On certain canonoid transformations and invariants for the parametric oscillator

In classical mechanics a transformation in phase space is said to be canonoid if it maps only some Hamiltonian systems into Hamiltonian systems. Once canonoid transformations are considered, these systems can be classically described by means of Lagrangians or Hamiltonians other than the conventional ones. In this context, a basic role is played by the dynamical invariants generated through the Poisson brackets of the new independent variables in phase space. Here we obtain the explicit form of special canonoid transformations of the polynomial type for systems which can be classically described by an equation of the parametric oscillator type and discuss some algebraic properties shown by the associated dynamical invariants.