scholarly journals Topological obstacles to the realizability of integrable Hamiltonian systems by billiards

2019 ◽  
Vol 488 (5) ◽  
pp. 471-475
Author(s):  
V. V. Vedyushkina ◽  
A. T. Fomenko
Keyword(s):  
Magnetic Field ◽  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Integrable Hamiltonian Systems ◽  
Two Degrees Of Freedom ◽  
Degenerate Hamiltonian Systems ◽  
Liouville Equivalence

We introduce the following classes of integrable billiards: elementary billiards, topological, books, with potential, magnetic field, geodesic billiards. These classes are used to test the A.T. Fomenko conjecture about the realizability up to Liouville equivalence by billiards of integrable non-degenerate Hamiltonian systems with two degrees of freedom. In the class of book billiards found topological obstacles to realizability.

On Critical Level Sets of Some two Degrees of Freedom Integrable Hamiltonian Systems

1998 ◽  
Vol 50 (1) ◽  
pp. 134-151
Author(s):  
Christine Médan
Keyword(s):  
Large Class ◽  
Hamiltonian Systems ◽  
Geodesic Flow ◽  
Level Sets ◽  
Degrees Of Freedom ◽  
Critical Level ◽  
Integrable Hamiltonian Systems ◽  
Two Degrees Of Freedom ◽  
Lagrange Top

AbstractWe prove that all Liouville's tori generic bifurcations of a large class of two degrees of freedom integrable Hamiltonian systems (the so called Jacobi–Moser– Mumford systems) are nondegenerate in the sense of Bott. Thus, for such systems, Fomenko's theory [4] can be applied (we give the example of Gel'fand–Dikii's system). We also check the Bott property for two interesting systems: the Lagrange top and the geodesic flow on an ellipsoid.

Sign in / Sign up

Export Citation Format

Share Document