Sergey V. Savel’kaev
Department of Special Devices and Technologies, Faculty of Optical and Special Systems Engineering, Siberian State University of Geosystems and Technologies, Novosibirsk, Russia.
DOI: 10.4236/jamp.2026.149179   PDF    HTML   XML   10 Downloads   75 Views   Citations

Abstract

This paper considers a mechanical system consisting of two bodies with masses $m_1$ and $m_2$ linked by a kinematic hinged-rod constraint in the form of a rigid straight rod $R$ , allowing them to rotate both relative to each other and to their center of mass. Based on D’ Alembert’s principle formulated taking into account the constraint axiom for a constrained material point, equations of motion are derived separately for each body ($m_1$ and $m_2$ ) and for their center of mass in the field of relative inertial forces of the diametrically opposite bodies $m_2$ and $m_1$ both in vacuum and in a dissipative medium. It is shown that time in the proper reference frames of these bodies flows differently than time in the center-of-mass frame, varying with the kinematic characteristics and mass ratio of the bodies. In the center-of-mass frame and the laboratory frame, time is absolute. In addition, the momentum distribution between the bodies of the closed mechanical system is analyzed, and the relationship between this distribution and its energy balance is established under various initial conditions, demonstrating that its motion in the relative inertial force field satisfies the laws of conservation of momentum and energy, as well as the principle of least action.

Keywords

Two-Mass Mechanical System, Complex Motion, D’ Alembert’s Principle, Constraint Axiom for Relative Inertial Forces, Equations of Motion, Relativity of Time, Conservation of Momentum and Energy, Minimum Action

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1. Introduction

In theoretical mechanics, dynamic problems involving two interacting material bodies $m_i$ and $m_j$ can be reduced to simpler static problems by representing Newton’s second law in the form of D’ Alembert’s principle. Taking into account the constraint theorem, this principle for a constrained material point can be written as follows [1]:

${}_{\overrightarrow{\text{Ф}}}+{\overrightarrow{R}}_{ji}+{\overrightarrow{F}}_i=0;i,j=1,2;i\ne j$. (1)

Here ${}_{\overrightarrow{\text{Ф}}}=-m_i{\overrightarrow{а}}_i$ is the inertial force of body $m_i$ in a fixed laboratory reference frame $K$ (${\overrightarrow{а}}_i$ is its absolute acceleration in this frame); ${\overrightarrow{R}}_{ji}$ is the reaction of the constraint $R$ directed from body $m_i$ to body $m_j$ , which is counteracted by the inertial force ${}_{\overrightarrow{\text{Ф}}}$ of body $m_i$ ; and $F_i$ is the external force applied to body $m_i$ in the laboratory frame $K$ .

In D’Alembert’s principle (1), the inertial force ${}_{\overrightarrow{\text{Ф}}}$ accounts for the non-inertial nature of the proper frame $K_i$ of body $m_i$ , so that this principle is invariant relative to both the frame $K_i$ and the laboratory frame $K$ . In mechanics, this force is considered fictitious [2], introduced solely to reduce dynamic problems to simpler static problems.

The aim of this work is to move from D’ Alembert’s principle (1) to the dynamic equation for each of the interacting bodies $m_i$ and $m_j$ individually in the field of relative inertial force ${}_{\overrightarrow{\text{Ф}}}$ of these bodies, replacing the unknown reactions ${\overrightarrow{R}}_{ji};i,j=1,2;i\ne j$ of their constraint $R$ .

We consider a closed mechanical system consisting of two bodies $m_i$ and $m_j$ (MS2) [3] [4] (see Figure 1(a)). The bodies $m_i$ and $m_j$ are connected by a rigid straight rod of length $R=r_{12}=r_{21}$ mounted to them via ideal hinges (kinematic hinged-rod constraint $R$ ).

Figure 1. Kinematic diagram of a two-mass mechanical system with a diametric mass distribution ${\theta }_{С}=-\pi$ (a) and a map of events $e_1$ and $e_2$ showing that for an invariant rotation of body $m_2$ by an angle $\Delta \phi =\left\{0;\pi /2\right\}$ in frames $K_1$ and $K_{С}$ , this body is located at the same point of the laboratory frame $K$ (b).

The MS2 is acted upon by two types of inertial forces:

1) The intrinsic inertial force ${}_{\overrightarrow{\text{Ф}}}=-m_i\frac{\text{d}}{\text{d}t}{\overrightarrow{\upsilon }}_i=-m_i{\overrightarrow{a}}_i;i=1,2$ of body $m_i$ ,

included in D’Alembert’s principle (1). It is a fundamental property of this body and characterizes its physical ability to counteract the bending of its uniform straight trajectory in the laboratory frame $K$ under the action of any forces (e.g., Hooke’s, Coulomb’s, or Newton’s forces). As applied to the two connected bodies $m_i$ and $m_j$ moving in circular trajectories relative to their center of mass (CM) $C$ , these are their centrifugal inertial forces.

2) The relative inertial force ${}_{\overrightarrow{\text{Ф}}}=-\frac{\text{d}}{\text{d}t}{m}_j{\overrightarrow{\upsilon }}_{ji}=-m_j{\overrightarrow{a}}_{ji};i,j=\text{1,2;}i\ne j$. It is

applied to the working body $m_j$ , which performs relative motion with relative velocity ${\overrightarrow{\upsilon }}_{ij}$ and acceleration ${\overrightarrow{а}}_{ij};i,j=\text{1,}2;i\ne j$ in the proper frame $K_i$ of the reference body $m_i$ . If the masses of the MS2 bodies are equal ($m_i=m_j$ ), either of them can be chosen as the reference body. If $m_i\gg m_j$ , the body with the greater mass should be chosen as the reference body.

2. Dynamic Equations for the Closed MS2

The dynamic equations for bodies $m_1$ and $m_2$ of the closed MS2 are derived based on the kinematic diagram shown in Figure 1(a). The kinematic hinged-rod constraint $R$ between bodies $m_1$ and $m_2$ ensures their planar relative translational motion along circular trajectories $s_{12}$ and $s_{21}$ (shown partially in Figure 1(a)) in the proper frames $K_2$ and $K_1$ of the diametrically opposite bodies $m_2$ and $m_1$ and the simultaneous complex planar translational relative motion along trajectory $s_{1,2}$ (which is common for the bodies when their masses $m_1=m_2$ ) in the reference frame $K_{С}$ attached to their CM $С$ . During these motions, the axes of the reference frames $K_{С}$ , $K_1$ , and $K_2$ and the fixed laboratory frame $K$ always remain parallel. The kinematic hinged-rod constraint $R$ is massless with ideal hinges, and the bodies $m_1$ and $m_2$ are point particles.

Since the kinematic hinged-rod constraint $R$ leads to bending of the trajectories $s_{1,2}$ , $s_{21}$ , and $s_{12}$ of bodies $m_2$ and $m_1$ , it follows that in the reference frames $K_{С},K_1$ , and $K_2$ , these bodies are acted upon by the intrinsic ${}_{\overrightarrow{\text{Ф}}}$ and relative ${}_{\overrightarrow{\text{Ф}}}$ inertial forces of these bodies, which counteract the bending of their trajectories.

The motion of bodies $m_j;j=1,2$ together with the proper reference frames $K_i$ of the diametrically opposite bodies $m_i;i=1,2;i\ne j$ is translational with respect to the reference frame $K$ and relative with respect to the proper reference frames $K_i$ of bodies $m_i$ . As a result, bodies $m_j$ perform complex translational relative motion along the trajectory $s_{1,2}$ in the reference frame $K$ . This motion generates a cross term $U$ that captures the mutual influence of the translational and relative motions, which will be taken into account when deriving the dynamic equations for the MS2.

The trajectories $s_{1,2}$ of bodies $m_1$ and $m_2$ in the reference frame $K_{С}$ are defined by radius vectors ${\overrightarrow{r}}_{1C}$ and ${\overrightarrow{r}}_{2C}$ , and the trajectories $s_{12}$ and $s_{21}$ of their relative motion in the proper reference frames $K_2$ and $K_1$ of the diametrically opposite bodies $m_2$ and $m_1$ are defined by radius vectors ${\overrightarrow{r}}_{12}$ and ${\overrightarrow{r}}_{21}$ with magnitudes $r_{12}=r_{21}=R$ . The radius vectors ${\overrightarrow{r}}_{1C}$ and ${\overrightarrow{r}}_{2C}$ are defined by the equations $m_1{\overrightarrow{r}}_{1С}+m_2{\overrightarrow{r}}_{2С}=0$ and ${\overrightarrow{r}}_{21}={\overrightarrow{r}}_{2C}-{\overrightarrow{r}}_{1C}$ [1] [5]:

${\overrightarrow{r}}_{1C}=-\frac{m_2}{m_1+m_2}{\overrightarrow{r}}_{21}$ ; ${\overrightarrow{r}}_{2C}=\frac{m_1}{m_1+m_2}{\overrightarrow{r}}_{21}$ . (2)

The kinetic energy of the MS2 in planar motion resulting from the two translational relative motions of its bodies $m_1$ and $m_2$ (with an alternate choice of their diametrically opposite bodies $m_2$ and $m_1$ as the reference body) is defined by the following generalized coordinates:

• Two polar coordinates $r_{1C}$ and ${\phi }_{1C}$ of the reference body $m_1$ in the frame $K_C$ of the CM $C$ and one polar coordinate ${\phi }_{21}$ of the working body $m_2$ in the proper frame $K_1$ of the reference body $m_1$ .

• Two polar coordinates $r_{2C}$ and ${\phi }_{2C}$ of the reference body $m_2$ in the reference frame $K_C$ of the CM $C$ and one polar coordinate ${\phi }_{12}$ of the working body $m_1$ in the proper frame $K_2$ of the reference body $m_2$ .

The polar coordinates ${\phi }_{12}$ and ${\phi }_{21}$ of the closed MC2 under consideration are kinematically dependent ${\phi }_{12}={\phi }_{21}+\pi$ (Figure 1(a)). This kinematic dependence satisfies the requirement for dynamic self-consistency ${}_{\overrightarrow{\text{Ф}}}+{}_{\overrightarrow{\text{Ф}}}=0$ of this closed MC2, which is important when the reactions ${\overrightarrow{R}}_{12}$ and ${\overrightarrow{R}}_{21}$ constraints $R$ of its bodies $m_1$ and $m_2$ are replaced by their relative inertial forces ${\overrightarrow{R}}_{12}\Rightarrow {}_{\overrightarrow{\text{Ф}}}$ and ${\overrightarrow{R}}_{21}\Rightarrow {}_{\overrightarrow{\text{Ф}}}$.

Thus, the total number of independent generalized coordinates describing the planar translational relative motion of the MS2 bodies $m_1$ and $m_2$ in the reference frame $K$ is $N=5$ , which equals the number of degrees of freedom of the system $s=3n-l=3\cdot 2-1=5$ . Here $n=2$ is the number of MS2 bodies $m_1$ and $m_2$ , each of which have three degrees of freedom in an unconstrained state in the reference frame $K$ ; $l=1$ is one kinematic hinged-rod constraint $R$ .

The hinges themselves do not introduce new independent constraints beyond the condition of constant length $R=\text{const}$ of the rod (they merely provide the kinematic possibility of rotation about their axes). The set of $N=5$ generalized coordinates determines the total energy $E$ of the MS2 in the reference frame $K$ , which includes the kinetic energy $T_{er}$ of the translational relative motion of bodies $m_1$ and $m_2$ and the cross term $U$ due to their two complex translational relative motions [3].

For the chosen generalized coordinates, the motion of the MS2 bodies $m_1$ and $m_2$ is characterized by the following momenta:

• Intrinsic momentum ${\overrightarrow{p}}_{iС}=m_i{\overrightarrow{\upsilon }}_{iС};i=1,2$ of body $m_i$ , which characterizes the motion of this body together with its proper reference frame $K_i$ in the frame $K_{С}$ . Here ${\overrightarrow{\upsilon }}_{iС}={\overrightarrow{\omega }}_{iС}\times {\overrightarrow{r}}_{iС}$ is the velocity of body $m_i$ in the frame $K_{С}$ and ${\overrightarrow{\omega }}_{iС}={\overrightarrow{\omega }}_{С}$ and ${\overrightarrow{r}}_{iС}$ are the angular velocity and radius vector (2) of the translational motion of this body in the frame $K_{С}$ along a circular trajectory $s_i;i=1,2$ (Figure 1(a)).

• Relative momentum ${\overrightarrow{p}}_{ij}=m_i{\overrightarrow{\upsilon }}_{ij};i,j=1,2;i\ne j$ of body $m_i$ , which characterizes the motion of this body along a trajectory $s_{ij}$ in the proper frame $K_j$ of the diametrically opposite body $m_j$ . Here ${\overrightarrow{\upsilon }}_{ij}={\overrightarrow{\omega }}_{ij}\times {\overrightarrow{r}}_{ij}$ is the relative velocity of body $m_i$ in the reference frame $K_j$ and ${\overrightarrow{\omega }}_{ij}$ and ${\overrightarrow{r}}_{ij}$ are the angular velocity and radius vector of the translational motion of this body in the frame $K_j$ .

• Translational momentum ${\overrightarrow{p}}_{ie}=m_i{\overrightarrow{\upsilon }}_{ie};i=1,2$ body $m_i$ , which characterizes the motion of this body together with the proper reference frame $K_j$ of the diametrically opposite body $m_j$ . Here ${\overrightarrow{\upsilon }}_{iе}={\overrightarrow{\upsilon }}_{jC}={\overrightarrow{\omega }}_{jC}\times {\overrightarrow{r}}_{jC}$ is the translational velocity of body $m_i$ together with the proper frame $K_j$ of the diametrically opposite body $m_j$ , expressed in terms of its intrinsic velocity ${\overrightarrow{\upsilon }}_{jC}={\overrightarrow{\omega }}_{jC}\times {\overrightarrow{r}}_{jC}$ in the frame $K_{С}$ .

For bodies $m_1$ and $m_2$ with initial momenta ${\overrightarrow{p}}_1\left(0\right)=-{\overrightarrow{p}}_2\left(0\right)$ , we establish the relationship between the intrinsic ${\overrightarrow{p}}_{iС}=m_i{\overrightarrow{\upsilon }}_{iС};i=1,2$ , relative ${\overrightarrow{p}}_{ij}=m_i{\overrightarrow{\upsilon }}_{ij};i,j=1,2;i\ne j$ , and translational ${\overrightarrow{p}}_{ie}=m_i{\overrightarrow{\upsilon }}_{ie};i=1,2$ momenta that satisfies the conservation of momentum (${\overrightarrow{p}}_{С}=\overrightarrow{\text{const}}$) of the closed MS2.

Since bodies $m_1$ and $m_2$ are linked by the kinematic hinged-rod constraint $R$ , it follows that for their initial momenta ${\overrightarrow{p}}_1\left(0\right)=-{\overrightarrow{p}}_2\left(0\right)$ (Figure 1(a)) under instantaneous transmission of interactions, their distribution in the reference frame $K_{С}$ is diametric:

${\theta }_{С}=\arccos \left(p_1\left(0\right)/p_2\left(0\right)\right)=-\pi$ . (3)

This corresponds to the concepts of optimal periodic motion [6] [7] for a closed MS2 [3] [4], where ${\theta }_{С}$ is the lag angle of the phase ${\phi }_{1C}={\phi }_{21}+{\theta }_C={\phi }_{21}-\pi$ of the translational motion body $m_1$ in the reference frame $K_{С}$ relative to the phase ${\phi }_{21}$ of the translational motion body $m_2$ in the proper reference frame $K_1$ of body $m_1$ (Figure 1(a)).

In the MS2 with five ($s=5$ ) degrees of freedom under instantaneous transmission of perfectly elastic interaction, the initial momentum ${\overrightarrow{p}}_i\left(0\right)=\left(0,1\right);i=1,2$ of each of its bodies $m_i$ is distributed equally between its relative ${\overrightarrow{p}}_{ij};i,j=1,2;i\ne j$ and intrinsic ${\overrightarrow{p}}_{iС};i=1,2$ momenta. Consequently, when $m_1=m_2$ , the relative and translational velocities are equal ${\overrightarrow{\upsilon }}_{ij}={\overrightarrow{\upsilon }}_{iC}$ , as for body $m_2$ shown in Figure 1(b). The magnitudes of these velocities ${\upsilon }_{12}={\upsilon }_{1C}$ and ${\upsilon }_{21}={\upsilon }_{2C}$ will be determined based on the kinetic energy conservation law for the MS2 during the analysis of its energy balance. The velocity ${\overrightarrow{\upsilon }}_C$ of the CM $С$ in the reference frame $K$ is ${\overrightarrow{\upsilon }}_C=\left({\overrightarrow{p}}_1\left(0\right)+{\overrightarrow{p}}_2\left(0\right)\right)/m=0$ , as shown in Figure 1(b). Acquiring the momenta ${\overrightarrow{p}}_{ij}$ and ${\overrightarrow{p}}_{iС}$ , the MS2 bodies $m_1$ and $m_2$ begin to perform complex translational relative motion in the reference frame $K$ with absolute velocities

${\overrightarrow{\upsilon }}_i={\overrightarrow{\upsilon }}_{ie}+{\overrightarrow{\upsilon }}_{ij};i,j=1,2;i\ne j$, (4)

where ${\overrightarrow{\upsilon }}_{iе}={\overrightarrow{\upsilon }}_{jC}={\overrightarrow{\omega }}_{jC}\times {\overrightarrow{r}}_{jC}$ and ${\overrightarrow{\upsilon }}_{ij}={\overrightarrow{\omega }}_{ij}\times {\overrightarrow{r}}_{ij}$ are the translational velocity of body $m_i$ together with the proper reference frame $K_j$ of the diametrically opposite body $m_j$ and its relative velocity in this frame.

The definition

$m{\overrightarrow{\upsilon }}_{С}={\displaystyle \sum _i{m}_i}{\overrightarrow{\upsilon }}_i=m_1\left({\overrightarrow{\upsilon }}_{1e}+{\overrightarrow{\upsilon }}_{12}\right)+m_2\left({\overrightarrow{\upsilon }}_{2e}+{\overrightarrow{\upsilon }}_{21}\right)={\overrightarrow{p}}_1\left(0\right)+{\overrightarrow{p}}_2\left(0\right)=0$ , (5)

obtained taking into account the absolute velocities ${\overrightarrow{\upsilon }}_i$ (4), implies that for the initial momenta ${\overrightarrow{p}}_1\left(0\right)=-{\overrightarrow{p}}_2\left(0\right)$ of bodies $m_1$ and $m_2$ , the velocity ${\overrightarrow{\upsilon }}_{С}$ of their CM $C$ is ${\overrightarrow{\upsilon }}_{С}=0$ .

Definition (5) yields the following relationship between the translational ${\overrightarrow{p}}_{ie}=m_i{\overrightarrow{\upsilon }}_{ie}=m_i{\overrightarrow{\upsilon }}_{jC}$ and relative ${\overrightarrow{p}}_{ij}=m_i{\overrightarrow{\upsilon }}_{ij}=m_i{\overrightarrow{\omega }}_{ij}\times {\overrightarrow{r}}_{ij};i,j=1,2;i\ne j$ momenta:

${\overrightarrow{p}}_{1e}+{\overrightarrow{p}}_{12}=0;{\overrightarrow{p}}_{2e}+{\overrightarrow{p}}_{21}=0$ . (6)

In view of the definition $m_1{\overrightarrow{r}}_{1С}+m_2{\overrightarrow{r}}_{2С}=0$ for the CM $C$ , the translational momenta ${\overrightarrow{p}}_{ie}=m_i{\overrightarrow{\upsilon }}_{ie}=m_i{\overrightarrow{\upsilon }}_{jC};i,j=1,2;i\ne j$ are linked by a similar relation:

${\overrightarrow{p}}_{1e}+{\overrightarrow{p}}_{2e}=0$ . (7)

Then, it follows from definition (5) that

${\overrightarrow{p}}_{12}+{\overrightarrow{p}}_{21}=0$ (8)

According to (6)-(8), the momentum distribution of the closed MS2 satisfies the equalities

${\overrightarrow{p}}_{1С}=-{\overrightarrow{p}}_{2С}$ ; ${\overrightarrow{p}}_{1С}=-{\overrightarrow{p}}_{21}$ ; ${\overrightarrow{p}}_{2С}=-{\overrightarrow{p}}_{12}$ ; ${\overrightarrow{p}}_{12}=-{\overrightarrow{p}}_{21}$ , (9)

which ensure the conservation of its momentum

${\overrightarrow{p}}_{С}={\overrightarrow{p}}_{1С}+{\overrightarrow{p}}_{2С}+{\overrightarrow{p}}_{12}+{\overrightarrow{p}}_{21}=\overrightarrow{\text{const}}=0$, (10)

for ${\overrightarrow{\upsilon }}_{С}=0$ (5). Equalities (9) will be used to analyze the energy balance of the MS2 for two sets of initial momenta ${\overrightarrow{p}}_1\left(0\right)=-{\overrightarrow{p}}_2\left(0\right)$ and ${\overrightarrow{p}}_1\left(0\right)=0$ , ${\overrightarrow{p}}_2\left(0\right)=\left(0,1\right)$ of its bodies $m_1$ and $m_2$ .

When $m_1=m_2$ , equalities (6) reduce the sums of the translational ${\overrightarrow{\upsilon }}_{ie}$ and relative ${\overrightarrow{\upsilon }}_{ij}$ velocities to zero (${\overrightarrow{\upsilon }}_{1e}+{\overrightarrow{\upsilon }}_{12}=0$ and ${\overrightarrow{\upsilon }}_{2e}+{\overrightarrow{\upsilon }}_{21}=0$ ), which is consistent with (5), resulting in zero absolute velocity ${\overrightarrow{\upsilon }}_i=0$ (4) for each of the bodies $m_i$ in the reference frame $K$ . However, since the translational ${\overrightarrow{\upsilon }}_{ie}$ and relative ${\overrightarrow{\upsilon }}_{ij}$ velocities of the bodies $m_i$ are not individually zero, they uniquely determine the kinetic $T_{er}$ and total $E$ energies of the MS2 in the reference frame $K$ , which will be used when defining them in terms of the zero absolute velocity ${\overrightarrow{\upsilon }}_i$ (4).

We temporarily reduce the number of degrees of freedom of the closed MS2 to two ($s=2$ ): along the Cartesian coordinate $x_i$ of the reference body $m_i;i=1,2$ in the reference frame $K$ and along the polar coordinate ${\phi }_{ji};i,j=1,2;i\ne j$ of the working body $m_j$ in the proper frame $K_i$ of the reference body $m_i$ . Then, the reaction ${\overrightarrow{R}}_{ji}$ of the kinematic hinged-rod constraint $R$ applied to any of the selected reference bodies $m_i$ can be replaced by the relative inertial force ${\overrightarrow{R}}_{ji}={}_{\overrightarrow{\text{Ф}}}=m_j{\omega }_{ji}^2{\overrightarrow{r}}_{ji}$ of the conditionally discarded working body $m_j$ (this will be subsequently mathematically proven using the example of MS2 with $s=2$ ). As a result, from (1) we obtain the dynamic equation for any of the selected reference bodies $m_i$ of the closed MS2 with $s=2$ and ${\overrightarrow{F}}_i=0$ in the form of D’Alembert’s principle or Newton’s second law in the field of the relative inertial force of the discarded working body $m_j$ :

${}_{\overrightarrow{\text{Ф}}}+{}_{\overrightarrow{\text{Ф}}}=0;m_i{\overrightarrow{a}}_i={}_{\overrightarrow{\text{Ф}}};i,j=1,2;i\ne j$, (11)

which does not contain the unknown reactions ${\overrightarrow{R}}_{ji}$ (1) of the kinematic hinged-rod constraint $R$ and where ${\overrightarrow{a}}_i$ is the absolute acceleration of the reference body $m_i$ in the laboratory frame $K$ .

We now determine the form of the relative inertial forces ${}_{\overrightarrow{\text{Ф}}}$ (11) for the closed MS2 with $N=s=5$ shown in Figure 1, a when its bodies $m_1$ and $m_2$ perform complex planar translational relative motion in the reference frame $K$ . In view of the absolute velocities ${\overrightarrow{\upsilon }}_i$ (4), the total kinetic energy of this simultaneous complex planar motion of the two bodies $m_1$ and $m_2$ can be represented as [3]

$T_{er}=\frac{1}{2}{\displaystyle \sum _i{m}_i{\overrightarrow{\upsilon }}_i^2}=\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{1e}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{2e}^2+\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{12}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{21}^2+U,$ (12)

where $U$ is the cross term of the translational relative motion, defined in [5] as the generalized kinetic potential

$U=m_1{\upsilon }_{1e}{\upsilon }_{12}\cos {\theta }_C+m_2{\upsilon }_{2e}{\upsilon }_{21}\cos {\theta }_C=-\left(m_1{\upsilon }_{2C}{\upsilon }_{12}+m_2{\upsilon }_{1C}{\upsilon }_{21}\right)$ , (13)

which characterizes the contribution of the relative motion (${\overrightarrow{\upsilon }}_{12r},{\overrightarrow{\upsilon }}_{21r}$ ) of bodies $m_1$ and $m_2$ to their translational motion (${\overrightarrow{\upsilon }}_{1e}={\overrightarrow{\upsilon }}_{2С}$ ,${\overrightarrow{\upsilon }}_{2e}={\overrightarrow{\upsilon }}_{1С}$ ); here ${\theta }_{С}=\widehat{{\overrightarrow{\upsilon }}_{2e}{\overrightarrow{\upsilon }}_{21}}=-\pi$ is the angle between the vectors of the translational ${\overrightarrow{\upsilon }}_{2e}$ and relative ${\overrightarrow{\upsilon }}_{21}$ velocities (Figure 1(a)).

The sum of the first and second terms included in $T_{er}$ (12) is the kinetic energy of the translational motion $T_{Сe}$ of bodies $m_1$ and $m_2$ in the reference frame since $K$ , taking into account the absolute velocities ${\overrightarrow{\upsilon }}_i$ (4), it characterizes the translational momentum ${\overrightarrow{p}}_{Ce}=\left(m_1+m_2\right){\overrightarrow{\upsilon }}_{С}={\displaystyle \sum _i{m}_i{\overrightarrow{\upsilon }}_i}=m_1{\overrightarrow{\upsilon }}_{1e}+m_1{\overrightarrow{\upsilon }}_{12}+m_2{\overrightarrow{\upsilon }}_{2e}+m_2{\overrightarrow{\upsilon }}_{21}=m_1{\overrightarrow{\upsilon }}_{1e}+m_2{\overrightarrow{\upsilon }}_{2e}$ of MS2 as a whole.

The sum of the first and second terms in $T_{er}$ (12) is the kinetic energy of the translational motion of bodies $m_1$ and $m_2$ . Taking into account the permutation $m_1\iff m_2$ , valid for equal masses $m_1=m_2$ , this sum satisfies the equality

$T_{Сe}=\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{1e}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{2e}^2=\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{2C}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{1С}^2=\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{1C}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{2С}^2$ , (14)

expressed in terms of the translational momenta ${\overrightarrow{p}}_{1e}=m_1{\overrightarrow{\upsilon }}_{1e}=m_1{\overrightarrow{\upsilon }}_{2C}$ and ${\overrightarrow{p}}_{2e}=m_2{\overrightarrow{\upsilon }}_{2e}=m_2{\overrightarrow{\upsilon }}_{1C}$ of bodies $m_1$ and $m_2$ together with the proper frames $K_2$ and $K_1$ of the diametrically opposite bodies $m_2$ and $m_1$ ; this defines the physical meaning of the sum $T_{Се}$ .

The sum of the third and fourth terms of the kinetic energy $T_{er}$ (12) is the kinetic energy $T_r=\frac{1}{2}{m}_1{\upsilon }_{12}^2+\frac{1}{2}{m}_2{\upsilon }_{21}^2$ of the relative motion of body $m_1$ in the frame $K_2$ and body $m_2$ in the frame $K_1$ .

The potential $U$ (13) defines the relative inertial forces of bodies $m_1$ and $m_2$ during their complex translational relative motion:

$\begin{array}{l}{\overrightarrow{F}}_{1С}=-\frac{\partial U}{\partial r_{2C}}\cdot {\overrightarrow{r}}_{2C}^0=m_1{\omega }_{2С}{\omega }_{12}{r}_{12}\cdot {\overrightarrow{r}}_{1C}^0;\\ {\overrightarrow{F}}_{2С}=-\frac{\partial U}{\partial r_{1C}}\cdot {\overrightarrow{r}}_{1C}^0=m_2{\omega }_{1С}{\omega }_{21}{r}_{21}\cdot {\overrightarrow{r}}_{2C}^0,\end{array}$ (15)

which, in view of equality (14), act on the diametrically opposite bodies $m_2$ and $m_1$ in the reference frame $K$ [3] [4]. Here ${\overrightarrow{r}}_{1С}^0=-{\overrightarrow{r}}_{2С}/r_{2С}$ and ${\overrightarrow{r}}_{2С}^0=-{\overrightarrow{r}}_{1С}/r_{1С}$ are the unit vectors opposing the radius vectors ${\overrightarrow{r}}_{2С}$ and ${\overrightarrow{r}}_{1С}$ (2) and defining the direction of the relative inertial forces ${\overrightarrow{F}}_{1С}$ and ${\overrightarrow{F}}_{2С}$ toward the CM $С$ (see Figure 1(a)).

Internal momenta, which take into account the thermal motion of molecules, and the rotation of atoms in the crystal lattice of bodies MC2 and their core are not considered in the kinetic energy of MC2, as their small contribution to the motion of its CM $С$ makes them extremely insignificant.

Thus, in the mathematical decomposition of kinetic energy $T_{er}$ (12) into transportive $T_{Сe}$ and relative $T_r$ components, with their cross term in the form of kinetic potential $U$ (13), they have different physical meanings, since they are expressed in terms of the transportive ${\overrightarrow{\upsilon }}_{1e}={\overrightarrow{\upsilon }}_{2С}$ , ${\overrightarrow{\upsilon }}_{2e}={\overrightarrow{\upsilon }}_{1С}$ and relative ${\overrightarrow{\upsilon }}_{12}$ , ${\overrightarrow{\upsilon }}_{21}$ velocities of bodies $m_1$ and $m_2$ MC2, so that when summed, the same momentum of these bodies is not considered more than once.

Using the equalities $p_{1C}=p_{12}$ and $p_{2C}=p_{21}$ (6) and the radii $r_{1С}$ and $r_{2С}$ (2), the angular velocities ${\omega }_{1С}={\omega }_{2С}={\omega }_{С}$ of bodies $m_1$ and $m_2$ in the reference frame $K_{С}$ and their angular velocities ${\omega }_{12}$ and ${\omega }_{21}$ in the proper frames $K_2$ and $K_1$ of the diametrically opposite bodies $m_2$ and $m_1$ (see (15)), can be expressed as follows [1] [3] [4]:

$\begin{array}{l}{\omega }_{С}={\omega }_{1С}={\omega }_{2С}={\omega }_{12}+{\omega }_{21};\\ {\omega }_{12}=\frac{m_2{\omega }_C}{m_1+m_2};{\omega }_{21}=\frac{m_1{\omega }_C}{m_1+m_2}.\end{array}$ (16)

Based on the kinematic feasibility of the motion of bodies $m_1$ and $m_2$ of the MS2 without its destruction, it is necessary that the angular displacements of these bodies in the reference frames $K$ , $K_{С}$ , $K_1$ , and $K_2$ be invariant:

$\Delta \phi ={\omega }_{С}\Delta t_C={\omega }_{12}\Delta t_2={\omega }_{21}\Delta t_1$ , (17)

where $\text{Δ}{t}_1=t_1-t_0$ , $\text{Δ}{t}_2=t_2-t_0$ , and $\text{Δ}{t}_{С}=t_{С}-t_0$ are the time increments and $t_0$ is the unified initial time in all reference frames; $t_1$ and $t_2$ is the proper time in the reference systems $K_1$ and $K_2$ ; $t_{С}$ is the proper time in the reference frame $K_{С}$ , which is absolute $t_{С}=t$ with respect to the time $t$ in the reference frame $K$ .

Physically, invariant (17) characterizes the fact that for any angular displacement $\Delta \phi =\left\{0;\pi /2\right\}$ of body $m_2$ in the reference frames $K_1$ and $K_{С}$ (which is also true for body $m_1$ in the reference frames $K_2$ and $K_{С}$ ), this body will be located at the same point of the trajectory $s_{1,2}$ of the reference frame $K$ , as shown, e.g., for this body by events $e_1$ and $e_2$ in Figure 1(b). In other words, it determines that for angular velocities ${\omega }_{21}$ and ${\omega }_{12}$ (16) in reference frames $K_1$ and $K_2$ , different from the angular velocity ${\omega }_{С}$ in reference frame $K_{С}$ , the time increments $\Delta t_{С}$ , $\Delta t_1$ and $\Delta t_2$ in these frames must be different. That is, in reference frame $K_1$ or $K_2$ , where the angular velocity ${\omega }_{21}$ or ${\omega }_{12}$ (16) is higher ${\omega }_{С}$ , the proper time $t_1$ or $t_2$ flows more slowly (the intervals between events are longer), and vice versa. Therefore, the clock rates in reference frames $K_1$ and $K_2$ , due to different angular velocities ${\omega }_{21}$ and ${\omega }_{12}$ , are determined through the proper time $t_1$ and $t_2$ . Since the proper times $t_1$ and $t_2$ are different in reference frames $K_1$ and $K_2$ , the observed clock rates of this process in these systems will also differ. It is important to note that for the considered MS2 model, relativistic time dilation is not introduced (as in Einstein’s special theory of relativity [8]). It operates within the framework of classical mechanics, but takes into account that during the relative motion of parts of the MC2 (rotation), the very concept of locality of time for reference systems $K_1$ and $K_2$ in relation to time $t_{С}=t$ in the reference system $K_{С}$ is derived on the basis of angular coordinates ${\phi }_{С}={\omega }_{С}{t}_C$ , ${\phi }_{21}={\omega }_{21}{t}_1$ and ${\phi }_{12}={\omega }_{12}{t}_2$ , associated with time $t_{С}$ , $t_1$ and $t_2$ in reference systems $K_{С},K_1$ and $K_2$ through the kinematics of the MC2—angular velocities ${\omega }_{С},{\omega }_{21}$ and ${\omega }_{12}$ , and is not postulated on the current model of the clock.

From the invariant $\Delta \phi$ (17) one can determine the proper time $t_1$ and $t_2$ in the reference systems $K_1$ and $K_2$ :

$t_1=\frac{m_1+m_2}{m_1}{t}_{С}$ ; $t_2=\frac{m_1+m_2}{m_2}{t}_{С}$ . (18)

where time $t_{С}$ in the reference frame $K_{С}$ is absolute $t_{С}=t$ .

The relativity of times $t_1$ and $t_2$ (18) is not associated with the relativity of time in Einstein’s general relativity theory [8], which is based on the constancy of the speed of light $c=\text{const}$ in all observer systems moving uniformly relative to a light source. A general property of the relativity of time (18) in comparison with Einstein’s special relativity theory [8] is that as the mass of the reference body $m_i$ increases (which is equivalent to the strengthening of its gravitational field), the rate of time flow in the proper frame $K_i$ of this body slows down and approaches the rate of time $t_{С}$ in the reference frame $K_{С}$ of the CM $C$ , which is absolute ($t_{С}=t$ ).

Taking into account the relative inertial forces ${\overrightarrow{F}}_{1С}$ and ${\overrightarrow{F}}_{2С}$ (15), D’Alembert’s principle and Newton’s second law (11) for bodies $m_1$ and $m_2$ of the closed MS2 under consideration take the form

${}_{\overrightarrow{\text{Ф}}}+{\overrightarrow{F}}_{jC}=0;m_i{\overrightarrow{a}}_i={\overrightarrow{F}}_{jC};i,j=1,2;i\ne j$. (19)

The first Equation in (19) characterizes the equilibrium of body $m_i$ in the laboratory reference frame $K$ during uniform motion along a geodesic trajectory $s_i,$ while the second characterizes its dynamics in the same frame $K$ .

The relative inertial forces $F_{jС}=m_j{\omega }_{iС}{\omega }_{ji}{r}_{ji};i,j=1,2;i\ne j$ in Equation (19) have the same dimensions as the relative inertial forces ${}_{\overrightarrow{\text{Ф}}}=-m_j{\omega }_j^2{r}_{ji};i,j=1,2;i\ne j$ in Equation (11). They act through the constraint $R$ on the diametrically opposite body $m_i$ , forming an inertial domain (ID) [3] [4] that excludes the unknown reactions ${\overrightarrow{R}}_{ji}$ (1) of this constraint $R$ from D’Alembert’s principle. Therefore, D’ Alembert’s principle (1) and the dynamic Equations (11) and (19) become invariant in the fixed laboratory reference frame $K$ , as the interaction of bodies $m_i$ and $m_j$ via the reactions ${\overrightarrow{R}}_{ji}$ of the constraint $R$ is equivalent to the action of the relative inertial forces ${}_{\overrightarrow{\text{Ф}}}$ and ${\overrightarrow{F}}_{ji}$ . Moreover, due to the equivalence of gravitational and inertial masses, the ID field can be viewed as a central dynamic attractive force field whose action on the MS2 is equivalent to that of the external force field.

The MS2 model, represented by Equation (19), explicitly excludes constraint $R$ reactions ${\overrightarrow{R}}_{ji}$ and Newton’s third law. It is constructed not using the geometry of a rigid rod $R$ per se, but through additional dynamic assumptions about the nature of the total momentum distribution and phase relationships. In this MS2 model, momentum transfer between its bodies is accomplished through the action of relative inertial forces ${\overrightarrow{F}}_{jС}$ (19). That is, the redistribution mechanism consists of the action of the relative inertial force ${\overrightarrow{F}}_{jС}$ of the working body $m_j$ through the constraint $R$ on the supporting body $m_i$ .

Moreover, the MS2 model is constructed under the following key assumptions:

• The transfer of momentum from the working body $m_j$ to the supporting body $m_i$ with a larger mass is carried out by means of the inertial force ${\overrightarrow{F}}_{jС}$ of the working body $m_j$ . This replaces “instantaneity” in the classical sense with a controlled redistribution, which can be described as instantaneous within the framework of the proposed MS2 model with a specific distribution of its total momentum ${\overrightarrow{p}}_{С}$ (10).

• To conserve the total momentum ${\overrightarrow{p}}_{С}=\overrightarrow{\text{const}}$ of a closed MS2, the condition is introduced that the phases of the interacting bodies in the reference frame of their center of mass are shifted by angle ${\theta }_{С}=-\pi$ (3). This is not simply a kinematic condition, but a dynamic condition that effectively defines the following rule: for any initial momentum distribution, a closed MS2 instantaneously tends to ensure the phase condition ${\theta }_{С}=-\pi$ , which is equivalent to an instantaneous redistribution within the framework of its adopted model.

From the perspective of classical mechanics, the introduction of instantaneous momentum redistribution through phase conditions and relative inertial forces are additional postulates, not consequences of the standard equations of classical mechanics. It is these postulates that make the proposed MS2 model internally consistent within its own logic, but simultaneously take it beyond the traditional approach.

Replacing $T_{Сe}=\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{1e}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{2e}^2$ in (12) by $T_{Сe}=\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{1C}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{2С}^2$ in accordance with equality to (14) and taking into account that the external potential energy of the closed MS2 is zero ($\Pi =0$ ) yields its Lagrangian

$\begin{array}{c}L=T_{er}=\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{1C}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{2C}^2+\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{12}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{21}^2+U\\ =\frac{1}{2}{m}_1{\omega }_{1C}^2{r}_{1C}^2+\frac{1}{2}{m}_2{\omega }_{2C}^2{r}_{2C}^2+\frac{1}{2}{m}_1{\omega }_{12}^2{r}_{12}^2+\frac{1}{2}{m}_2{\omega }_{21}^2{r}_{21}^2\\ -\left(m_1{\omega }_{2C}{r}_{2C}{\omega }_{12}{r}_{12}+m_2{\omega }_{1C}{r}_{1C}{\omega }_{21}{r}_{21}\right)\end{array}$ (20)

in the reference frame $K$ for $m_1=m_2$ .

The partial derivatives of the Lagrangian function $L$ (20) of the form $-\frac{\partial L}{r_{1C}}=0$ and $-\frac{\partial L}{r_{2C}}=0$ yield the generalized forces

$\begin{array}{l}-m_1{\omega }_{1C}^2{r}_{1С}+m_2{\omega }_{1C}{\omega }_{21}{r}_{21}=0;\\ -m_2{\omega }_{2C}^2{r}_{2C}+m_1{\omega }_{2C}{\omega }_{12}{r}_{12}=0\end{array}$ (21)

in the form of the D’ Alembert principle (19) in curvilinear coordinates, in the field of relative inertial forces ${\overrightarrow{F}}_{jC}$ , which establishes their equivalence to reactions ${\overrightarrow{R}}_{}{}_{ij}$ .

Indeed, the first terms of (21) define the intrinsic inertial forces ${}_{\overrightarrow{\text{Ф}}}={}_{\overrightarrow{\text{Ф}}}=-m_1{\omega }_{1C}^2{\overrightarrow{r}}_{1С}$ and ${}_{\overrightarrow{\text{Ф}}}={}_{\overrightarrow{\text{Ф}}}=-m_2{\omega }_{1C}^2{\overrightarrow{r}}_{2С}$ of bodies $m_1$ and $m_2$ (Figure 1(a)), which are directed away from CM $С$ .

The second terms define the relative inertial forces ${\overrightarrow{F}}_{2C}=m_2{\omega }_{1C}{\omega }_{21}{r}_{21}\cdot {\overrightarrow{r}}_{2C}^0$ and ${\overrightarrow{F}}_{1C}=m_1{\omega }_{2C}{\omega }_{12}{r}_{12}\cdot {\overrightarrow{r}}_{1C}^0$ of bodies $m_2$ and $m_1$ during their complex translational relative motion, where the choice of direction for the unit vectors ${\overrightarrow{r}}_{1С}^0=-{\overrightarrow{r}}_{2С}/r_{2С}$ and ${\overrightarrow{r}}_{2С}^0=-{\overrightarrow{r}}_{1С}/r_{1С}$ toward the CM $С$ (Figure 1(a)) is due to the fact that the forces ${\overrightarrow{F}}_{2C}$ and ${\overrightarrow{F}}_{1C}$ are counteracted by the intrinsic inertial forces ${}_{\overrightarrow{\text{Ф}}}$ and ${}_{\overrightarrow{\text{Ф}}}$ of bodies $m_1$ and $m_2$ (21).

Equation (19), expressed in terms of the relative inertial forces ${\overrightarrow{F}}_{1С}$ and ${\overrightarrow{F}}_{2С}$ (15), allow an independent analysis of the dynamics of each of MS2 bodies $m_1$ and $m_2$ without taking into account the unknown reactions ${\overrightarrow{R}}_{ji}$ (1) of their constraint $R$ . Since these forces form a balanced system of forces ${\overrightarrow{F}}_{1С}+{\overrightarrow{F}}_{2С}=0$ , their action on the closed MS2 system satisfies the conservation of its momentum (${\overrightarrow{p}}_C=\stackrel{\rightharpoonup }{\text{const}}$).

We rederive the dynamic Equation (19) for the MS2 for the case where the initial momenta of its bodies $m_1$ and $m_2$ are ${\overrightarrow{p}}_1\left(0\right)=0$ and ${\overrightarrow{p}}_2\left(0\right)=\left(0,1\right)$ , respectively, as shown in Figure 2(a).

In the MS2 with five ($s=5$ ) degrees of freedom, the initial momentum ${\overrightarrow{p}}_2\left(0\right)=\left(0,1\right)$ at time $t=0$ is instantaneously redistributed between the bodies $m_1$ and $m_2$ of the MS2 and its CM $C$ so that the translational velocities ${\overrightarrow{\upsilon }}_{1e},{\overrightarrow{\upsilon }}_{2e}$ (${\overrightarrow{\upsilon }}_{1e}={\overrightarrow{\upsilon }}_{2C}$ , ${\overrightarrow{\upsilon }}_{2e}={\overrightarrow{\upsilon }}_{1C}$ ) and the relative velocities of its bodies $m_1$ and $m_2$ ${\overrightarrow{\upsilon }}_{21},{\overrightarrow{\upsilon }}_{12}$ become equal to each other: ${\overrightarrow{\upsilon }}_{12}={\overrightarrow{\upsilon }}_{1C}$ and ${\overrightarrow{\upsilon }}_{21}={\overrightarrow{\upsilon }}_{2C}$ (Figure 2(b)). Their magnitudes ${\upsilon }_{12}={\upsilon }_{1C}$ and ${\upsilon }_{21}={\upsilon }_{2C}$ will be determined from the energy conservation law of the closed MS2 in the subsequent analysis of its energy balance. The velocity ${\overrightarrow{\upsilon }}_C$ of the CM $С$ in the reference frame $K$ is ${\overrightarrow{\upsilon }}_C=p_2\left(0\right)/m$ , as shown in Figure 2(a). Acquiring the momenta ${\overrightarrow{p}}_{ij}$ and ${\overrightarrow{p}}_{iС}$ , bodies $m_1$ and $m_2$ begin to perform complex translational relative motion along the circle $s_{1,2C}$ relative to the frame $K_{С}$ (Figure 2(b)). Due to the simultaneous rectilinear and uniform motion of the CM $С$ of these bodies in the reference frame $K$ , this motion follows spiral trajectories (not shown in Figure 2).

Based on the definition

${\overrightarrow{\upsilon }}_{С}=\frac{{\displaystyle \sum _i{m}_i{\overrightarrow{\upsilon }}_i}}{m_1+m_2}=\frac{m_1{\overrightarrow{\upsilon }}_{ie}+m_1{\overrightarrow{\upsilon }}_{12}+m_2{\overrightarrow{\upsilon }}_{2e}+m_2{\overrightarrow{\upsilon }}_{21}}{m_1+m_2}=\frac{m_1{\overrightarrow{\upsilon }}_{ie}+m_2{\overrightarrow{\upsilon }}_{2e}}{m_1+m_2}$ , (22)

obtained taking into account the absolute velocities ${\overrightarrow{\upsilon }}_i$ (4), the motion of the CM $C$ can be considered translational ${\overrightarrow{\upsilon }}_{С}\equiv {\overrightarrow{\upsilon }}_{Сe}$ .

Figure 2. Kinematics of the MS2 system whose bodies $m_1$ and $m_2$ have initial momenta ${\overrightarrow{p}}_1\left(0\right)=0$ and ${\overrightarrow{p}}_2\left(0\right)=\left(0,1\right)$ .

Then, the translational velocities ${\overrightarrow{\upsilon }}_{ie}$ of bodies $m_1$ and $m_2$ are the sum

${\overrightarrow{\upsilon }}_{ie,abs}={\overrightarrow{\upsilon }}_{С}+{\overrightarrow{\upsilon }}_{iе},i=1,2$ , (23)

whose cross terms vanish when it is squared:

$m_1{\overrightarrow{\upsilon }}_{С}\cdot \left({\overrightarrow{\omega }}_C\times {\overrightarrow{r}}_{1C}\right)+m_2{\overrightarrow{\upsilon }}_{С}\cdot \left({\overrightarrow{\omega }}_C\times {\overrightarrow{r}}_{1C}\right)={\overrightarrow{\upsilon }}_{С}\cdot \left({\overrightarrow{\omega }}_C\times \left(m_1{\overrightarrow{r}}_{1C}+m_2{\overrightarrow{r}}_{2C}\right)\right)=0$ . (24)

Replacing the translational velocity ${\overrightarrow{\upsilon }}_{iе}$ in (12) by ${\overrightarrow{\upsilon }}_{ie,abs}$ (23) yields the kinetic energy $T_{er}$ of the MS2 in the form

$T_{er}=\frac{1}{2}m{\overrightarrow{\upsilon }}_{С}^2+\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{1e}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{2e}^2+\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{12}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{21}^2+U,$ (25)

where $T_{С}=\frac{1}{2}m{\overrightarrow{\upsilon }}_{С}^2$ is the kinetic energy of the rectilinear uniform motion of the CM $С$ with velocity ${\overrightarrow{\upsilon }}_{С}$ in the reference frame $K$ (Figure 2(b)).

The kinetic energy (25) differs from Koenig’s kinetic energy by containing the translational kinetic energy term $T_e=\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{1e}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{2e}^2$ and the generalized kinetic potential $U=-\left(m_1{\upsilon }_{2C}{\upsilon }_{12}+m_2{\upsilon }_{1C}{\upsilon }_{21}\right)$ , which defines the relative inertial forces ${\overrightarrow{F}}_{1C}$ and ${\overrightarrow{F}}_{2C}$ (15).

Based on Galileo’s principle of relativity, when ${\overrightarrow{p}}_C=\overrightarrow{\text{const}}$, the term $T_C$ in (25) can be omitted when constructing the Lagrangian of the MS2 under consideration. Then, taking into account the permutation of masses $m_1\iff m_2$ (14), allowable when $m_1=m_2$ , its Lagrangian can be written in the form

$\begin{array}{c}L=T_{er}=\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{2C}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{1C}^2+\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{12}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{21}^2+U\\ =\frac{1}{2}{m}_1{\omega }_{2C}^2{r}_{2C}^2+\frac{1}{2}{m}_2{\omega }_{1C}^2{r}_{1C}^2+\frac{1}{2}{m}_1{\omega }_{12}^2{r}_{12}^2+\frac{1}{2}{m}_2{\omega }_{21}^2{r}_{21}^2\\ -\left(m_1{\omega }_{2C}{r}_{2C}{\omega }_{12}{r}_{12}+m_2{\omega }_{1C}{r}_{1C}{\omega }_{21}{r}_{21}\right),\end{array}$ (26)

similar to (20).

From the Lagrangian $L$ (26), one can determine the relative inertial forces ${\overrightarrow{F}}_{1C}$ and ${\overrightarrow{F}}_{2C}$ (15) and the Equation of motion (19) for each of the MS2 bodies ($m_1$ and $m_2$ ).

3. Dynamic Equations for the MS2 System with $s=2$ Degrees of Freedom in a Dissipative Medium

The absolute momenta of the reference and working bodies $m_1$ and $m_2$ ($m_1>m_2$ ) of the MS2 in the reference frame $K$ can be expressed in terms of their Cartesian coordinates

$x_1=x_1$ and $x_2=x_1+R\cos {\phi }_{21}$ (27)

as follows [3] [4]:

$p_{1x}=m_1\frac{\text{d}}{\text{d}t}{x}_1$ ; $p_{2x}=m_2\frac{\text{d}}{\text{d}t}{x}_2=m_2\frac{\text{d}}{\text{d}t}{x}_1+m_2\frac{\text{d}}{\text{d}t}R\cos {\phi }_{21}$ . (28)

The generalized independent coordinates of the MS2 can be chosen to be the Cartesian coordinate $x_1$ of the reference body $m_1$ moving along the $x$ axis of the reference frame $K$ and the polar coordinate ${\phi }_{21}={\omega }_{21}{t}_1$ of the working body $m_2$ rotating relative to the reference body $m_1$ in its proper frame $K_1$ , in which time $t_1$ is absolute ($t_1=t_{С}=t$ ), according to (16) and (17) with ${\omega }_{12}=0$ and the relative velocity ${\omega }_{21}=\text{const}$ , which is maintained constant by the internal active pair of forces [3] [4].

Then, the absolute momentum MS2 can be represented as

$p_x={\displaystyle \sum _i{p}_{ix}}=p_{Сx}+p_{21x}=\left(m_1+m_2\right)\frac{\text{d}}{\text{d}t}{x}_1+m_2\frac{\text{d}}{\text{d}t}R\cos {\phi }_{21}$ . (29)

It is composed of the intrinsic momentum of the reference body $m_1$ with the reduced mass $m=m_1+m_2$ , which, together with proper reference frame $K_1$ performs absolute motion in the reference frame $K$ , and the relative momentum of the working body $m_2$ in the reference frame $K_1$ :

$p_{Сx}=\left(m_1+m_2\right)\frac{\text{d}}{\text{d}t}{x}_1$ , $p_{21x}=m_2\frac{\text{d}}{\text{d}t}R\cos {\phi }_{21}$ . (30)

Using the momenta $p_{1x}$ and $p_{2x}$ (28) and accounting for the constraint axiom for the constrained material point and the dissipative force

$F_{1x}=-{\mu }_1{\dot{x}}_1$ (31)

acting on the reference body $m_1$ moving with velocity ${\dot{x}}_1$ along the $x$ axis of the reference frame $K$ , Newton’s second law for each of the interacting bodies $m_1$ and $m_2$ individually can be expressed as

$\text{d}{p}_{1x}/\text{d}t=X_{12}+F_{1x};\text{d}{p}_{2x}/\text{d}t=X_{21},$ (32)

where $X_{12}=-X_{21}$ are the projections of the reactions ${\overrightarrow{R}}_{ij}$ of the constraint $R$ of bodies $m_1$ and $m_2$ onto the $x$ axis of the reference frame $K$ , which satisfy Newton’s third law [1].

Summing the left and right sides of Equation (32) and taking into account that${\displaystyle \sum _{i,j=1,2;i\ne j}{X}_{ij}}=0$ yields D’ Alembert’s principle and Newton’s second law

${\text{Ф}}_{1x}+{\text{Ф}}_{21x}+F_{1x}=0$ ; $m\frac{{\text{d}}^2{x}_1}{\text{d}{t}^2}={\text{Ф}}_{21x}+F_{1x}$ , (33)

in the form (11) in the reference frame $K$ for the dissipative external medium, where${\text{Ф}}_{1x}=-\frac{\text{d}{p}_{Сx}}{\text{d}t}=-\left(m_1+m_2\right)\frac{{\text{d}}^2{x}_1}{\text{d}{t}^2}$ is the intrinsic inertial force of the reference body $m_1$ with the reduced mass $m=m_1+m_2$ and ${\text{Ф}}_{21x}=-\frac{\text{d}{p}_{21x}}{\text{d}t}=-m_2\frac{{\text{d}}^{2}R\cos {\phi }_{21}}{\text{d}{t}^2}=m_2{\omega }_{21}^{2}R\cos {\phi }_{21}$ is the intrinsic inertial force of the working body $m_2$ in the reference frame $K_1$ .

Equation (33) are expressed in terms of the relative inertial force ${}_{\overrightarrow{\text{Ф}}}$ of the working fluid $m_2$ and can be obtained directly from the Lagrangian MC2, written relative to the Cartesian coordinate $x_1=f\left({\phi }_{21}\right)$ , which does not contain constraint $R$ reactions ${\overrightarrow{R}}_{ij}$ [3], which establishes the equivalence of constraint $R$ reactions ${\overrightarrow{R}}_{ij}$ and the relative inertial force ${}_{\overrightarrow{\text{Ф}}}$ of the working fluid $m_2$ . Equation (33) also show that under $m_1\ne m_2$ MC2 it is reduced to a single body with mass $m=m_1+m_2$ with $s\le 3$ degrees of freedom, moving in a dissipative medium in a field of relative inertial force ${}_{\overrightarrow{\text{Ф}}}$ along a rectilinear trajectory.

The solution of the inhomogeneous differential equation in the form of Newton’s second law (33) was obtained in [3] [4] for the initial momenta ${\overrightarrow{p}}_1=0$ and ${\overrightarrow{p}}_2=\left(0,1\right)$ of the MS2 bodies $m_1$ and $m_2$ . It is the sum of the general solution of the homogeneous Equation (33) (with its right-hand side set to zero) and the particular solution of this inhomogeneous equation.

The general solution characterizes the transient process occurring in the MS2 at the beginning of its motion and determines the position $x_O$ of the force center $O$ of the MS2 ID on the $x$ axis of the reference frame $K$ after the completion of the transient process.

The particular solution describes the steady-state linear harmonic oscillation of the reference body $m_1$ with reduced mass $m=m_1+m_2$ along the $x$ axis of the reference frame $K$ in the vicinity of the force center $O$ of the ID:

$x_{1O}=A_1\cos {\phi }_{1O}=-A_1\cos \left({\phi }_{21}+\epsilon \right),$ (34)

where

$A_1=a/\sqrt{1+{\xi }^2};\phi ={\theta }_{С}+\epsilon =-\pi +\epsilon ;\epsilon =\text{arctg}\xi ;$ (35)

${\phi }_{1O}={\phi }_{21}+\phi$ and $A_1$ are the phase and amplitude of the linear harmonic oscillation of the reference body $m_1$ ; $\phi ={\phi }_{1O}-{\phi }_{21}$ is the lag angle of the phase ${\phi }_{1O}$ of the linear harmonic oscillation of the reference body $m_1$ relative to the phase ${\phi }_{21}$ of the rotational motion of the working body $m_2$ , including the dissipative angle $\epsilon$ that takes into account the viscosity of the external medium; $a=m_{2}R/\left(m_1+m_2\right)$ is the amplitude coefficient; $\xi =2{\gamma }_1/{\omega }_{21}$ is the dissipative parameter; ${\gamma }_1={\mu }_1/\left[2\left(m_1+m_2\right)\right]$ is the motion damping coefficient of the reference body $m_1$ with reduced mass $m=m_1+m_2$ ; and ${\mu }_1$ is the linear viscous drag coefficient of the external medium to the motion of the reference body $m_1$ .

The known coordinate $x_{1O}$ (34) of the reference body $m_1$ and the coordinate transformations (27) ensure the determination of the coordinate $x_{2О}$ of the working body $m_2$ in the reference frame $K_O$ attached to the force center $O$ . Then, the coordinate $x_{СO}$ of the CM $С$ of the MS2 in this reference frame can be determined by the formula

$x_{СO}=\left(m_1{x}_{10}+m_2{x}_{20}\right)/m$ , (36)

where $m_1$ is the actual mass of the reference body $m_1$ .

4. Numerical Analysis of the Energy Balance and Least Action for the MS2

The closed MS2 shown in Figure 1 and Figure 2 has the following intrinsic parameters: $m_1=m_2=1$ kg; $R=1$ m; $r_{1C}=m_{2}R/\left(m_1+m_2\right)=0.5$ m and $r_{2C}=m_{1}R/\left(m_1+m_2\right)=0.5$ m.

1) Initial conditions and kinematic characteristics of the MS2 (Figure 1(a))

The magnitudes of the initial momenta ${\overrightarrow{p}}_1\left(0\right)=-{\overrightarrow{p}}_2\left(0\right)$ of bodies $m_1$ and $m_2$ are $p_1\left(0\right)=m_1{\upsilon }_1\left(0\right)=1\text{ kg}\cdot \text{m}/\text{s}$ and $p_2\left(0\right)=m_2{\upsilon }_2\left(0\right)=1\text{ kg}\cdot \text{m}/\text{s}$ , where ${\upsilon }_1\left(0\right)=1\text{m}/\text{s}$ and ${\upsilon }_2\left(0\right)=1\text{m}/\text{s}$ are the magnitudes of the velocities of bodies $m_1$ and $m_2$ in the reference frame $K$ before their interaction.

The kinetic energy $T_{er}$ (12) of the MS2 under perfectly elastic interactions must satisfy the conservation relation

$\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_1^2\left(0\right)+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_2^2\left(0\right)=\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{2C}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{1C}^2+\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{12}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{21}^2$ . (37)

Taking into account the MS2 momentum distribution (9), for $m_1=m_2$ we can write

${\upsilon }_{1С}={\upsilon }_{2С}={\upsilon }_{12}={\upsilon }_{21}$ . (38)

Then, for the initial velocities ${\upsilon }_1\left(0\right)={\upsilon }_2\left(0\right)=1\text{m}/\text{s}$ , relation (37) yields

${\upsilon }_{1С}={\upsilon }_{12}={\upsilon }_{2С}={\upsilon }_{21}=\frac{\sqrt{2}}{2}=0.707\text{m}/\text{s}$ . (39)

After bodies $m_1$ and $m_2$ begin to interact (at $t>0$ ), the kinematic characteristics of the MS2 system take the following values: ${\omega }_{12}=\frac{{\upsilon }_{21}}{R}=0.707{\text{ s}}^{-1}$ ; ${\omega }_{21}=\frac{{\upsilon }_{21}}{R}=0.707{\text{ s}}^{-1}$ and ${\omega }_{С}={\omega }_{1С}={\omega }_{2С}={\omega }_{12}+{\omega }_{21}=1.414{\text{ s}}^{-1}$ (Figure 1(a)).

In the reference frame $K$ , components of the total energy of the MS2 are

$T_e=\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{1e}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{2e}^2=\frac{1}{2}{m}_1{\omega }_{С}^2{r}_{2C}^2+\frac{1}{2}{m}_2{\omega }_{С}^2{r}_{1C}^2=0.5\text{ kg}\cdot {\text{m}}^{\text{2}}/\text{s}$; (40)

$T_r=\frac{1}{2}{m}_1{\upsilon }_{12}^2+\frac{1}{2}{m}_2{\upsilon }_{21}^2=\frac{1}{2}{m}_1{\omega }_{12}^2{R}^2+\frac{1}{2}{m}_2{\omega }_{21}^2{R}^2=0.5\text{ kg}\cdot {\text{m}}^{\text{2}}/\text{s}$ ; (41)

$\begin{array}{c}U=-\left(m_1{\upsilon }_{2C}{\upsilon }_{12}+m_2{\upsilon }_{1C}{\upsilon }_{21}\right)\\ =-\left(m_1{\omega }_C{r}_{2C}{\omega }_{12}R+m_2{\omega }_C{r}_{1C}{\omega }_{21}R\right)=-1\text{ kg}\cdot {\text{m}}^{\text{2}}/\text{s}\text{.}\end{array}$ (42)

The energy balance of the MS2 for initial momenta ${\overrightarrow{p}}_1\left(0\right)=-{\overrightarrow{p}}_2\left(0\right)$ is shown in Table 1.

Table 1. MS2 energy balance.

Initial conditions	$T_C$	$T_e$	$T_r$	$U$	$E_D$	$E$
${\overrightarrow{p}}_1\left(0\right)=-{\overrightarrow{p}}_2\left(0\right)$	0	0.5	0.5	–1	0	0
${\overrightarrow{p}}_1\left(0\right)=0;{\overrightarrow{p}}_2\left(0\right)=\left(0,1\right)$	0.25	0.125	0.125	–0.25	0	0.25

Since the kinetic energy $T_{er}=T_e+T_r=1$ of the MS2 (Table 1) is equal to the initial kinetic energy $T_{er}=T\left(0\right)=m_1{\upsilon }_1^2\left(0\right)/2+m_2{\upsilon }_2^2\left(0\right)/2=1$ of its bodies $m_1$ and $m_2$ , its kinetic energy is conserved (37). The zero total energy of the MS2 $E=E_D=0$ , where $E_D=T_{er}+U$ , implies that the kinetic energy $T_{er}$ of bodies $m_1$ and $m_2$ is periodically converted into their generalized kinetic potential $U$ and vice versa. For example, relative to the $x$ axis (Figure 1(a)), the kinetic potential of bodies $m_1$ and $m_2$ is at a maximum ($U=\max$ ) and their kinetic energy is at a minimum ($T_{er}=\min$ ). When the bodies rotate by an angle of$\pi /2$ , the situation is reversed. The global minimum of the total energy $E=E_D=0$ MS2 corresponds to the least action ($L=\min$ ) (20).

2) Initial conditions and kinematic characteristics of the MS2 (Figure 2(a))

The magnitudes of the initial momenta ${\overrightarrow{p}}_1\left(0\right)=0$ ; ${\overrightarrow{p}}_2\left(0\right)=\left(0,1\right)$ of bodies $m_1$ and $m_2$ are $p_1\left(0\right)=0$ and $p_2\left(0\right)=m_2{\upsilon }_2\left(0\right)=1\text{ kg}\cdot \text{m}/\text{s}$ , where ${\upsilon }_2\left(0\right)=1\text{m}/\text{s}$ is the magnitude of the velocity of body $m_2$ in the reference frame $K$ before its interaction with body $m_1$ .

The kinetic energy $T_{er}$ (12) of the MS2 under perfectly elastic interaction must satisfy the conservation relation

$\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_2^2\left(0\right)-\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{С}^2=\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{2C}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{1C}^2+\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{12}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{21}^2$ . (43)

Taking into account the momentum distribution of the MS2 (9) for $m_1=m_2$ we can write ${\upsilon }_{1С}={\upsilon }_{2С}={\upsilon }_{12}={\upsilon }_{21}$ (30). Then, for an initial velocity ${\upsilon }_2\left(0\right)=1\text{m}/\text{s}$ of

body $m_2$ and a velocity ${\upsilon }_{С}=\frac{m_2{\upsilon }_2\left(0\right)}{m_1+m_2}=0.5\text{m}/\text{s}$ of the CM $C$ , relation (43) yields

${\upsilon }_{1С}={\upsilon }_{12}={\upsilon }_{2С}={\upsilon }_{21}=\frac{1}{\sqrt{8}}=0.354\text{m}/\text{s}$ . (44)

After bodies $m_1$ and $m_2$ begin to interact (at $t>0$ ), the kinematic characteristics of the MS2 are as follows: ${\omega }_{12}=\frac{{\upsilon }_{12}}{2R}=0.354{\text{ s}}^{-1}$ ; ${\omega }_{21}=\frac{{\upsilon }_{21}}{2R}=0.354{\text{ s}}^{-1}$ and ${\omega }_{С}={\omega }_{1С}={\omega }_{2С}={\omega }_{12}+{\omega }_{21}=0.708{\text{ s}}^{-1}$ (Figure 2(a) and Figure 2(b)).

In the reference frame $K$ , the components of the total energy of the MS2 are

$T_{С}=\frac{1}{2}\left(m_1+m_2\right){\overrightarrow{\upsilon }}_{С}^2=0.25\text{ kg}\cdot {\text{m}}^{\text{2}}/\text{s}$; (45)

$T_e=\frac{1}{2}{m}_1{\overrightarrow{\upsilon }}_{1e}^2+\frac{1}{2}{m}_2{\overrightarrow{\upsilon }}_{2e}^2=\frac{1}{2}{m}_1{\omega }_{С}^2{r}_{2C}^2+\frac{1}{2}{m}_2{\omega }_{С}^2{r}_{1C}^2=0.125\text{ kg}\cdot {\text{m}}^{\text{2}}/\text{s}$; (46)

$T_r=\frac{1}{2}{m}_1{\upsilon }_{12}^2+\frac{1}{2}{m}_2{\upsilon }_{21}^2=\frac{1}{2}{m}_1{\omega }_{12}^2{R}^2+\frac{1}{2}{m}_2{\omega }_{21}^2{R}^2=0.125\text{ kg}\cdot {\text{m}}^{\text{2}}/\text{s}$ ; (47)

$\begin{array}{c}U=-\left(m_1{\upsilon }_{2C}{\upsilon }_{12}+m_2{\upsilon }_{1C}{\upsilon }_{21}\right)\\ =-\left(m_1{\omega }_C{r}_{2C}{\omega }_{12}R+m_2{\omega }_C{r}_{1C}{\omega }_{21}R\right)=-0.25\text{ kg}\cdot {\text{m}}^{\text{2}}/\text{s}\text{.}\end{array}$ (48)

Table 1 shows the energy balance of the MS2 system when its bodies $m_1$ and $m_2$ have initial momenta ${\overrightarrow{p}}_1\left(0\right)=0$ and ${\overrightarrow{p}}_2\left(0\right)=\left(0,1\right)$ kg∙m²/s. For these initial momenta, the kinetic energy of the MS2 contains an additional term in the form of the kinetic energy $T_C=\left(m_1+m_2\right){\upsilon }_C^2/2=0.25$ of its CM $C$ . This term depends on the initial conditions ${\overrightarrow{p}}_1\left(0\right)=0$ ; ${\overrightarrow{p}}_2\left(0\right)=\left(0,1\right)$ and can be omitted since ${\overrightarrow{p}}_{С}=\overrightarrow{\text{const}}$. Then, the kinetic energy $T_{er}=T_e+T_r=0.25$ of the MS2 (Table 1) is equal to the initial kinetic energy $T_{er}=T\left(0\right)=m_1{\upsilon }^2\left(0\right)/2+m_2{\upsilon }^2\left(0\right)/2-m_1{\upsilon }_{С}^2/2$ of its bodies $m_1$ and $m_2$ minus the omitted term $T_C$ , so that the kinetic energy of the MS2 in the form (43) is conserved. The total energy of the MS2 for the initial conditions ${\overrightarrow{p}}_1\left(0\right)=0$ and ${\overrightarrow{p}}_2\left(0\right)=\left(0,1\right)$ is not zero: $E=T_C+E_D=0.25$ , where $E_D=T_{er}+U=0$ is a term characterizing the periodic conversion of the kinetic energy $T_{er}$ of bodies $m_1$ and $m_2$ into their generalized kinetic potential $U$ and vice versa. For example, relative to the $x$ axis (Figure 2(b)), the kinetic potential of bodies $m_1$ and $m_2$ is at a maximum ($U=\max$ ) and their kinetic energy is at a minimum ($T_{er}=\min$ ). When the bodies rotate by an angle of $\pi /2$ , the situation is reversed. The local minimum of the total energy $E=T_C+E_D=0.25$ of the MS2 for $E_D=T_{er}+U=0$ corresponds to the least action ($L=\min$ ) (26).

5. Conclusions

Thus, based on D’ Alembert’s principle (1) formulated taking into account the constraint axiom for a constrained material point, equations of motion for each of bodies $m_1$ and $m_2$ of the closed MS2 system were derived describing their dynamics in a fixed laboratory reference frame $K$ in the field of the relative inertial forces of the diametrically opposite bodies $m_2$ and $m_1$ . The obtained results are extended to the case of their complex translational relative translational motion for the closed MS2 system with five ($s=5$ ) degree of freedom, as well as to their motion for the MS2 with two ($s=2$ ) degrees of freedom in a dissipative medium.

It is shown that for the MS2 with five ($s=5$ ) degrees of freedom, time ($t_1=\frac{m_1+m_2}{m_1}{t}_C$ and $t_2=\frac{m_1+m_2}{m_2}{t}_C$ ) in the proper reference frames $K_1$ and

$K_2$ of the interacting bodies $m_1$ and $m_2$ flows differently than time in the reference frame $K_{С}$ of their CM $C$ . In the frame $K_{С}$ and in the laboratory frame $K$ , time is absolute ($t_C=t$ ).

By constructing the Lagrangian of the closed MS2 with five ($s=5$ ) degrees of freedom, it is shown that its motion in the field of relative inertial forces satisfies the laws of conservation of momentum and energy, as well as the principle of least action under the specified initial conditions.

The relevance of this study lies in the fact that it establishes the relationship between the relativity of kinematic variables (including time) and the dynamics of each of the interacting bodies $m_i$ and $m_j$ of the MS2 in the stationary laboratory reference frame$K$ through the structure of the system Lagrangian that defines the intrinsic (${}_{\overrightarrow{\text{Ф}}}$) and relative (${}_{\overrightarrow{\text{Ф}}}$ and ${\overrightarrow{F}}_{iC}$ ) inertial forces of its bodies $m_i$ and $m_j$ . The relative inertial forces ${}_{\overrightarrow{\text{Ф}}}$ and ${\overrightarrow{F}}_{iC}$ of the interacting bodies $m_i$ and $m_j$ act are equivalent to the reaction ${\overrightarrow{R}}_{ij}$ of their constraint $R$ in D’ Alembert’s principle (1), which defines the dynamic Equations (11) and (19), thereby simplifying the dynamic analysis of complex mechanical systems.

The results of this work can find useful in applications for the synchronization and dynamic analysis of complex motions in robotics, the design of propulsion devices in low-dissipation media [3] [4] [9], as well as for the synchronization of complex orbital motions of artificial and natural celestial bodies interacting via a gravitational field.