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

Ambiguities on the Hamiltonian formulation of the free falling particle with quadratic dissipation