The conservation of momentum is often used in controlling underactuated mechanical systems with symmetry. If a symmetry-breaking force is applied to the system, then the momentum is notconserved any longer in general. However, there exist forces linear in velocity such as the damping force that breakthe symmetry but induce a new conserved quantity in place of the original momentum map. We formalize the new conserved quantity which can be constructed by combining the time integral of a general damping force and the original momentum map associated with the symmetry. From the perspective of stability theories, the new conserved quantity implies the corresponding variable possesses the self recovery phenomenon, i.e. it will be globally attractive to the initial condition of the variable. We discover that what is fundamental in the damping-induced self recovery is not the positivity of the damping coefficient but certain properties of the time integral of the damping force. The self recovery effect and theoretical endings are demonstrated by simulation results using the two-link planar manipulator and the torque-controlled inverted pendulum on a passive cart. (This is an outcome of the collaboration with Soo Jeon at the University of Waterloo).