## 2 Materials and Methods  
As is known, electron-optical systems with a straight optical axis are most often used in electronic instrument making. At the same time, in practice, there are many cases when it becomes necessary to use focusing systems with a curvilinear optical axis [29; 30; 31; 32]. In this paper, we will first consider the problem of constructing a method for studying cathode lenses with a curvilinear optical axis. From the general theoretical materials obtained in this case, results can be derived that can also be used to study emission systems with a straight optical axis. The motion of a charged particle with charge \(e\) and mass \(m\) in an electromagnetic field with electric field strength \(\overset{\rightarrow}{E}\) and magnetic field strength \(\overset{\rightarrow}{H}\), as is known, satisfies the equation  
\[\frac{d\left(m\overset{\rightarrow}{V}\right)}{dt}=e\overset{\rightarrow}{E} +\frac{e}{c}\left[\overset{\rightarrow}{V},\overset{\rightarrow}{H}\right] \tag{1}\]  
where \(\overset{\rightarrow}{V}\)--the speed of the particle, \(t\)--time. Note that in the relativistic case, according to the special theory of relativity, the value of the mass \(m\) of a particle moving at high speed and the value of the rest mass of the same charged particle \(m_{0}\) are related by the equation  
\[m=\frac{m_{0}}{\sqrt{1-\frac{V^{2}}{c^{2}}}}, \tag{2}\]  
where \(c\) is the speed of light and \(V\) is the absolute value of the speed of a charged particle. The magnitudes of the electric field strength and the magnetic field strength are generally determined as functions of spatial coordinates and time  
\[\overset{\rightarrow}{E}=\overset{\rightarrow}{E}\left(\overset{ \rightarrow}{R},t\right)\text{, }\overset{\rightarrow}{H}=\overset{ \rightarrow}{H}\left(\overset{\rightarrow}{R},t\right). \tag{3}\]  
Here, the spatial coordinates are specified using the radius vector \(\overset{\rightarrow}{R}\). Vectors \(\overset{\rightarrow}{E}\) and \(\overset{\rightarrow}{H}\), according to field theory, satisfy the equations  
\[\overset{\rightarrow}{div}\overset{\rightarrow}{H}=0\text{, }\rot\overset{ \leftarrow}{E}=-\frac{1}{c}\frac{\partial\overset{\rightarrow}{H}}{\partial t}, \tag{4}\]  
\[\overset{\rightarrow}{div}\overset{\rightarrow}{E}=4\pi\rho\text{, }\rot \overset{\rightarrow}{H}=-\frac{1}{c}\frac{\partial\overset{\rightarrow}{E}}{ \partial t}+\frac{4\pi}{c}\overset{\rightarrow}{j}. \tag{5}\]  
In the last equations, the symbol \(\rho\) denotes the space charge distribution function, \(\overset{\rightarrow}{j}\)--beam current density. Note that Equation (4) constitutes the first pair of [PERSON]'s equations, and Equation (5) constitutes the second pair of [PERSON]'s equations. Electromagnetic fields, however, are also characterized by scalar electrostatic potential \(\phi\) and vector magnetic potential \(\overset{\rightarrow}{A}\), which are related to \(\overset{\rightarrow}{E}\) and \(\overset{\rightarrow}{H}\) as follows  
\[\overset{\rightarrow}{E}=-\frac{1}{c}\frac{\partial\overset{\rightarrow}{A }}{\partial t}-grad\phi\text{, }\overset{\rightarrow}{H}=rot\overset{ \rightarrow}{A}. \tag{6}\]  
The magnetic field strength in the absence of a vortex component can also be determined through the value of the scalar magnetic potential \(\omega\)  
\[\overset{\rightarrow}{H}=-grad\omega. \tag{7}\]  
Equation (1), taking into account (6), takes the form  
\[\frac{d\left(\overset{\rightarrow}{mV}\right)}{dt}=-e\left(\frac{\partial \overset{\rightarrow}{A}}{\partial t}+grad\phi\right)-\frac{e}{c}\left[ \overset{\rightarrow}{V},\overset{\rightarrow}{H}\right]. \tag{8}\]  
Equation (8) is written in vector form and represents, as is known, the general equation of motion of charged particles in electromagnetic fields. For constant electromagnetic fields, the vectors \(\overset{\rightarrow}{E}\) and \(\overset{\rightarrow}{H}\) do not depend on time and are functions of spatial coordinates only.