Restricted Constant of Motion for the One-Dimensional Harmonic Oscillator with Quadratic Dissipation and Some Consequences in Statistic and Quantum Mechanics

A restricted constant of motion, Lagrangian and Hamiltonian, for the harmonic oscillator with quadratic dissipation is deduced. The restriction comes from the consideration of the constant of motion for the velocity of the particle either for v 0 or for v < 0. A study is done about the implications that these restricted variables have on the specific heat of a thermodynamical system of oscillators with this dissipation, and on the quantization of this dissipative system.