_{1}

^{*}

In work, dynamics of the spherical loaded clots is studied. For the self-coordinated description of non-stationary processes model representation of potential, obviously time-dependent and allowing construction movement integral is used. Classical and quantum tasks are considered.

At research of the self-coordinated systems of charged particles in theories of accelerators, in physical electronics, and in physics of plasma movement, integrals in some cases play the defining role theories of bunches. It is possible to specify the theory of electronic rings, the theory of rigidly focusing systems, the kinetic theory of quasistationary conditions of bunches (see [

We will consider, further, spherically a symmetric task. Hamilton-Jacobi’s equation in this case has an appearance (see [

Here

where

Size of

We will pass from to the invariant remaining at a certain dependence of potential function from r and t.

We will consider a look Hamiltonian:

Here

where

Density of particles is expressed by integral in phase space:

We will present an element of phase space in the form:

Averaging on

Thus density of current of

When using variables:

In order that in both members of equation of Poisson there was an identical dependence on multiplier

Then follows from Poisson’s equation:

Constant

Density of particles can be written down in a look:

and density of current:

where

Decisions for the potential of

The given expressions for density and current satisfy to the continuity equation:

Because

Conditions under which there could be states described here demand special research.

To Hamiltonian (1.2), there corresponds Shrӧdinger’s equation of the following look:

If to enter new variables

That for

As well as in the previous section, it is considered that

possible to find the private solution of the difficult non-stationary self-coordinated problem on dynamics of the charged quantum ensemble. Thus Schrӧdinger’s equation should be added with Poisson’s equation for the potential of field determined by own charge.

The left member of equation after transition from r to variable

was carried out at a certain dependence of function of distribution on the interfaced movement integral, in the case described by Schrodinger’s equation, existence of the specified multiplayer can be reached at a certain way of division of variables. We will put in (2.2)

These equations have to be added with Poisson’s Equation. Density of a charge has an appearance:

We will enter, further, dimensionless sizes:

We will put, further,

Results are represented on

where

4-D spherical clot

Schrödinger’s equation for a particle in the non-stationary field described by the potential of a look (1.1) has an appearance:

We will consider a 4-dimensional case. We will enter, as well as in the previous sections of variables

In (3.2) point means derivative

We will put, further,

This equation differs from the considered equation for a 3-dimensional case first, that the size equivalent to a square of the full moment is considered equal to zero here, and, above all-in the right part don’t have composed,

proportional

The charge density determined by function

and the equation for potential

As in variables

Thus, we have nonlinear system of the ordinary differential equations of the 4th order:

We will enter dimensionless variables in (3.4): we will designate:

As it is possible to see in

The problems studied above were considered also in works [

Alexander Chikhachev, (2016) Dynamics of the Spherical Charged Clots. Journal of Modern Physics,07,543-551. doi: 10.4236/jmp.2016.76057