The foundations of mechanics were created by generations of mathematicians and astronomers trying to describe the motion of such complex mechanical system as the planet system. Lagrange occupies a special place among them. Benefiting from his achievements W. R. Hamilton in 1834 stated that [1] ”Among the successors of those illustrious men (autor’s remark Galileo, Newton), Lagrange has perhaps done more than any other analyst to give extent and harmony to such deductive researches, by showing that the most varied consequences respecting the motions of systems of bodies may be derived from one radical formula; the beauty of the methods so suiting the dignity of the results as to make of his great work a kind of scientific poem”.

Euler also played a great role in the creation of the mechanics of many bodies [2]. However, it was Lagrange who formulated the function called his name, which is a difference between kinetic energy and potential energy.
$$\begin{array}{}fL={E}_{k}-{E}_{p}\end{array}$$(1)

where: *E*_{k} - kinetic energy, *E*_{p} - potential energy.

The kinetic energy is the sum of the kinetic energy of all components and, similarly, potential energy is the sum of the potential energy of all components. It is worth noting that Lagrange was considering a conservative arrangement. This means that there are no energy dissipation components in the Lagrange function. For the electromechanical system, a distinction between kinetic electrical and mechanical energy, as well as potential electric and mechanical energy, can be made. For electromechanical rotating devices, the independent (input) mechanical coordinate, variable of the model, is the rotor rotation angle, and the electric input coordinate is the electric charge of the electrical circuit. A characteristic feature of such a system is the lack of potential energy, both electrical and mechanical. But in the Lagrange function, there is a mechanical coordinate in the form of the dependence of the inductance of the electrical circuit from the angle of rotation. In addition, there are the electrical and mechanical coordinate time derivatives, velocities occurring in the expressions of kinetic energy. After the distinction of the generalised coordinates and their velocities, the Lagrange equation of the first kind (Lagrange - d’Alembert equation) is derived:
$$\begin{array}{}{\displaystyle \left(\frac{d}{dt}\frac{\mathrm{\partial}fL}{\mathrm{\partial}{\dot{q}}_{e}}\right)\delta {q}_{e}+\left(\frac{d}{dt}\frac{\mathrm{\partial}fL}{\mathrm{\partial}{\dot{q}}_{m}}\right)\delta {q}_{m}=0}\end{array}$$(2)

Above equation is the sum of products derived from Euler’s variation products of Lagrange function and virtual displacements of individual generalised variables. When these displacements are independent, the system is holonomic and the Euler variation derivatives of the Lagrange function describing the virtual forces balances of the elements of the considered system are called Euler - Lagrange equations:
$$\begin{array}{}{\displaystyle \frac{d}{dt}\frac{\mathrm{\partial}fL}{\mathrm{\partial}{\dot{q}}_{e}}=0}\end{array}$$(3)
$$\begin{array}{}{\displaystyle \frac{d}{dt}\frac{\mathrm{\partial}fL}{\mathrm{\partial}{\dot{q}}_{m}}-\frac{\mathrm{\partial}fL}{\mathrm{\partial}{q}_{m}}=0}\end{array}$$(4)

The equations above are called the equations of motion and they fully describe the dynamics of the system. But if the virtual displacements are not independent, then the system is a nonholonomic one. The Lagrange - d’Alembert equation should be used in the description of the system and on this basis, the character of interactions between mechanical and electrical system should be determined, “there is no doubt that the correct equations of motion for nonholonomic mechanical systems are given by the Lagrange - d’Alembert principle” [3] (page 236).

From Ref. [3] it also follows that experimental studies should be considered for the analysis of interactions between elements of a complex dynamic system, “the question of applicability of the nonholonomic model … cannot, in any specific situation, be solved within the framework of an axiomatic scheme without recourse to experimental results” [3] (page 244).

The Lagrange function is most commonly used to describe motion equations of electromechanical systems [4]. It is described by generalized coordinates and velocities as the difference between kinematic and potential energy (1).

In general, potential energy is often not present in electromechanical systems. The Lagrange function does not describe energy flow. It describes relation between particular generalized coordinates and their derivatives. Hence, the holonomicity of the system should be discussed on the basis of Lagrange function of homogeneous and conservative systems. If, electric coordinates (electric charges) *q*_{e} and mechanical coordinates (angles) *q*_{m} are discriminated among generalized coordinates the Lagrange function of the system may be described as:
$$\begin{array}{}{\displaystyle fL=\frac{1}{2}L({q}_{m})\cdot {\left(\frac{d{q}_{e}}{dt}\right)}^{2}+\frac{1}{2}J\cdot {\left(\frac{d{q}_{m}}{dt}\right)}^{2}}\end{array}$$(5)

Based on the Lagrange function, Lagrange - d’Alembert equation may be formulated [5]. For the electromechanical system, it is in form presented as Equation (2).

According to Ref. [5] (page 90), the variations *δq*_{e} and *δq*_{m} from Lagrange - d’Alembert equation are independent only for the holonomic system, and only for that system, we obtain the Lagrange equations of the second kind (Euler - Lagrange equations) in form (3) and (4).

In order to analyze the motion equation of the electromechanical system, the reluctance motor with a single pair of stator and rotor poles was chosen – Figure 1.

Figure 1 A general diagram of the reluctance motor

A construction of a reluctance motor (RM) is very simple because it has no brushes, magnets and rotor windings. Consequently, a reluctance motor is highly reliable and fault-tolerant. The second advantage of reluctance motors is that the friction torque of the motor depends mainly on friction of bearing, it is rather small and can easily be regulated.

The major drawback of the motor is that it starts to operate only for particular angles of the rotor versus stator. However, the electric circuit is described only by means of one equation. Therefore, it facilitates the analysis and experimental verification of the mathematical model.

The mathematical model of an electromechanical device consists of electrical and mechanical parts, described only by two equations. The analysis of systems is based on the measurement of the electric circuit and the mechanical system parameters. The mechanical and electric equations are most often formulated as follows [6, 7]:
$$\begin{array}{}{\displaystyle J\frac{d\omega}{dt}+k(\omega )+{T}_{L}=Te}\end{array}$$(6)
$$\begin{array}{}{\displaystyle L(i,\phi )\frac{di}{dt}+\frac{dL(i,\phi )}{d\phi}i\omega +{R}_{s}i={U}_{s}}\end{array}$$(7)

where: *J* - moment of inertia, *k*(*ω*) - torque of friction, *T*_{L} - torque of load, *∂Lφ*/*∂φ* - derivative of inductance versus the rotor rotation angle, *U*_{s} - terminal voltage, *i* - phase current, *R*_{s} - winding resistance and *T*_{e}, the electromagnetic torque, is defined as [8]:
$$\begin{array}{}{\displaystyle {T}_{e}=\frac{1}{2}\frac{\mathrm{\partial}L(\phi )}{\mathrm{\partial}\phi}{i}^{2}}\end{array}$$(8)

Multiplying the equation (6) by the angular velocity *ω*, gives a power balance equation for the mechanical part (9). Multiplying the equation (7) by the current *i*, yields the power equation of electric equation (10):
$$\begin{array}{}{\displaystyle J\frac{d\omega}{dt}\omega +k(\omega )\omega +{T}_{L}\omega =\frac{1}{2}\frac{\mathrm{\partial}L(\phi )}{\mathrm{\partial}\phi}\omega {i}^{2}}\end{array}$$(9)
$$\begin{array}{}{\displaystyle L(\phi )\frac{di}{dt}i+\frac{\mathrm{\partial}L\phi}{\mathrm{\partial}\phi}\omega {i}^{2}+{R}_{s}{i}^{2}={U}_{s}i}\end{array}$$(10)

A term on the right side of the equation (9) represents the power which is transferred from the electrical part of motor. Terms on the left side of the equation (9) represent the power related to the changes of rotor kinetic energy, friction power, and load power.

The first term on the left side of the equation (10) defines the rate of change of magnetic energy which is cumulated in the motor inductance. The second term describes the intensity of energy which is transformed to the mechanical system. The last term represents the thermal loss energy of stator winding. A term on the right side describes the electric power supplied to the motor. It should be emphasized, that the power transferred to the mechanical system from the electric system in the equation (10) is different than the intensity of energy transferred to the mechanical system in the equation (9).

Some explanation of the power difference may be found in [9, 10, 11]. In these papers, the power transformed from electric circuit is split into two parts:
$$\begin{array}{}{\displaystyle \frac{1}{2}L\frac{d{i}^{2}}{dt}+\frac{1}{2}{i}^{2}\frac{dL}{d\phi}\omega +\frac{1}{2}{i}^{2}\frac{dL}{d\phi}\omega +{R}_{s}{i}^{2}={U}_{s}i}\end{array}$$(11)

and then it is written as:
$$\begin{array}{}{\displaystyle \frac{d}{dt}(\frac{1}{2}L{i}^{2})+\frac{1}{2}{i}^{2}\frac{dL}{d\phi}\omega +{R}_{s}{i}^{2}={U}_{s}i}\end{array}$$(12)

The first term is interpreted as the rate of magnetic energy accumulation, the second one is described as power transformed to the mechanical system. It should be noted, that in a steady state, the energy stored in the coil has two terms: the first one - $\frac{1}{2}L\frac{d{i}^{2}}{dt},$ in the steady state equals zero, and the second one - $\frac{1}{2}{i}^{2}\omega \frac{dL}{d\phi},$ has the same form as the power transferred to the mechanical system. In a steady state the transfer of electric power to mechanical output power is continuous. It means that accumulation rate of magnetic energy is also constant and the magnetic energy of motor increases continuously. Is it possible?

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