When one considers the dynamics of mechanical systems with a friction proportional to the velocity one obtains a system with the remarkable property that a symplectic form is transformed into a multiple of itself. The same phenomenon happens when one minimizes the action after discounting it by an exponential factor (these models are very common in economics when one minimizes the present cost and includes inflation). Then, the Euler Lagrange equations, lead also to conformally symplectic systems. We will present several results for this systems: 1) A KAM theory for these systems that leads to efficient algorithms. 2) Absence of Birkhoff invariants near Lagrangian tori. 3) Numerical experiments at the breakdown of tori 4) Analyticity domains of expansions for KAM tori in weakly dissipative systems (it is a singular perturbation, so series diverge and the domains do not include a ball). All these works are in collaboration with R. Calleja and A. Celletti, L. Corsi (the numerical work reported is by R. Calleja, A. Celletti, J.L - Figueras)