Seminarium KMMF "Teoria Dwoistości"

We provide a method of deformation of a class of Hamiltonian systems via Poisson—Hopf algebras that allows for the algebraic derivation of their constants of the motion and the geometric description of their dynamical properties. More specifically, we start by a non-autonomous Hamiltonian system on a Poisson manifold N whose dynamic is determined by a t-dependent Hamiltonian taking values in a finite-dimensional Lie algebra of functions isomorphic to an abstract Lie algebra g. The Hamiltonian system is then attached to the universal enveloping algebra U(g) and the Poisson algebra C∞(g*) relative to the Kirillov—Kostant—Souriau bracket. The deformed quantum algebras Uz(g) and Poisson—Hopf algebras Cz∞(g*), along with the induced symplectic foliations on g* for each z, allow for the deformation of the t-dependent Hamiltonian function of our original system. This originates a z-parametrized family of Hamiltonian systems on N whose constants of the motion can be derived through Casimir elements of Uz(g) and Cz∞(g*). The co-algebra structure of Cz∞(g*) enables us to derive multi-dimensional generalizations of the original Hamiltonian system. Among other applications, we deform a one-dimensional Winternitz—Smorodisnky oscillator to obtain a family of oscillators with a position-dependent mass and we provide their generalizations to other higher-dimensional manifolds.