scholarly journals Force Evolutionary Billiards and Billiard Equivalence of the Euler and Lagrange Cases

2021 ◽  
Author(s):  
V. V. Vedyushkina ◽  
A. T. Fomenko
Keyword(s):  
Hamiltonian Systems ◽  
Integrable Systems ◽  
Integrable Hamiltonian Systems

Abstract A class of force evolutionary billiards is discovered that realizes important integrable Hamiltonian systems on all regular isoenergy 3-surfaces simultaneously, i.e., on the phase 4-space. It is proved that the well-known Euler and Lagrange integrable systems are billiard equivalent, although the degrees of their integrals are different (two and one).

DEFORMATIONS OF LOOP ALGEBRAS AND CLASSICAL INTEGRABLE SYSTEMS: FINITE-DIMENSIONAL HAMILTONIAN SYSTEMS

2004 ◽  
Vol 16 (07) ◽  
pp. 823-849 ◽  
Author(s):  
T. SKRYPNYK
Keyword(s):  
Lie Algebras ◽  
Hamiltonian Systems ◽  
Spectral Parameter ◽  
Integrable Systems ◽  
Integrable Hamiltonian Systems ◽  
Infinite Dimensional ◽  
Loop Algebras ◽  
Finite Dimensional ◽  
Quasigraded Lie Algebras ◽  
Classical Integrable

We construct a family of infinite-dimensional quasigraded Lie algebras, that could be viewed as deformation of the graded loop algebras and admit Kostant–Adler scheme. Using them we obtain new integrable hamiltonian systems admitting Lax-type representations with the spectral parameter.

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.

MODIFIED ALGEBRAIC STRUCTURES FOR SOME COMPLETELY INTEGRABLE SYSTEMS

2001 ◽  
Vol 16 (38) ◽  
pp. 2457-2462
Author(s):  
STANISŁAW P. KASPERCZUK
Keyword(s):  
Hamiltonian Systems ◽  
Quantum Groups ◽  
Algebraic Structure ◽  
Integrable Systems ◽  
Algebraic Structures ◽  
Completely Integrable Systems ◽  
Integrable Hamiltonian Systems ◽  
Completely Integrable ◽  
Completely Integrable Hamiltonian Systems

We modify the algebraic structure of a Poisson bialgebra by considering the deformed coproduct [Formula: see text] and deformations of an algebra e(2). These modifications lead to the quantum groups and provide new classes of completely integrable Hamiltonian systems.

Effects of symmetry on instability-driven chaotic energy transport in near-integrable, Hamiltonian systems

1994 ◽  
Vol 37 (4-5) ◽  
pp. 323-339 ◽  
Author(s):  
M.Gregory Forest ◽  
Christopher G. Goedde ◽  
Amarendra Sinha
Keyword(s):  
Hamiltonian Systems ◽  
Energy Transport ◽  
Integrable Hamiltonian Systems

scholarly journals Two-Photon Algebra and Integrable Hamiltonian Systems

2001 ◽  
Vol 8 (sup1) ◽  
pp. 18-22 ◽  
Author(s):  
Angel Ballesteros ◽  
Francisco J Herranz
Keyword(s):  
Hamiltonian Systems ◽  
Integrable Hamiltonian Systems ◽  
Two Photon
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