scholarly journals The reduction of the degree of integrals of hamiltonian systems with the help of billiards

2019 ◽  
Vol 486 (2) ◽  
pp. 151-155
Author(s):  
V. V. Vedyushkina ◽  
A. T. Fomenko
Keyword(s):  
Lie Algebra ◽  
Hamiltonian Systems ◽  
Integrable Systems ◽  
Degrees Of Freedom ◽  
Integrable Hamiltonian Systems ◽  
Two Degrees Of Freedom ◽  
Quadratic Integral ◽  
High Degree

In the theory of integrable Hamiltonian systems with two degrees of freedom there are widely known integrable systems whose integrals have a high degree, namely 3 and 4: the Kovalevskaya system and its generalizations - the Kovalevskaya - Yahya system and the Kovalevskaya system on the Lie algebra so(4), Goryachev-Chaplygin-Sretensky, Sokolov and Dullin-Matveyev. The article shows that using integrable billiards bounded by arcs of confocal quadrics decreases the degree of integrals 3 and 4 of these systems fo some isoenergy 3-surfaces. Moreover, the integrals of degree 3 and 4 reduce to the same canonical quadratic integral on billiards.

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.

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