By Mohsen Razavy

This publication discusses concerns linked to the quantum mechanical formula of dissipative platforms. It starts with an introductory evaluate of phenomenological damping forces, and the development of the Lagrangian and Hamiltonian for the damped movement. it truly is proven, as well as those tools, that classical dissipative forces is additionally derived from solvable many-body difficulties. an in depth dialogue of those derived forces and their dependence on dynamical variables can be awarded. the second one a part of this booklet investigates using classical formula within the quantization of dynamical structures below the effect of dissipative forces. the implications express that, whereas a passable option to the matter can't be chanced on, diversified formulations symbolize diverse approximations to the entire answer of 2 interacting platforms. The 3rd and ultimate a part of the booklet specializes in the matter of dissipation in interacting quantum mechanical platforms, in addition to the relationship of a few of those versions to their classical opposite numbers. a few very important functions, comparable to the speculation of heavy-ion scattering and the movement of a radiating electron, also are mentioned.

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De and D. Sperber, Phys. Rev. Lett. 40, 1123 (1978). V. I. D. A. Komar (Nova Science, Commack, 1988) p. 57. A. Pars, A Treatise on Analytical Dynamics, (John Wiley & Sons, New York, 1965). [16] H. Helmholtz, J. reine angew. Math. 100, 137 (1887). [17] J. Douglas, Trans. Am. Math. Soc. 50, 71 (1940). [18] M. Henneaux, Ann. Phys. (NY) 140, 45 (1982). [19] V. I. D. Skarzhinsky, Lebedev Physical Institute Preprint No (216), Moscow (1978). D. M. Lifshitz, Mechanics, (Pergamon Press, London, 1960) p.

E. 1: Scattering of the projectile P from the target T showing the degrees of freedom of the motion. Both P and T are assumed to be rigid spheres. 31) ^ {MP + MT)' and Vcoui and Vs are Coulomb and nuclear potentials respectively. Associated with these four degrees of freedom we can introduce a symmetric 4 x 4 matrix for the friction tensor kij. However to fit the observed data it seems that we need a coefficient of friction kr for the radial motion, and a coefficient kt for the relative sliding motion of the two nuclear surfaces [12] [13].

Note that L has the dimension of energy, but C has the dimension of power. 23) 18 Classical Dissipative Motion T as the sum of the quadratic terms in velocities, 1 3 2 ! C i<-M*J'*J' ( 3 - 24 ) and in addition we can have a "gyroscopic" term, 3 1 — y ^ XiGijXj. 25) Here the matrix elements M^ and Vij are real and symmetrical, while Gy s are asymmetrical, but all are constants. 2,3, (3-26) where B^ = Rij + Gij. 27) The theory of damped heavy ion scattering offers an interesting example of the application of the above-mentioned Lagrangian formulation.