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Here subscripts t, y and denote the partial derivatives with respect to time t and coordinates y, , respectively.

On the right of Eq.(6) the expressions of both square brackets are equal to zero because of following relations:

i) of the classical equation of motion

where is the same potential, that is into (3), and

ii) of the expression for the classical Lagrang function L(t)

so that the function

makes a sense of an action integral.

Into Eq.(6)

By deduction of Eq.(6) we made use of an potential energy expansion in the form

It is obvious that the expansion (11) is correct in the case when a classical trajectory is close to a quantum one.

Thus we get the equation for the function in the form

We pay attention here to three originating moments: 1) Equation (12) is the Schrödinger equation again, but without an external force. 2) We have the system of two equations of motion: quantum Eq.(12) and classical Eq.(7). In a general case these equations make up the system of bound equations, because the coefficient k can be a function of classical trajectory, . As we show below a connection between Eqs. (12) and (7) arises in the case, if classical Eq. (7) is nonlinear. 3) Classical Eq.(7) contains some dissipative term, and so makes sense of a dissipative coefficient. The arising of dissipation just into the classical equation is looked quite naturally - a dissipation has the classical character.

Let us assume that is the potential energy of a linear harmonic oscillator

where is the certain constant. Then we have

and

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