CCT251545 analogue Then Eq can be rewritten in the

Then Eq. (3) can be rewritten in the following form, replacing the differentiation with respect to the dimensionless time by a variable dotted:
Since the parameter T is arbitrary, let us set the following relationship between the newly-emerged multipliers: which separates the common multiplier out of Eq. (7), and this multiplier can be omitted. The Hamilton equation in dimensionless quantities takes the form
The proposed procedure for synthesizing new potential structures with useful properties (perfect angular focusing in the chosen plane) is represented by an algorithm with the following steps.
The Hamilton–Jacobi equation for the action, which can be defined as truncated because it depends only on the coordinates, but not on time, is in the case of planar motion (9) converted to conformal coordinates u and v by means of an analytic function of the complex variable ω:
Thus, the old coordinates x and y are the functions of the new ones, i.e., u and v. Eq. (10) can be interpreted physically as an actual Hamilton–Jacobi equation in the new coordinates u and v, describing the motion in the plane, which occurs under the action of a field with the potential φ**(u, v) at a zero total CCT251545 analogue = 0. All trajectories on the (х, у) plane of fixed energy h are converted by this transformation to trajectories with a zero total energy in the (u, v) plane. For finding these trajectories it is sufficient to use the transformation formulae by substituting the expressions for the х(t) and у(t) trajectories on the (х, у) plane into them. The families of isoenergetic trajectories change their forms depending on z(ω), but preserve some of their properties. Conformal mappings are known to retain the intersection angles of curves and decrease (or increase) short segments depending on the magnitude of the analytic function derivative in the location of the segment [9]. Two important physical conclusions follow from here. Firstly, a set of trajectories emerging from a point is mapped onto the (u, v) plane to another set emerging from a point and having the same angular width as the original one. Secondly, if the set on the (х, у) plane has been focused to a short segment, then the transformed set is focused in the (и, v) plane to another short segment, decreased (or increased) by a factor equal to the magnitude of the mapping function derivative z(ω). In those cases when the (х, у) and the (u, v) planes are assumed to be the planes of symmetry of two different three-dimensional harmonic fields, and the functions φ(x, y) and φ**(u, v) to be the potential distributions in these planes, the afore-mentioned procedure of transforming one type of fields into another allows to flexibly reconstruct the structure of a field in the vicinity of the plane of symmetry. Let us illustrate this assertion by examples.
Field transformation examples Let us choose a two-dimensional function as the initial potential. De facto, such a distribution can be obtained for a harmonic field of hyperboloids of revolution Φ = x2 + y2 − 2z2 in the plane of symmetry φ = Φ = 0. It is easy to verify that the particles that emerged from the point (x0, y0) at various angles will get to the point (−x0, −y0) symmetrical with respect to the origin in time . For example, if we take (1, 0) as a starting point, then after a specified period of time, all the particles will focus into the point (−1, 0). The particle trajectories calculated in the Mathematica software [10] are shown in Fig. 1. Variant 1. Let us transform potential (13) by a logarithmic function according to Eq. (10). Setting the total energy h = 1 and taking into account that , we obtain:
The trajectories in the new potential can be determined both by the transformation formulae (12) and by integrating the equations of motion. The particle trajectories are shown in Fig. 2. It can be seen that the conformal mapping (14) changes the divergent flux of particles into a parallel one.