Geometrical dynamics and hidden symmetries as tools to describe chaotic systems

In this work we consider a particle with abelian gauge charge in curved space as a model to describe some hamiltonian systems. We can use it as a toy model for characterization of chaotic regime for some systems in terms of the tensors related to metric space. In this case a difierent criteria for unstable behavior instead of Lyapunov one is used, since there is a dynamical curvature and others related quantities, associated with metric tensor, as a tool to determine this behavior. The formalism to achieve that is through obtainment of hidden symmetries of the phase space, generated by either symmetric or antisymmetric tensor flelds constructed from Killing tensor and Killing-Yano equations respectively in curved background space. For symmetric case it is obtained conserved quantities in terms of higher powers of particle momenta while for antisymmetric one there is indication of conserved currents. From the symmetric tensor, by means of symmetry generated by flrst order momenta combinations, it is deflned a dual metric, corresponding to the inverse of the second rank Killing tensor, whose geometrical invariants are constructed. This dual metric, whose meaning will be discussed, permits to flnd the singular structure of some related phase space operators which has connections with the hidden symmetries mentioned. Using these quantities we obtain a geometrical formulation for indication of chaotic regime in some dynamical systems.