Mechanics historically developed in two main directions One

Mechanics historically developed in two main directions. One branch, which is customarily called classical mechanics, comes directly from Newton\'s laws of motion. The problem is in determining the behavior of a charged particle if the acting forces are known at any specific time, and it possesses an unambiguous solution. According to the second approach, commonly referred to as analytical mechanics [7], the study of equilibrium and motion is based on two main quantities: the kinetic 8-Bromo-cAMP, sodium salt and the force function; the latter is often replaced by potential energy. These two fundamental scalars contain the total dynamics of the most complex material systems, provided, however, that these scalars are assumed as a basis of some principle, rather than just an equation. Let a particle occupy definite positions at some instants of time t1 and t2, characterized by two sets of coordinate values P1 and P2. Then the particle moves between these points in such a way that the time integral of the difference between the kinetic and potential energies should be of the least possible value. The integrand is called the Lagrange function of the system, and the integral is called the action. The principle of least action states the following: The actual motion occurring in nature is that for which the action takes the least value. Our study operates within the framework of variation mechanics, as this ideology seems more natural for the analytical approach to the synthesis of new corpuscular-optical systems. Furthermore, it should be noted that all calculations are carried out with dimensionless quantities. The expediency of this choice, and the different character of the quantities introduced depending on the type of problem is detailed in monographs [5,8]. We are going to consider a particular case with direct relevance for our investigation where only the electrostatic field is involved in the problem. The actual motion of the particle in Euclidean space with Cartesian coordinates X, Y, Z and in real time t depends on the particle\'s charge and mass, the effective electric fields and the initial conditions. All of this is expressed in dimensional quantities and as a whole makes up a physical model of the corpuscular-optical system. Meanwhile, calculations and analysis of the results can be easily simplified by eliminating the standard physical units and introducing new special ones unifying all the relationships and making them purely dimensionless. The change to dimensionless quantities replaces the actual physical problem by a simpler mathematical model of motion and allows conducting our study with as much generality as possible. Each physical trajectory X(t), Y(t), Z(t) is put in correspondence with a dimensionless curve x(τ), y(τ), z(τ). To find the true dimensional trajectory from the dimensionless curve, it is enough to multiply the functions x(τ), y(τ), z(τ) by the given linear scaling factor l. Let us show the transition to the dimensionless model of motion for our problem. Let us write the Hamilton–Jacobi equation for the motion of the particle with the charge q and the mass m in an electrostatic field with the potential Ф(x, y) in the plane (x, y): where E is the total energy of the particle and S is the action (a scalar quantity). Given that the partial derivatives of the action with respect to the coordinates are equal to the respective momenta
Let us introduce a dimensional quantity l which will serve as a characteristic dimension of the device, and use it as a connection between the dimensional Cartesian coordinates X and Y and the dimensionless scalars x and y:
Aside from l, let us set a characteristic value of the potential Ф0, for example, the maximum potential in magnitude at the boundary of the system:
Let us introduce the dimensionless time τ and the dimensionless total energy h by the formulae where T is some characteristic time (that we have not defined yet) and H0 is the total energy of the particle.