On algebraic construction of certain integrable and super-integrable systems

Abstract

a b s t r a c tWe propose a new construction of two-dimensional natural bi-Hamiltonian systems associated with a very simple Lie algebra. The presented construction allows us to distinguish three families of superintegrable monomial potentials for which one additional first integral is quadratic, and the second one can be of arbitrarily high degree with respect to the momenta. Many integrable systems with additional integrals of degree greater than two in momenta are given. Moreover, an example of a super-integrable system with first integrals of degree two, four and six in the momenta is found.

Substantiation of the structure theory of design of technological machines and devices

“…According to [8][9][10][11][12][13], in theory the necessary condi tion is the existence of an idea, from which follows the goal (goals), the perspective in the direction from the study to the practical application of the results. In the opinion of the authors [3,4,7,[14][15][16][17] the whole theory potentially consists in concepts (terms), treating the con cept as an abstraction [18,19], a mental reflection of the general essential features of the object, the phenomenon, the scientific fact.…”

“…For this reason, for further research in the direction of creating a theory of design structures, one should use the features of constructing algebraic systems [10,11]. Using these mathematical tools it will be possible:…”

On a Trivial Family of Noncommutative Integrable Systems

“…4) which are the sum of the geodesic Hamiltonian T and potential energy V (q). According to [15,24,31,32], for natural Hamiltonians there is other representation for the vector field Y (1.2) Y = Λ 0 0 Π dq dp .…”

“…The next step is a search of integrals of motion for this Hamilton function H. If the recursion operator N at every point has n distinct functionally independent eigenvalues, we can say that M is a regular bi-Hamiltonian manifold. If the recursion operator N does not have this property then we can say that bi-Hamiltonian manifold M is irregular [24,31]. So, there are three different cases: 1) recursion operator produces the necessary number of integrals of motion;…”

“…Following [24] let us consider tensor field A, which is proportional to P A = ρ(q, p)P, 12) then equations (2.7) have the following partial solution…”