Augmented Perpetual Manifolds and Perpetual Mechanical Systems-Part II: Theorem and a Corollary for Dissipative Mechanical Systems Behaving as Perpetual Machines


 Perpetual points in mathematics defined recently, and their significance in nonlinear dynamics and their application in mechanical systems is currently ongoing research. The perpetual points significance relevant to mechanics so far is that they form the perpetual manifolds of rigid body motions of mechanical systems. The concept of perpetual manifolds extended to the definition of augmented perpetual manifolds that an externally excited multi-degree of freedom mechanical system is moving as a rigid body. As a continuation of this work, herein the internal force’s and their associated energies, for a motion of multi-degree of freedom dissipative flexible mechanical system with solutions in the exact augmented perpetual manifolds, leads to the proof of a theorem that based on a specific decomposition with respect to their state variables dependence, all the internal forcing vectors are equal to zero. Therefore there is no energy storage as potential energy, and the process is internally isentropic. This theorem provides the conditions that a mechanical system behaves as a perpetual machine of a 2nd and a 3rd kind. Then in a corollary, the behavior of a mechanical system as a perpetual machine of third kind further on is examined. The developed theory leads to a discussion for the conditions of the violation of the 2nd law of thermodynamics for mechanical systems that their motion is described in the exact augmented perpetual manifolds. Moreover, the necessity of a reversible process to violate the 2nd thermodynamics law, which is not valid, is shown. The findings of the theorem analytically and numerically are verified. The energies of a perpetual mechanical system in the exact augmented perpetual manifolds for two types of external forces have been determined. Then, in two examples of mechanical systems, all the analytical findings with numerical simulations, certified with excellent agreement. This work is essential in physics since the 2nd law of thermodynamics is not admitting internally isentropic processes in the dynamics of dissipative mechanical systems. In mechanical engineering, the mechanical systems operating in exact augmented perpetual manifolds, with zero internal forces, there is no degradation of any internal part of the machine due to zero internal stresses. Also, the operation of a machine in the exact augmented perpetual manifolds is of extremely high significance to avoid internal damage, and there is no energy loss.


INTRODUCTION
The perpetual points have been defined recently by Prasad in [1] as the sets of points that arise when the accelerations and jerks, describing the motion in a mechanical system, for non-zero velocity vector, are equal to zero. In [2], the experimental investigation of perpetual points in mechanical systems in a tilted pendulum is reported. So far, relevant to perpetual points, there are four main research directions. The first one is directly related to the perpetual points with the strict mathematical formulation of them [1,[3][4], including the experimental research for identifying the perpetual points of mechanical systems [2]. In the second research direction, the perpetual points are used to advance nonlinear dynamics, such as locating hidden and chaotic attractors [5][6][7][8][9][10][11][12][13]. The third research direction is through perpetual points to identify dissipative systems [14][15][16][17][18] and the fourth one with their significance in mechanics [19][20][21][22].
This article is a continuation of the research relative to mechanics, that already, there are three theorems relevant to perpetual points, proved in [19][20][21]. The first two correlate the perpetual points of linear mechanical systems with rigid body motions, in [19] for conservative mechanical systems and in [20] for dissipative systems. Also, the perpetual points of mechanical systems that are forming perpetual manifolds in [20] is shown. In [21], based on the perpetual manifolds concept, the augmented perpetual manifolds are defined as those manifolds that arise in a multi-degrees of freedom flexible mechanical system that all the accelerations are equal but not necessarily zero. In the exact augmented perpetual manifolds, a multi-degree of freedom system is moving as a rigid body, since all inertia elements are having the same generalized coordinates. A theorem is proved by indicating the conditions that a solution of a nonlinear non-autonomous mechanical system can be in the exact augmented perpetual manifolds [21]. More precisely, the forces in the equations of motion are separated as internal forces and external forces. The internal forces correspond to a perpetual mechanical system by means this system admits exact rigid body motions as a solution. Then, the form of the external forces and initial conditions have been defined, such as the perpetual mechanical subsystem's motion is described by one ordinary differential equation with solutions in the exact augmented perpetual manifolds. The theorem's outcome leads to a corollary stating that the application of harmonic external forcing in a perpetual mechanical system leads to particle-wave motion.
The standing and the longitudinal wave-particle motion of a multi-degrees of freedom mechanical system in [21] is shown. Another immediate outcome of the theorem in [21][22] is a corollary proved in [22] that in the exact augmented perpetual manifolds, the sum of the internal forces is zero.
This article is divided into two sections, the theoretical and the numerical section with the examples, and this work is a continuation of the work done in [21][22] by examining the internal forces and the energies at the exact augmented perpetual manifolds of natural mechanical systems incorporating the view of the 2nd law of Thermodynamics through the entropy definition given in [23].
In §2.1, the preliminary definitions, such as the relevant theory of the exact augmented perpetual manifolds developed in [21], separating the forces from internal and external and analysing them, lead to the definitions of the different types of energy of the natural mechanical system. Finally the entropies definitions for a closed system are shown.
In §2.2 a theorem is stated proved, by defining the conditions that all the individual internal forces, but not only as a sum that in [22] is shown, are zero when the exact augmented perpetual manifolds describe the dynamics of the perpetual mechanical system. Therefore dissipative forces too, which leads to noloss of internal energy as a heat to the environment which lead to a discussion about the existence of perpetual machines of 2 nd , 3 rd kind, and the validity of the 2nd law of thermodynamics, e.g., is it actual requirement the reversibility of the process, and the 'arrow of time' associated with the entropy, is discussed.
As a note, for an out of the content, with the current article, science, the research of a pioneered team lead by Ilia Prigogine, examined the 2nd law of thermodynamics for systems far away from thermodynamic equilibrium in chemistry [24]. In §2.3, a corollary is stated and proved, whether or not a mechanical system behaves as a perpetual machine of 3 rd kind in a reversible process.
In §2.4, the validity of the developed theory in sections 2.2 and 2.3 is examined analytically using two types of external forces. In section 2.4.1 the analytical determination of the different types of energies of a natural mechanical system in the exact augmented perpetual manifolds for the two types of external forcing is shown. In section 2.4.2, there is a preliminary analytical investigation, whether or not the two types of external forces lead to a perpetual machine of third kind behaviour.
The 2nd law of thermodynamics states that 'in a process, the entropy of a system is increasing' [23,[25][26].
In section 3, the analytical findings with two examples are numerically. The first example is a perpetual mechanical system, and the 2 nd example is a non-perpetual mechanical system, but within some boundaries, there is a perpetual mechanical subsystem, under conditions.

Preliminaries
In some mechanical systems, the perpetual points are not just a few points, but they form the perpetual manifolds [19][20]. In [21], the exact perpetual manifolds of rigid body motions were defined as the sets of perpetual points that correspond to exact rigid body motions, whereas all masses have the same displacement.
Moreover, in [21], the perpetual mechanical systems were defined as the unforced systems that admit exact perpetual manifolds solutions. Also, in [21], the concept of augmented perpetual manifolds is introduced, and these are manifolds defined by the solutions of the equations of the forced perpetual mechanical systems in the state-space when all the time depended accelerations have the same but not necessarily zero values. In forced systems, the solutions, that all the generalized velocities are the same and all the generalized coordinates are the same, form the exact augmented perpetual manifolds of rigid body motions. In mathematical form are defined as follows [21], The 2N +1 dimensional Exact Augmented Perpetual Manifolds e.g. of a N-dof mechanical discrete system, with generalized coordinates , that admits solutions of perpetual manifolds, arise when, ̈( ) =̈( ), for = 1, … , , whereas the overdot means the derivatives in time, and based on the above definitions, a theorem in [22] is written and in [21] is proven, and it is about the conditions that a mechanical system has a solution in an exact augmented perpetual manifold.
In this article, the developed theory is restricted to natural mechanical systems, and the theorem of [21][22] can be written as follows: Any (≥ 2)-degrees of freedom discrete natural perpetual mechanical systems with constant inertia matrix− , described by the following equations of motion, that lead to the following characteristic differential equation, describing the motion, with vector field G, for a set of initial conditions, at the time instant 0 , given by equations, defines the generalized coordinates− and their velocities−̇ in the exact augmented perpetual manifold as described by equation (2).
The sign ′ × ′ for multiplication of matrices is used, and the sign ′ • ′ for the scalar product is used.
Forces analysis for the development of the theory -The generalized coordinate's only dependent forces (linear and nonlinear), at least in the exact augmented perpetual manifolds, they must arise by an elastic potential. The nonlinear generalized coordinate's dependent forces, in their general form, since they can be nonsmooth functions, it might be impossible to be defined, even by using sub-differentials of an elastic potential function. In the exact augmented perpetual manifolds, they must be single-valued forces, and their elastic potential function becomes mathematically possible to be defined.
-The existence of 'external' forces− that are on the right-hand side of equations (3) means that the perpetual natural mechanical system is not isolated, but it is a subsystem. These forces in the exact augmented perpetual manifolds either correspond to more 'grounded' (connected with rigid places) elements, e.g.
springs, dashpots, etc., that build up the overall system or external forces of a field or any other source by the environment. Therefore the 'external' forces− are the forces that connect the perpetual natural mechanical subsystem with the 'environment'. The boundaries for perpetual mechanical system internal energy analysis are defined by its limits and are associated with the forces on the lefthand side of equations (3). These elements are not considered part of the perpetual natural mechanical subsystem because their consideration does not lead to a perpetual natural mechanical subsystem. Therefore these elements are treated as external forces, and they are separated on the right-hand side of equations (3). The energy of all the internal forces included in equation (4) in the exact augmented perpetual manifolds is given by integrating the internal forces power of the system, as follows: which means that the sum of the energies of the different types of internal forces is equal to zero. Although the energies associated with the two types of linear internal forces in linear perpetual mechanical systems is shown in [20] that are zero, but this is not the case for the arbitrarily defined nonlinear forces since the energy associated with each one of them might change form between them with a sum equal to zero.
In order to understand clearly the energies associated with each type of nonlinear internal forces (might be nonsmooth too) of equation (4) whereas, the first term is nonlinear generalized coordinates dependent forces and must be associated with potential energy, the second term are nonlinear generalized velocities depended forces and the third term being the generalized nonlinear forces with the condition that none part of the third term can be decomposed as a linear combination of any of the other two types of nonlinear forces.
In the following part of this section, the energies of natural mechanical systems with a motion described by equations (3) are defined as follows [27][28][29]: -Kinetic energy, for smooth functions of velocities, is given by ( ), whereas for natural mechanical systems is limited to the so-called " 2 " kinetic energy [29].
-Potential energy ( ) of the perpetual mechanical system, whereas, the first term is the potential energy associated with the linear forces.
Since the nonlinear forces are not necessarily smooth functions, therefore the second term of the potential of the nonlinear forces in this general form might be impossible to be defined even through sub-differentials of forces Nevertheless, the restrictions of these types of forces, for a solution in the exact augmented perpetual manifolds, lead that these potentials can be defined.
-Energy (̇) associated with the velocity dependent forces is given by [28], whereas, the first term corresponds to linear forces. This integral is not always defined in the general nonlinear and nonsmooth form of the velocities dependent nonlinear forces. This general case that the integral might not be defined is not a problem for developing this theory. The reason is that the examination of the dynamics is restricted in the exact augmented perpetual manifolds that certain restrictions of the functional form of these forces are given, and these restrictions lead to the existence of the integral of equation (10c). Upon each energy explicit functional form term, these forces, linear and nonlinear, might be dissipative or not (flutter) [28]. If any term of the energy of the generalized velocities dependent forces in equation (10c) is negative, then the system is gaining mechanical energy associated with this force (linear or nonlinear). If it is positive, the system is losing mechanical energy, and the associated forces are dissipative. The initial energy associated with this type of forces is related to the system's dynamics from previous time instants and represents the energy lost or gained in the system's mechanical energy, which is the reason that is disregarded.
-Energy of the generalized forces ( ) [28], Similarly, with the nonlinear generalized velocity-dependent forces, due to the force's general type, which might be nonsmooth, this integral is not always defined, e.g., at the force's discontinuities. Herein, the theory is developed around the augmented perpetual manifolds and lead to integrable functions in equation (10d). As mentioned previously, if the energy of generalized forces in equation (10d) is negative, then the system is gaining mechanical energy. In the case of being positive, the system loses mechanical energy, and they are dissipative forces. The generalized force's initial energy represents the mechanical energy gained or lost from the system's previous dynamic evolution and is disregarded.
-Power ( ), and the work done ( ) by the external forces is given by [28], and, whereas, for any functional form of the external forces, which might be nonsmooth functions, the integral of equation (10f) cannot always be defined, e.g.
in time instants e.g. with such discontinuities that the integral given by the equation (10f) is not defined. The external force's initial energy is the energy gained or lost of the mechanical energy throughout previous time instances. That is the reason that is disregarded.
In the exact augmented perpetual manifolds, considering a solution in the form of equation (2), the kinetic energy is taking the form, The power of the external forces is defined by equation (10e), and considering a correlation of external forces given by equation (5), then is taking the form, • , , � ( ),̇( ), �, (11b) whereas the last term � , , � is the power of the −external force of the −reference mass used in the derivation of the characteristic differential equation.
Considering equation (11b) in equation (10f) lead that the total work done by the external forces � , � is given by, whereas the last term � , , � is the work done by the −external force of the −reference mass.
The work done by the external forces may cause either decrease or increase of the system's energy, and at each time instant, the positive (negative) sign of the power associated with the external forces indicates if it is actually input (output) energy.
The equations (3) for smooth forces can be easily derived using the least action principle, Newton's law or Lagrange equation. In a system with nonsmooth nonlinear forces (internal and external), the derivation of equations of motion through the least action principle has some limitations, and as representative constraint, the integrands must be nonsmooth but convex functions [30][31][32]].
The mechanical system described by equations (3) is not necessarily a representation of a specific configuration of a mechanical structure but generally a system that the perspective of the theory of mechanics is applied, e.g. a solid-state lattice [33]. In that respect, the mechanical systems' dynamics in the exact augmented perpetual manifolds through thermodynamics is examined in the following section.
More precisely, the perpetual natural (without any gyroscopic effects) mechanical systems in the exact augmented perpetual manifolds, through the individual energies of the system, are examined by considering the 2nd law using the entropy definition [23].
In the following part of this section, some preliminary definitions from thermodynamics are introduced.
The perpetual mechanical systems are closed systems since the number of degrees of freedom remains constant [23,[25][26].
In a process the total change of entropy (Δ ) of a system is comprised by two terms [23], whereas, the first term is associated with the entropy of the surroundings of the system, and for closed systems (perpetual mechanical subsystem adding the source of the external forces) is given by, with ∆ being the heat difference in an absolute temperature ( ). The difference of the surroundings entropy is zero for an adiabatic process.
The second term in equation (12a) is associated with the internal change of entropy, The 2 nd law of thermodynamics through the entropy definition states that, and, with the equality to zero in case of reversible process.
In the considered mechanical configuration, the thermal environmental source of energy is not examined. Only the loss of mechanical energy in the form of heat due to dissipation within the perpetual mechanical system is examined, which leads to the examination of the internal entropy difference (Δ ) in a process described by the dynamics of a perpetual mechanical subsystem in the exact augmented perpetual manifolds.
In an environment of a perpetual mechanical system that the source of energy provides the external forces to the perpetual mechanical system through an adiabatic process which means that the entropy in the environment is constant, and in case that the internal entropy is constant, then the system behaves as a perpetual machine of 2nd kind [23].
The perpetual machine of 3 rd kind is not related to the 2 nd law of thermodynamics [26]. The entropy of the surrounding environment is not considered but only the internal entropy of the system, and the perpetual machine of 3 rd kind is defined when there is no energy loss, e.g. due to friction in the mechanical system [26].
In the next section, a theorem and a corollary are proved for perpetual mechanical subsystems when the exact augmented perpetual manifolds define their motion.

Theorem
In this section, the theoretical developments in [21][22] are extended by examining energies in the exact augmented perpetual manifolds of the perpetual natural mechanical systems. The examination of the individual energies, associated with each type of internal forces of a mechanical system, when its motion is described in the exact augmented perpetual manifolds, leads to the proof of a theorem.
whereas, the external forces are on the right-hand side of equation (13) In the exact augmented perpetual manifolds with a solution defined by the differential equation (6), the system of equations (13) is taking the form, taking common factor the acceleration, and considering equations (5) lead to, Since the motion in the exact augmented perpetual manifolds is defined by equation (6) which is valid for any combination of arbitrary values of generalized coordinates, and of their non-zero velocities leading to, and, Taking into account equations (15a,b) into equation (14c) leads to, In the exact augmented perpetual manifolds all the nonlinear internal force's vectors required to be orthogonal to each other, and multiplying equation (15c) with the transpose of the −element of the first sum of vectors of equation (15c) leads to, Repeating the same pre-multiplication of equation (15c) using the ℎ −element of the second sum of vectors of the nonlinear forces of equation (15c) leads to, And with the −element of the third sum of vectors of the nonlinear forces of equation (15c) leads to, Therefore in the exact augmented perpetual manifolds, all the individual terms of internal forces vectors, based on the decompositions indicated in equation (14c), they are zero. b) Considering the definition of power and equations (15a,15d), then the potential energy in the exact augmented perpetual manifolds is given by, and remains constant, with it's initial value in the exact augmented perpetual manifolds, that is associated with the nonlinear forces. Therefore during the dynamic evolution there is no energy storage as potential energy on the mechanical system.
c) The energy associated with the velocities' depended linear forces is given by the first term of equation (10c), and in the exact augmented perpetual manifolds considering equation (15b) is taking the form, The energy associated with the velocity depended nonlinear forces, is given by the second term of equation (10c), and in the exact augmented perpetual manifolds considering equation (15e) leads to, The energy associated with the generalized force is given by equation (10d), and considering equation (15f) that is valid in the exact augmented perpetual manifolds, then is taking the form, Therefore, in the exact augmented perpetual manifolds, all the individual types of energies associated with each internal force, are either zero or constant with the initial value. Therefore there is no energy loss as heat due to dissipation, which lead that the internal entropy is, and the process in the exact augmented perpetual manifolds is isentropic [23]. d) In the augmented perpetual manifold, considering the overall system that the external forces are provided to the perpetual mechanical system through an adiabatic process then, and by adding equation (20a) leads to, therefore, the perpetual mechanical system with the surroundings is behaving as a perpetual machine of 2 nd kind [23,[25][26].
e) Rearranging the characteristic differential equation (6), multiplying it with the velocity of the associated generalized velocity−̇, and then integrate in time lead to, In the middle of equation (22), the terms inside the integral correspond to the power associated with the −external force ( ) and therefore, the function arising with integration is the work ( , , ) that is done by the −external force ( ) and it is the energy provided to or removed from the system. The integral of equation (22) exists for smooth characteristic differential equations of the augmented perpetual manifolds. In the case of a nonsmooth characteristic differential equation, herein must form a Filippov's system. In Filippov's systems, the velocities are smooth, and these integrals, integration for locally bounded functions of external forces at discontinuities, can be determined with Lebesgue-Stieltjes [34]. The notation at discontinuity, occurring at the time instant ∈ [ − , + ], is a superscript with negative sign (−) indicating the time instant just before the discontinuity and a superscript with positive sign (+) just after the discontinuity. Examining the locally bounded variation of the power of the external force at a discontinuity, the vector fields since is forming a Filippov's system have locally bounded variation at discontinuities, therefore, The integration by parts of the left-hand side of equation (22) using Riemman's integral requires that the velocity and the acceleration must be continuous functions. In the case of non-smooth systems, to simplify this proof, at the discontinuities, the Y-integral additional terms are considered, that is given in the '19.3.13 theorem for integration by parts' in [35]. The theorem's validity requires that the integrand functions ( , ) must have bounded variation in discontinuities, but in this case the integrands = =̇ are continuous functions, therefore the existence of acceleration discontinuities leads that in the integration by parts, the Yintegral additional terms are zero. Therefore the same formula given by the Reimann integral is applied and leads to, Therefore, taking into account equation (24) into equation (22) lead to, whereas, the last term arise with the use of equation (11c).

Rearrangement of equation (25) lead to,
Also, the 2 −kinetic energy � 2 , � of the system is given by, Therefore, the total work � , � done by the external forcing is equal to the change of the kinetic energy � , � from the system's initial state.
When the work done from the external force increases, the same amount of energy is given in the system, and becomes kinetic energy� , �. In case that the work � , � done from the external force is decreasing then the system's kinetic energy decreases with the same amount, and there is mechanical energy loss of the system outside to the 'environment' equal to the work done by the external forces. On each equation of the system (13), admitting rigid body motion that the terms of the nonlinear forces' are non-zero, their functional form can be in three forms.
The first one is constants (non-zero) that might be cancelled out with each other, 2) linearly dependent on the generalized coordinates and/or velocities that might be cancelled out with the linear forces, and c) nonlinear terms that might be canceled out by each other. The orthogonality condition for the three terms of equation (9) is imposed to avoid the cancelling out of the nonlinear vectors by each other, accompanied by the necessity that their sum must be equal to zero (perpetual mechanical system). Initially the case that each vector of equation (9) is having 'single' elements and not comprised by any sum vectors is examined.
In the following example, the associated linear system is a perpetual one, with the following form of nonlinear forces, whereas each force's vector with linear elements, in rigid body motions, is equal to zero. In the exact augmented perpetual manifolds the first two nonlinear forces are not necessarily zero but they are given by, and, summing them up, it certifies that the mechanical system is perpetual, but the energy is transferred from the elastic potential of the nonlinear forces to another form. The orthogonality conditions are not fulfilled, and this case in the theorem is excluded through the imposed orthogonality conditions.
The 2 nd case that the nonlinear forces in rigid body motions might lead to cancelling out of them by the linear forces in the next example is shown. As a second example, the following system is considered, whereas the negative sign of the ,2 parameter in the diagonal terms of the matrix multiplying the velocities indicates flutter. In the case of rigid body motions, the internal forces are taking the form, Summing equations (28e-h) up lead to zero-sum of the internal forces, this means that the system is perpetual. Noting that the two vectors associated with the nonlinear forces in the case of rigid body motions are orthogonal to each other, Therefore whereas the nonlinear forces for rigid body motion are linearly dependent on the generalized coordinates and/or velocities, they might be cancelled by the linear forces. Setting up a requirement that the underlying linear system must be perpetual, there is no such chance, and then the theorem is valid.
Moreover the aforementioned can be extended easily in the case that each vector with nonlinear forces of equation (9) is comprised of a sum. Then each term might cancel out each other of the same vector e.g. softening and stiffening nonlinearities within the vector with generalized coordinates depended forces, but imposing the orthogonality condition to be valid for each element of each vector of equation (9) there is not such a chance.

Remarks:
1. In mechanical systems, the velocity is always finite, but the requirement is included for any further application of this formalism.
2. In case that the external forces are zero, then there is no work done, and therefore, the system cannot be considered as a perpetual machine.
3. In the statement of the 2 nd law of thermodynamics, the change of a system's entropy is zero only in the case of a reversible process [23].
Following the above theorem, in a motion of a perpetual mechanical system described by the exact augmented perpetual manifolds, the change of the internal entropy is zero irrespective of the reversibility of the dynamics and also the existence of an irreversible adiabatic process that can provide the external forces. Elaborating further on, a reversible process of a system occurs, when the system is in one state with evolution either backwards or forward the system in both cases could be in the same state, therefore the states corresponds to limit cycles, or periodic motions.
The exact augmented perpetual manifolds of a mechanical system can be verifying the isentropic character of the process from the statistical mechanics perspective [23].

A natural mechanical system behaves as a perpetual machine based on the combination of the 'environment' and the mechanical system configuration itself
(perpetual).
In the augmented perpetual manifolds the input energy to the system is taking the form, whereas in case of zero initial potential energy associated with the nonlinear forces, the input energy is equal to the kinetic energy and also to the mechanical energy to the system.

Corollary
After the proof of the theorem of the previous section, a corollary herein is stated and proved.

Corollary
A perpetual natural mechanical discrete system, in reversible dynamics, and solution in the exact augmented perpetual manifolds, although each individual internal forces cannot behave for the total time interval as a perpetual machine of 3rd kind. Proof: In case that the system is starting from a given position and a velocity that either increase (decrease) considering periodicity, the system must come back to the original state with the initial velocity, therefore at a certain time instant the velocity must decrease (increase) which means that for certain time intervals the velocity increase and some others of the same cycle decrease. Since the velocity is straightforwardly correlated with the kinetic energy that in the exact augmented perpetual manifolds for zero initial potential energy is equal to the mechanical energy this means the perpetual mechanical system in certain time intervals is losing energy (in any form that is associated with the work done by the external forces) and therefore on these time intervals cannot behave as a perpetual machine of 3 rd kind. Also, if the external forces are zero, the velocities are constant, and then there is no work transferred to the mechanical system, and it cannot be considered as a perpetual machine of 3 rd kind. _ There are many books in thermodynamics discussing possible ideal ways that the 2 nd thermodynamics law is valid, and the 'ideal' perpetual machines. This corollary gives more insight in dynamical processes of mechanical systems that behave as perpetual machines.

Analytical examination of energies
In [21], for two types of external forces, the explicit form of the analytical solutions in the exact augmented perpetual manifolds are derived, and in Table   A � associated with a velocity given by the equation (A.2a) for the 1 st type of external forces � (1) ( )� in Table A.1, is given by, , -The kinetic energy � , � associated with a velocity given by the equation (A.2b) for the 2 nd type of external forces � (2) ( )� in Table A.1, is given by, , The power � , � of the external forces in the exact augmented perpetual manifolds is given by equation (11b). The power associated with the two types of the external forces, given in Table A.1, is, � of all the external forces, associated with the 1 st type of external forces � (1) ( )� given by equation (A.1a) in Table A -The power � , � for the 2 nd type of external forces � The work done � , � by the external forces in the exact augmented perpetual manifolds is given with integration of the external forces power in time and in general form by the equation (11c). The explicit form of the external work done by the two types of the external forces, given in Table A � done by all the external forces of the 1 st type � The last term obtained using equation (30a) for any time instant− and for the time instant− 0 that the motion with the 1 st type of external forces is starting, and certifies equation (27) that is the e-part of the theorem.
-The work � , � done by all the external forces of the 2 nd type � The last term of equation (32b)  In this part of the section,

Analytical examination of external forcing positive power
This subsection, presents a brief analysis of the conditions that the power of the two types of external forces, given in Table A Ia.
In case of ( , ) ∈ ℝ >0 × ℝ >0 the external forces are positive, and the velocity is given by, whereas in the case of positive initial velocity �̇, 1 ( 0 ) ∈ ℝ >0 �, then all terms in equation (34a), in all time instants, are positive, resulting in positive power.

Ib.
In case of zero initial velocity with positive external force (positive acceleration), the velocity increases and becomes positive and codirected with the external force which lead to positive power.

Ic.
In case of negative initial velocity, then equation (34a) is taking the form, whereas the last term can be easily verified that is negative and as long as the time passes (increasing t) the velocity increase and becomes positive, after the time instant, that the last parenthesis term in equation (34b) becomes positive, and is given by, and ever since the velocity is positive resulting in positive power.

Id.
In case of ( , ) ∈ ℝ <0 × ℝ <0 , the external forces are negative, and the velocity is given by, whereas in the case of negative initial velocity �̇, 1 ( 0 ) ∈ ℝ <0 �, all terms in equation (35a) are negative, and the velocity is decreasing in all time instants resulting in positive power.

Ie.
In zero initial velocity with negative external force (acceleration), the velocity decreases and becomes negative, which means co-directed with force and leads to positive power. If.
In case of positive initial velocity�̇, 1 ( 0 ) ∈ ℝ >0 � with negative external force (acceleration), then equation (35a) is taking the form, whereas it is profound that the force's sign is alternating every half of the excitation period ( = 2 � = � ). When this occurs, since there is a phase difference in the sinusoidal time-dependent function of the velocity, it wouldn't necessarily occur simultaneously.
Since the general case examination is rather complicated, to simplify things without losing the general perspective, some assumptions are made to examine a certain occasion.
Standing waves motion for ∈ ℝ >0 is examined. Choosing initial conditions that lead to zero wave velocity then, and the power is given by, , It is profound that the power could be positive or negative based on the phase of the sinusoidal function. For the time instants that the phase is, on these time intervals the power of the external force is positive, and the system behaves as a perpetual machine of 3 rd kind. For all the other phases the power is negative or zero.
In the next section, the validity of the theory in two different mechanical systems with motion in exact augmented perpetual manifolds is examined.

A perpetual mechanical system, model of a five wagons train
A train with five wagons modeled as five degrees of freedom mechanical is considered, as in Figure 1 is shown.
The equations of motion are given by, The mass matrix of the train is defined by, and ( = 1, . . ,5) are positive constants.
The linear forces based on stiffness matrix is defined by, and the linear forces based on the damping matrix defined using Rayleigh damping as follows, with = 0. The nonlinear forces vector associated with an elastic potential of nonlinear forces is given by, and the nonlinear damping forces vector is given by, noting that the internal nonlinear force between the 4 th and 5 th wagon, in equation (40e) is smoothed dry friction force.
The external forcing vector is given by, In [21], the same system has been examined, and that it forms a perpetual mechanical system is shown. The same ratio of external forces is used that leads to exact augmented perpetual manifolds solutions, and they are described by the following form of equation (6), Since in the exact augmented perpetual manifolds each one internal forcing vector of equation (39) with explicit form of matrices given by the equations (31a-e) is zero, leads that the nonlinear mechanical system is perpetual. Also, the underlying linear system is a perpetual mechanical system, and that the nonlinear forcing vectors are all orthogonal to each other, can easily certified. Therefore the conditions of the theorem are fulfilled.

Boundary limits of the Perpetual Mechanical System
The applied external forces � ,1 � for each time interval with the indication of the equations that define the solutions, the wave velocity and the type of motion in Table 1 are shown. It should be highlighted that the system, for the considered total time interval from 0-3 sec, since there are various non-convexified discontinuities of the external forces at the beginning of each time-subinterval, the system is not forming a Filippov's system.
The different types of energies used in numerical simulations are given by equations (10) in §2.1.
-The potential energy terms that are associated with the nonlinear forces in equation (40d) can be given by at least two definitions; either they can be defined by, or, In this example, to simplify things, the elastic potential for the nonlinear forces given by equations (42b) is used, which lead to zero reference value for the elastic potential in the exact augmented perpetual manifolds. Therefore based on equation (29) in the exact augmented perpetual manifolds, the input energy comprised by the initial kinetic energy adding the work done by the external forces is equal to the kinetic energy; and the mechanical energy of the system.
In Table 2, for each time interval, the equations that define the analytical solutions of the kinetic, of the power of the external forces, their work they have done, and the type of system in terms of energy behavior, are shown.
The parameters that define the mass, stiffness, damping and nonlinear forces in Table 3 are shown. The natural frequencies of the underlying linear system obtained by solving the eigenvalue problem in Table 4 are shown, and they are used to determine the damping matrices.
The external forcing parameters and the initial conditions used for each time interval in Table 5 are given. After the first time interval, the initial conditions, on each time interval, arise by the last state from the previous time-interval motion.

Analysis of the system dynamics, for each time interval, in terms of the energy exchange with the environment
This can be done, based on the type of external forcing (Tables 1, A.1), with their parameters (Table 5), and considering the analysis of the section 2.3.2 about the examination of the sign of the external forces power as follows: 1) For ∈ (0, 1], the external forcing is (1) (Table A.1), and for zero initial velocity, the power is given by the equation (31a). Since the parameters defining the force in Table 4 are positive �( , ) ∈ ℝ >0 × ℝ >0 �, the analysis (Ia) in subsection 2.4.2, leads to the conclusion that the system behaves as a perpetual machine of 3 rd kind. Considering the values indicated in Table 5, the velocity is changing sign at the time instant = 1.4150406 . Ever since on this time interval the velocity is co-directed with the force, resulting in the positive power of the external forcing, and the system behaves as a perpetual machine of 3 rd kind.
3) For ∈ (2,3], the external forcing is (2) , and based on analysis (II), the power is having alternated sign. As indicated in Table 3, the wave velocity is zero and therefore, the system returns after the excitation period to the original state. As shown in the analysis (II) of subsection 2.4.2, the system behaves as a perpetual machine of 3 rd kind for the time subintervals defined by the equation (38b), which is taking the following form, the power is positive, and the system behaves as a perpetual machine of 3 rd kind, and they are indicated in Table 8. Therefore, the system is not behaving as a perpetual machine of 3 rd kind in all time instants, which certifies the corollary.  Considering 1% damping ratio for the 2 nd natural frequency then the Rayleigh damping coefficient is 1 = 1.0850479 • 10 −3 , and is used in equation (30c) to obtain the values of the damping coefficients ( Table 3). The nonlinear damping coefficients are of the same arithmetic values as the linear damping coefficients, and also, in smoothed dry friction nonlinear damping forces, the following value = 10 6 / is used.   The external forces � ,1 �, for each time interval in Table 1 are indicated, with the general form given in Table A.1, and they are not state depended but only time depended. Therefore, for consistency using the time resolution of the numerical simulations, and the values of the parameters given in Table 5, they are   In a-part of the theorem the individual terms of the internal forces are equal to zero. This is certified, using the numerically determined generalized coordinates and velocities, that all the individual internal forces given by the equations (40b-e) have been evaluated. The numerically determined generalized coordinates and velocities, used to determine all the individual internal forces given by the equations (40b-e). In Table 7 the maximum absolute values of the timeseries of each element of each internal force are shown. The maxima of the linear forces in the first two columns of Table 7 are shown. In the last two columns, the values of the maxima of the nonlinear forces are shown. The maximum of the maxima is in linear stiffness forces but in -9 order of magnitude, which is very minimal. In order to make sense of the determined values of the internal forces, the ratio of the time-series of them with the external forces given by the equations (40f) and also their explicit form is shown in Table 1, have been determined. The maximum absolute value of the timeseries ratios of each individual internal force with the external force in Table 8 is shown. The maximum of the maxima is of -10 order of magnitude; therefore all the internal forces are, due to numerical errors, almost zero. The eliminated internal forces certify the a-part of the theorem.    This is certifying the f-part of the theorem that co-directed force with velocity leads to the positive power of the external forces that the system's energy is increasing.
Examining the zone between the magenta dash-dot lines for the time instants ∈ [2,2.6818] , that the motion is periodic, it is clear that due to periodicity in some time intervals, the perpetual mechanical system is earning energy (cyan regions) and behaves as a perpetual machine of 3 rd kind. In some other time intervals, the system is losing energy, outside to the environment, therefore cannot behave as a perpetual machine of 3 rd kind, and this certifies the corollary.

A two degrees of freedom non-perpetual mechanical system
A 2-degrees of freedom non-perpetual mechanical system, as shown in Figure 4, including dry friction non-smooth external forces is considered. The equations of motion, setting the boundaries of the perpetual mechanical subsystem, and separating the forces associated with each element of the perpetual mechanical subsystem on the left side with other forces on the right-hand side, are given by, and, The application of the theorem proved in [21] on this mechanical system, by neglecting the external forces on the right-hand side of equations (45a), requires that the system of differential equations with the remaining terms, that are given by, should form a perpetual mechanical system. A perpetual mechanical system is defined, when accepts as perpetual points, the generalized coordinates and velocities for rigid body motion. Therefore the following correlation of generalized coordinates and velocities, perpetual mechanical system, leads to zero accelerations and jerks of this system.
Considering equations (46b-c) in the equation (46a) leads to, The equations of jerks arise by taking the time derivative of equations (46a) and they are given by, Considering the equations (46b-c) in the equations of jerks (46e) lead to,  In the right-hand side of equations (45a), there are nonsmooth forces with terms defined by equations (45d-e) that correspond to dry friction. Each dry friction force has a discontinuity when the associated velocity is zero [32].
Considering the first as −mass, then from equations (45b,c) arise that, The normal forces amplitude in dry friction components, are correlated by, which is exactly in the form of equation (5) that leads to an exact augmented perpetual manifolds solution.
The solution of the system (45a-e), in the augmented perpetual manifolds is given by the following form of the equation (6), In this example the considered time depended external force is given by, This characteristic differential equation (49) is forming ROM of the original system, and, herein, numerically is solved.
The considered two masses have values ,1 = 2000 and ,2 = 1000 , and for the perpetual mechanical system the linear stiffness is = 10 6 / , the nonlinear stiffness is , = 5 • 10 5 / 3 and, the parameter that defines the nonlinear generalized force is 1 = 10 6 • / 3 . The considered damping coefficient is = 516.398 • / , and for the nonlinear damping coefficient the same numerical value is considered with , = 516.398 , and also = 10 6 / (significantly high to approximate well a dry friction force). The parameters defining the other elements considered in external forcing are a linear spring with stiffness = 3.2 • 10 6 / , a nonlinear spring with stiffness The numerical simulations for this section were performed using Scilab 5.5.2 64bit [36] with the 'Adams' solver. The time step is 5 • 10 −4 s, and the relative and the absolute tolerance are 5 • 10 −14 .
In Figure 5a, the displacements of the two masses incorporating the ROM solution are depicted. They look like that they are in good agreement, and this can be  Table   10 is shown. The maximum of the maxima of the absolute values of each individual ratio has the minimal -10 order of magnitude, corresponding to the nonlinear damping forces. The nonzero values can be attributed to numerical errors, and through these minimal values, the a-part of the theorem is certified.
In Figure 6a, the power of the external forces obtained by the original system and the ROM solution is depicted, and it seems that they are in good agreement. This can be certified by examining the maximum absolute difference between the original system external forces power with this one from the ROM solution with a minimal value of 1.234×10 -2 W. The cyan regions indicate the time intervals that the power of the external forces is positive, and they are obtained by examining adjacent points of the original system time-series of the power that they have opposite sign.
The maximum absolute value of the potential energy associated with the linear forces is 1.541×10 -25 J, and with nonlinear forces is 1.187×10 -56 . Therefore they are almost zero.
The almost zero values of the elastic potential energies lead to the conclusion that the b-part of the theorem on this overall non-perpetual mechanical system is verified.
The maximum absolute value of the dissipated energy a) through the linear dissipative forces is 3.673×10 -18 J, and b) through the nonlinear dissipative forces      Moreover, on the time intervals indicated with cyan regions, the work done by the sum of the external forces becomes kinetic energy to the perpetual mechanical subsystem. Therefore, the perpetual mechanical subsystem behaves as a perpetual machine of 3 rd kind on these time intervals, and this certifies f-part of the theorem.
Although the overall mechanical system is not perpetual, limiting the boundaries to the perpetual mechanical subsystem, the theorem and the corollary are certified.

CONCLUSIONS
As a continuation of previous work relative to perpetual mechanical systems and their motion in the exact augmented perpetual manifolds, a theorem is proved, mainly stating that in motions described by exact augmented perpetual manifolds, each term of the individual the internal forces is zero, the internal entropy of the perpetual mechanical system and the elastic potential function remain constant.
Further on, this conclusion leads to the conditions that a perpetual mechanical system can behave as a perpetual machine of 2 nd kind and 3 rd kind. Therefore the violation of the 2 nd law of thermodynamics for mechanical systems with solutions in the exact augmented perpetual manifolds is shown. Theorem's outcome is that there is no need for periodic processes for a mechanical system to behave as a perpetual machine of 2 nd kind. Therefore, the well-known requirement of the reversibility of the processes for the optimal behavior of mechanical systems with the view of the 2 nd law of thermodynamics as it is shown, it is not valid, and 'the arrow of time' is not necessarily related with the entropy of a mechanical system.
A corollary is proved that in periodic exact augmented perpetual manifolds solutions, the perpetual mechanical system cannot behave as a perpetual machine of 3 rd kind for all the time instants.
The theorem is verified with the analytical determination of the kinetic energy, the external forces power, and the external forces work done in the exact augmented perpetual manifolds for two types of external forces. A preliminary theoretical investigation for the conditions, through some analysis of these two types of external force's power, that a perpetual mechanical system behaves as a perpetual machine of 3 rd kind is performed.
In the last section, the validity of all the theoretical findings in two mechanical systems is examined, in a perpetual mechanical system and in a non-perpetual mechanical system that includes a perpetual mechanical subsystem. The theorem, the corollary, and the mathematical formalism for the two types of forces through these examples numerically verified with a very good agreement between the theoretical and numerical results.
This work is significant in physics and mechanical engineering. In physics relevant to the 2 nd law of thermodynamics, which requires reversibility of dynamics for the isentropic process and also the perpetual machine's existence was not so far under consideration.
In mechanical engineering, machines operating without any internal forces are the ultimate ones since there is no degradation by internal stresses, and of course, the perpetual machines are the machines with the ultimate operation with zero loss of energy. Therefore the outcome is also of very high importance in mechanical engineering.
A continuation of this work can be done in several directions, e.g., statistical mechanics view of the dynamics in augmented perpetual manifolds, design of machines with zero internal forces, design of perpetual machines of 2 nd kind, the examination of the conditions that different type of external forces provides positive power to the system, and many others different directions.

Declaration Funding
The author did not receive support from any organization for the submitted work. More precisely in a few simulations and draft plots of a simple nonlinear system, that is not included herein, the equality between the external forces work done with the kinetic energy of the system, is examined.

Availability of data and material
Data and material are available upon request to the author.

Code availability
All written codes are available upon request to the author.

Ethics approval
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APPENDIX-A
The solutions in exact augmented perpetual manifolds for two types of external forces provided in [21] in the following Table A

APPENDIX-B
In this appendix, the nonsmooth equations of motion of the mechanical system in §3.2 are written in a certain form for their numerical solution. Since they have differential inclusion, the algorithm for two switch hypersurface boundary functions of [21] based on the switch model developed in [32], is used.
Considering the equations of motion (45a-e) with the following change of variables,