scholarly journals Singularities of integrable Liouville systems, reduction of integrals to lower degree and topological billiards: Recent results

2019 ◽  
Vol 46 (1) ◽  
pp. 47-63 ◽  
Author(s):  
A.T. Fomenko ◽  
V.V. Vedyushkina
Keyword(s):  
Differential Geometry ◽  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Lower Degree ◽  
State University ◽  
Integrable Hamiltonian Systems ◽  
Two Degrees Of Freedom ◽  
Moscow State University

In the paper we present the new results in the theory of integrable Hamiltonian systems with two degrees of freedom and topological billiards. The results are obtained by the authors, their students, and participants of scientific seminars of the Department of Differential Geometry and Applications, Faculty of Mathematics and Mechanics at Lomonosov Moscow State University.

scholarly journals Complete commutative subalgebras in polynomial poisson algebras: A proof of the Mischenko-Fomenko conjecture

2016 ◽  
Vol 43 (2) ◽  
pp. 145-168 ◽  
Author(s):  
Alexey Bolsinov
Keyword(s):  
Poisson Bracket ◽  
Lie Algebras ◽  
Hamiltonian Systems ◽  
Dual Space ◽  
State University ◽  
Integrable Hamiltonian Systems ◽  
Finite Dimensional ◽  
Complete Set ◽  
Commutative Subalgebras ◽  
Moscow State University

The Mishchenko-Fomenko conjecture says that for each real or complex finite-dimensional Lie algebra g there exists a complete set of commuting polynomials on its dual space g*. In terms of the theory of integrable Hamiltonian systems this means that the dual space g* endowed with the standard Lie-Poisson bracket admits polynomial integrable Hamiltonian systems. This conjecture was proved by S. T. Sadetov in 2003. Following his idea, we give an explicit geometric construction for commuting polynomials on g* and consider some examples. (This text is a revised version of my paper published in Russian: A. V. Bolsinov, Complete commutative families of polynomials in Poisson?Lie algebras: A proof of the Mischenko?Fomenko conjecture in book: Tensor and Vector Analysis, Vol. 26, Moscow State University, 2005, 87?109.)

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