Integrabilidad de sistemas no lineales hamiltonianos con N grados de libertad

The $\mathfrak{sl}_{2}(\mathbb{R})$ coalgebra symmetry and the superintegrable discrete-time systems

“…Finally, we note that the present construction can be applied to other Lie-Poisson algebras, especially the ones related to classified Lie algebras for which the Casimir functions are known, see [35,[44][45][46][47][48][49]. In the continuous setting this was done by Ballesteros and Blasco [4,16]. We note that the most natural extension would be the two-photon or h 6 Lie-Poisson algebra [66], a Lie-Poisson algebra containing many other interesting Lie-Poisson algebras as subalgebras, including sl 2 (R), whose associated Hamiltonian integrable systems have been discussed in [6,16] The figures in this paper are pdf and were produced in python using the libraries numpy [30] and mathplotlib [32].…”

“…Moreover, this realization is generic, since N = 1, m = 3 and r = 1. So, following the literature (see for example [11,16]), we will call this realization the generic one d.o.f. symplectic realization of sl 2 (R).…”

“…Although the many advances pursued in the framework of superintegrable systems with an arbitrary number of d.o.f. using the method described above, which has also been applied to the analysis of many other Lie-Poisson coalgebras [4,16], the problem of finding MS subcases is usually left open. This is due to the fact that additional integrals, not directly obtainable following this algebraic approach, may arise.…”

The sl2(R) coalgebra symmetry and the superintegrable discrete-time systems

“…Moreover, this realisation is generic, since N = 1, m = 3 and r = 1. So, following the literature (see for example [7,16]), we will call this realisation the generic one d.o.f. symplectic realisation of ( ) sl …”

“…Although the many advances pursued in the framework of superintegrable systems with an arbitrary number of d.o.f. using the method described above, which has also been applied to the analysis of many other Lie-Poisson coalgebras [8,16], the problem of finding MS subcases is usually left open. This is due to the fact that additional integrals, not directly obtainable following this algebraic approach, may arise.…”