Petar Topalov

Acad. G. Bonchev Str., bl. 8 Sofia, 1113, Bulgaria
Institute of Mathematics and Informatics, BAS


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
Topalov P. I.
In this paper we investigate the Poincare map in the regular neighbourhood of a critical leaf of the Liouville foliation of an integrable Hamiltonian system with two degrees of freedom. It was proved in [3], that for an arbitrary surface transversal to the trajectories, the Poincare map is a one-time-map along the flow of some Hamiltonian, which is defined on the considering surface (this Hamiltonian is called "the Poincare Hamiltonian"). In the paper [4] it was proved that for every transversal surface the Poincare map is a restriction to the surface of some smooth function, which is defined on the regular neighbourhood of the critical leaf.
Citation: Topalov P. I.,  The Poincare Map in the Regular Neighbourhoods of the Liouville Critical Leaves of an Integrable Hamiltonian System, Regular and Chaotic Dynamics, 1997, vol. 2, no. 2, pp. 79-86
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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