pp. 106-112
Matveev V. S.
We classify integrable geodesic flows on the torus and on the Klein bottle, which in addition admit a quadratic in momenta first integral. We also study cases if there exists an additional linear integral.
Citation: Matveev V. S.,  Quadratically Integrable Geodesic Flows on the Torus and on the Klein Bottle, Regular and Chaotic Dynamics, 1997, vol. 2, no. 1, pp. 96-102
Matveev V. S., Topalov P. I.
Jacobi Vector Fields of Integrable Geodesic Flows
1997, vol. 2, no. 1, pp.  103-116
We show that an invariant surface allows to construct the Jacobi vector field along a geodesic line and construct the formula for the normal part of the Jacobi field. If a geodesic line is the transversal intersection of two invariant surfaces (such situation we have, for example, if the geodesic line is hyperbolic) than we can construct the fundamental solution of Jacobi equation $\ddot{u}=-K(t)u$. That was done for quadratically integrable geodesic flows.
Citation: Matveev V. S., Topalov P. I.,  Jacobi Vector Fields of Integrable Geodesic Flows, Regular and Chaotic Dynamics, 1997, vol. 2, no. 1, pp. 103-116

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