Jacobi structures on real two- and three-dimensional Lie groups and their Jacobi–Lie systems

Integrable bi-Hamiltonian systems by Jacobi structure on real three-dimensional Lie groups

“…The general solution for the Jacobi equations yields the general form of the Jacobi structures on a manifold M [10]. We have obtained the Jacobi structures on real three-dimensional Lie groups as a smooth manifold in [4]. To find these Jacobi structures, one must determine the vielbein e µ a for the Lie groups, and for this, one must find the left-invariant one-form on the Lie group:…”

“…( 9) and (10), one can obtain the Jacobi structures Λ and E on the Lie group. In [4] we have obtained all Jacobi structures on three-dimensional Lie algebras and their Lie groups. Here, we will consider those structures such that after Poissonization of them (see relation (2) ) the resulting Poisson structures are nondegenerate.…”

“…Poisson manifolds play an important role as phase spaces of classical mechanics. In [4], we have classified all Jacobi structures on real three-dimensional Lie groups. In this work, using the Poissonization of the Jacobi structure (Λ, E) on M [5], we convert the Jacobi structure (Λ, E) on M into the Poisson structure on M ×R and we consider those Poisson structures that are non-degenerate, and so define symplectic structures.…”

“…Note that previously in[4] we have obtained all equivalence classes of Jacobi structure on a three-dimensional Lie group.Here we use only one of those equivalence classes, because on other classes after Poissonization the obtained Poisson brackets are degenerate or singular[13].…”

Contact Lie systems

“…In this context, this work investigates Lie systems possessing a VG Lie algebra of Hamiltonian vector fields relative to a contact structure. Contact Lie systems can be considered as a particular case of Jacobi-Lie systems (see [2,3,27]), which where first introduced in [27], but that work just contained one example of contact Lie systems, without analysis those properties typical for contact Lie systems. In fact, [27] was mostly dealing with Jacobi-Lie systems on one and two-dimensional manifolds, which do not retrieve contact Lie systems, apart from the trivial ones given by a one-dimensional VG Lie algebra on R.…”