  For a free falling particle moving in a media which has quadratic velocity
force effect on the particle, two equivalent constants of motion, with units of
energy, two Lagrangians, and two Hamiltonians are deduced. These quantities
describe the dynamics of the same classical system. However, their quantization
and the associated statistical mechanics (for an ensemble of particles)
describe two completely different quantum and statistical systems. This is
shown at first order in the dissipative parameter.
