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]:

Here Ф → i = − m i а → i is the inertial force of body m i in a fixed laboratory reference frame K ( а → i is its absolute acceleration in this frame); R → j i is the reaction of the constraint R directed from body m i to body m j , which is counteracted by the inertial force Ф → i 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 Ф → i 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 Ф → j i of these bodies, replacing the unknown reactions R → j i ; i , j = 1 , 2 ; i ≠ 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 θ С = − π (a) and a map of events e 1 and e 2 showing that for an invariant rotation of body m 2 by an angle Δ φ = { 0 ; π / 2 } 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 Ф → i = − m i d d t υ → i = − m i 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 Ф → j i = − d d t m j υ → j i = − m j a → j i ; i , j = 1,2; i ≠ j . It is

applied to the working body m j , which performs relative motion with relative velocity υ → i j and acceleration а → i j ; i , j = 1, 2 ; i ≠ 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 ≫ m j , the body with the greater mass should be chosen as the reference body.

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 Ф → i C and relative Ф → j i 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 ≠ 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 r → 1 C and r → 2 C , 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 r → 12 and r → 21 with magnitudes r 12 = r 21 = R . The radius vectors r → 1 C and r → 2 C are defined by the equations m 1 r → 1 С + m 2 r → 2 С = 0 and r → 21 = r → 2 C − r → 1 C [1][5]:

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 1 C and φ 1 C of the reference body m 1 in the frame K C of the CM C and one polar coordinate φ 21 of the working body m 2 in the proper frame K 1 of the reference body m 1 .Two polar coordinates r 2 C and φ 2 C of the reference body m 2 in the reference frame K C of the CM C and one polar coordinate φ 12 of the working body m 1 in the proper frame K 2 of the reference body m 2 .

The polar coordinates φ 12 and φ 21 of the closed MC2 under consideration are kinematically dependent φ 12 = φ 21 + π (Figure 1(a)). This kinematic dependence satisfies the requirement for dynamic self-consistency Ф → 12 + Ф → 21 = 0 of this closed MC2, which is important when the reactions R → 12 and R → 21 constraints R of its bodies m 1 and m 2 are replaced by their relative inertial forces R → 12 ⇒ Ф → 12 and R → 21 ⇒ Ф → 21 .

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 = 3 n − l = 3 ⋅ 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 = 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 e r 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 p → i С = m i υ → i С ; 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 υ → i С = ω → i С × r → i С is the velocity of body m i in the frame K С and ω → i С = ω → С and r → i С 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 p → i j = m i υ → i j ; i , j = 1 , 2 ; i ≠ j of body m i , which characterizes the motion of this body along a trajectory s i j in the proper frame K j of the diametrically opposite body m j . Here υ → i j = ω → i j × r → i j is the relative velocity of body m i in the reference frame K j and ω → i j and r → i j are the angular velocity and radius vector of the translational motion of this body in the frame K j .Translational momentum p → i e = m i υ → i e ; 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 υ → i е = υ → j C = ω → j C × r → j C 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 υ → j C = ω → j C × r → j C in the frame K С .

For bodies m 1 and m 2 with initial momenta p → 1 ( 0 ) = − p → 2 ( 0 ) , we establish the relationship between the intrinsic p → i С = m i υ → i С ; i = 1 , 2 , relative p → i j = m i υ → i j ; i , j = 1 , 2 ; i ≠ j , and translational p → i e = m i υ → i e ; i = 1 , 2 momenta that satisfies the conservation of momentum ( p → С = 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 p → 1 ( 0 ) = − p → 2 ( 0 ) (Figure 1(a)) under instantaneous transmission of interactions, their distribution in the reference frame K С is diametric:

This corresponds to the concepts of optimal periodic motion [6][7] for a closed MS2 [3][4], where θ С is the lag angle of the phase φ 1 C = φ 21 + θ C = φ 21 − π of the translational motion body m 1 in the reference frame K С relative to the phase φ 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 p → i ( 0 ) = ( 0 , 1 ) ; i = 1 , 2 of each of its bodies m i is distributed equally between its relative p → i j ; i , j = 1 , 2 ; i ≠ j and intrinsic p → i С ; i = 1 , 2 momenta. Consequently, when m 1 = m 2 , the relative and translational velocities are equal υ → i j = υ → i C , as for body m 2 shown in Figure 1(b). The magnitudes of these velocities υ 12 = υ 1 C and υ 21 = υ 2 C will be determined based on the kinetic energy conservation law for the MS2 during the analysis of its energy balance. The velocity υ → C of the CM С in the reference frame K is υ → C = ( p → 1 ( 0 ) + p → 2 ( 0 ) ) / m = 0 , as shown in Figure 1(b). Acquiring the momenta p → i j and p → i С , the MS2 bodies m 1 and m 2 begin to perform complex translational relative motion in the reference frame K with absolute velocities

where υ → i е = υ → j C = ω → j C × r → j C and υ → i j = ω → i j × r → i j 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

obtained taking into account the absolute velocities υ → i (4), implies that for the initial momenta p → 1 ( 0 ) = − p → 2 ( 0 ) of bodies m 1 and m 2 , the velocity υ → С of their CM C is υ → С = 0 .

Definition (5) yields the following relationship between the translational p → i e = m i υ → i e = m i υ → j C and relative p → i j = m i υ → i j = m i ω → i j × r → i j ; i , j = 1 , 2 ; i ≠ j momenta:

In view of the definition m 1 r → 1 С + m 2 r → 2 С = 0 for the CM C , the translational momenta p → i e = m i υ → i e = m i υ → j C ; i , j = 1 , 2 ; i ≠ j are linked by a similar relation:

Then, it follows from definition (5) that

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

which ensure the conservation of its momentum

for υ → С = 0 (5). Equalities (9) will be used to analyze the energy balance of the MS2 for two sets of initial momenta p → 1 ( 0 ) = − p → 2 ( 0 ) and p → 1 ( 0 ) = 0 , p → 2 ( 0 ) = ( 0 , 1 ) of its bodies m 1 and m 2 .

When m 1 = m 2 , equalities (6) reduce the sums of the translational υ → i e and relative υ → i j velocities to zero ( υ → 1 e + υ → 12 = 0 and υ → 2 e + υ → 21 = 0 ), which is consistent with (5), resulting in zero absolute velocity υ → i = 0 (4) for each of the bodies m i in the reference frame K . However, since the translational υ → i e and relative υ → i j velocities of the bodies m i are not individually zero, they uniquely determine the kinetic T e r 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 υ → 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 φ j i ; i , j = 1 , 2 ; i ≠ j of the working body m j in the proper frame K i of the reference body m i . Then, the reaction R → j i 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 R → j i = Ф → j i = m j ω j i 2 r → j i 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 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 :

which does not contain the unknown reactions R → j i (1) of the kinematic hinged-rod constraint R and where 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 Ф → j i (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 υ → 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]

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

which characterizes the contribution of the relative motion ( υ → 12 r , υ → 21 r ) of bodies m 1 and m 2 to their translational motion ( υ → 1 e = υ → 2 С , υ → 2 e = υ → 1 С ); here θ С = υ → 2 e υ → 21 ^ = − π is the angle between the vectors of the translational υ → 2 e and relative υ → 21 velocities (Figure 1(a)).

The sum of the first and second terms included in T e r (12) is the kinetic energy of the translational motion T С e of bodies m 1 and m 2 in the reference frame since K , taking into account the absolute velocities υ → i (4), it characterizes the translational momentum p → C e = ( m 1 + m 2 ) υ → С = ∑ i m i υ → i = m 1 υ → 1 e + m 1 υ → 12 + m 2 υ → 2 e + m 2 υ → 21 = m 1 υ → 1 e + m 2 υ → 2 e of MS2 as a whole.

The sum of the first and second terms in T e r (12) is the kinetic energy of the translational motion of bodies m 1 and m 2 . Taking into account the permutation m 1 ⇔ m 2 , valid for equal masses m 1 = m 2 , this sum satisfies the equality

expressed in terms of the translational momenta p → 1 e = m 1 υ → 1 e = m 1 υ → 2 C and p → 2 e = m 2 υ → 2 e = m 2 υ → 1 C 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 С е .

The sum of the third and fourth terms of the kinetic energy T e r (12) is the kinetic energy T r = 1 2 m 1 υ 12 2 + 1 2 m 2 υ 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:

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 r → 1 С 0 = − r → 2 С / r 2 С and r → 2 С 0 = − r → 1 С / r 1 С are the unit vectors opposing the radius vectors r → 2 С and r → 1 С (2) and defining the direction of the relative inertial forces F → 1 С and F → 2 С 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 e r (12) into transportive T С e 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 υ → 1 e = υ → 2 С , υ → 2 e = υ → 1 С and relative υ → 12 , υ → 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 1 C = p 12 and p 2 C = p 21 (6) and the radii r 1 С and r 2 С (2), the angular velocities ω 1 С = ω 2 С = ω С of bodies m 1 and m 2 in the reference frame K С and their angular velocities ω 12 and ω 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]:

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:

where Δ t 1 = t 1 − t 0 , Δ t 2 = t 2 − t 0 , and 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 Δ φ = { 0 ; π / 2 } 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 ω 21 and ω 12 (16) in reference frames K 1 and K 2 , different from the angular velocity ω С in reference frame K С , the time increments Δ t С , Δ t 1 and Δ t 2 in these frames must be different. That is, in reference frame K 1 or K 2 , where the angular velocity ω 21 or ω 12 (16) is higher ω 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 ω 21 and ω 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 φ С = ω С t C , φ 21 = ω 21 t 1 and φ 12 = ω 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 ω С , ω 21 and ω 12 , and is not postulated on the current model of the clock.

From the invariant Δ φ (17) one can determine the proper time t 1 and t 2 in the reference systems K 1 and K 2 :

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 = 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 F → 1 С and F → 2 С (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

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 j С = m j ω i С ω j i r j i ; i , j = 1 , 2 ; i ≠ j in Equation (19) have the same dimensions as the relative inertial forces Ф → j i = − m j ω j 2 r j i ; i , j = 1 , 2 ; i ≠ 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 R → j i (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 R → j i of the constraint R is equivalent to the action of the relative inertial forces Ф → j i and F → j i . 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 R → j i 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 F → j С (19). That is, the redistribution mechanism consists of the action of the relative inertial force F → j С 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 F → j С 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 p → С (10).

• To conserve the total momentum p → С = 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 θ С = − π (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 θ С = − π , 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 С e = 1 2 m 1 υ → 1 e 2 + 1 2 m 2 υ → 2 e 2 in (12) by T С e = 1 2 m 1 υ → 1 C 2 + 1 2 m 2 υ → 2 С 2 in accordance with equality to (14) and taking into account that the external potential energy of the closed MS2 is zero ( Π = 0 ) yields its Lagrangian

in the reference frame K for m 1 = m 2 .

The partial derivatives of the Lagrangian function L (20) of the form − ∂ L r 1 C = 0 and − ∂ L r 2 C = 0 yield the generalized forces

in the form of the D’ Alembert principle (19) in curvilinear coordinates, in the field of relative inertial forces F → j C , which establishes their equivalence to reactions R → i j .

Indeed, the first terms of (21) define the intrinsic inertial forces Ф → 1 = Ф → 1 C = − m 1 ω 1 C 2 r → 1 С and Ф → 2 = Ф → 2 C = − m 2 ω 1 C 2 r → 2 С 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 F → 2 C = m 2 ω 1 C ω 21 r 21 ⋅ r → 2 C 0 and F → 1 C = m 1 ω 2 C ω 12 r 12 ⋅ r → 1 C 0 of bodies m 2 and m 1 during their complex translational relative motion, where the choice of direction for the unit vectors r → 1 С 0 = − r → 2 С / r 2 С and r → 2 С 0 = − r → 1 С / r 1 С toward the CM С (Figure 1(a)) is due to the fact that the forces F → 2 C and F → 1 C are counteracted by the intrinsic inertial forces Ф → 1 and Ф → 2 of bodies m 1 and m 2 (21).

Equation (19), expressed in terms of the relative inertial forces F → 1 С and F → 2 С (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 R → j i (1) of their constraint R . Since these forces form a balanced system of forces F → 1 С + F → 2 С = 0 , their action on the closed MS2 system satisfies the conservation of its momentum ( p → C = 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 p → 1 ( 0 ) = 0 and p → 2 ( 0 ) = ( 0 , 1 ) , respectively, as shown in Figure 2(a).

In the MS2 with five ( s = 5 ) degrees of freedom, the initial momentum p → 2 ( 0 ) = ( 0 , 1 ) 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 υ → 1 e , υ → 2 e ( υ → 1 e = υ → 2 C , υ → 2 e = υ → 1 C ) and the relative velocities of its bodies m 1 and m 2 υ → 21 , υ → 12 become equal to each other: υ → 12 = υ → 1 C and υ → 21 = υ → 2 C (Figure 2(b)). Their magnitudes υ 12 = υ 1 C and υ 21 = υ 2 C will be determined from the energy conservation law of the closed MS2 in the subsequent analysis of its energy balance. The velocity υ → C of the CM С in the reference frame K is υ → C = p 2 ( 0 ) / m , as shown in Figure 2(a). Acquiring the momenta p → i j and p → i С , bodies m 1 and m 2 begin to perform complex translational relative motion along the circle s 1 , 2 C 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

obtained taking into account the absolute velocities υ → i (4), the motion of the CM C can be considered translational υ → С ≡ υ → С e .

Figure 2. Kinematics of the MS2 system whose bodies m 1 and m 2 have initial momenta p → 1 ( 0 ) = 0 and p → 2 ( 0 ) = ( 0 , 1 ) .

Then, the translational velocities υ → i e of bodies m 1 and m 2 are the sum

whose cross terms vanish when it is squared:

Replacing the translational velocity υ → i е in (12) by υ → i e , a b s (23) yields the kinetic energy T e r of the MS2 in the form

where T С = 1 2 m υ → С 2 is the kinetic energy of the rectilinear uniform motion of the CM С with velocity υ → 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 = 1 2 m 1 υ → 1 e 2 + 1 2 m 2 υ → 2 e 2 and the generalized kinetic potential U = − ( m 1 υ 2 C υ 12 + m 2 υ 1 C υ 21 ) , which defines the relative inertial forces F → 1 C and F → 2 C (15).

Based on Galileo’s principle of relativity, when p → C = 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 ⇔ m 2 (14), allowable when m 1 = m 2 , its Lagrangian can be written in the form

similar to (20).

From the Lagrangian L (26), one can determine the relative inertial forces F → 1 C and F → 2 C (15) and the Equation of motion (19) for each of the MS2 bodies ( m 1 and m 2 ).

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

as follows [3][4]:

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 φ 21 = ω 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 ω 12 = 0 and the relative velocity ω 21 = const , which is maintained constant by the internal active pair of forces [3][4].

Then, the absolute momentum MS2 can be represented as