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Variational principles for systems with linear friction in quantum mechanics and hydrodynamics

Wagner H. J.

Zeszyty Naukowe Politechniki Świętokrzyskiej. Mechanika

1998

Z. 66

257-267

Employing the method of Frechet differentials, it is shown that variational pronciples can be given for certain phenomenological field wquatons of dissipative quantum theory. In particular, the Caldirola-Kanai equation and Kostin's nonlinear Schrodinger-Langevin equation are dealt with. As can be seen in the Madelung hydrodynamic picture, these equations describe irrotational probability flows under the influence of frictional force densities proportional to the flow velocity. The results obtained are then generalized to the case of the so-called Takabayasi-Schonberg extension of quantum mechanics where probability flows with nonzero vorticity are also permitted. In the Madelung picture, the field equations of the Takabayasi-Schonberg extension of quantum mechanics are highly similar to classical hydrodynamical ones. Utilizing this similarity, the considerations for the dissipative quantum systems are carried over to an analogus frictional system in classical hydrodynamics. The respective flow eqation constitutes a nonstationary extension of Darcy's law for the seepage of fluids in porous media.