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The role of nonlinearities in the dynamics of the non-equilibrium systems is considered. Various types of nonlinearities for dynamical systems with holonomic and nonholonomic constraints are studied. Features of the nonlinearity responsible for the evolution of the systems to the equilibrium are considered. The interrelation of the nonlinearities with the concepts of symmetry breaking is analyzed. Mathematical justification of the irreversibility of the dynamics of systems is offered. Peculiarities of nonlinear nonequilibrium systems connected with hierarchical structure of natural objects are studied.

Natural systems in generally are open and nonequilibrium. Evolutionary pro- cesses of emergence and development of new structures are inherent to them. These processes are nonlinear because they are associated with increasing of the entropy but the increment of the entropy is determined by the infinitesimal quadratic nonlinear terms [

As a rule, for solving systems of nonlinear differential equations, they reduced to integrable equations by replacing the variables. The choice of the independent variables is determined by the nature of the symmetry of equations. Most of the nonlinear equations are simplified by the linearization. Sometimes the simplifications of the models of the systems under study, or hypothesis for simplification of the systems of equations are used. As a rule, with such simplifications, the nonlinear effects that determine the evolution of the system are lost. For example, the use of the hypothesis about holonomicity constraints in the construction of the canonical formalisms for a system of material points (MP) has led to the exclusion of nonlinear terms, responsible for dissipative processes [

The development of computer technology helps to solve nonlinear equations using numerical methods. However, numerical methods are convenient for pra- ctical applications or checking theoretical conclusions but they almost did not disclose the physical nature of the process.

The difficulty of the analytical solutions of nonlinear equations led to the development of qualitative methods of analysis. They, in particular, used to identify the statistical regularities of systems dynamics for the study of their phase portraits. These methods are effective in the study of dynamical chaos. They are also widely used for bifurcation analysis, for studying the characteristics of the nonlinear equations and for seeking of new nonlinear effects [

The presence of the universal laws of evolution of systems, regardless of whe- ther these are objects of the universe, or it is the atomic system [

The classification of the different types of the nonlinearity is possible to realize by starting from the analysis of the physical nature of the corresponding nonlinearities. The most natural criterion for separating the different types of nonlinearities, inherent for the dynamical systems of classical mechanics, is divide these systems in connection with holonomic and nonholonomic constraints. Accordingly, the first type of the nonlinearities consists from those nonlinearities, that inherent for the conservative Hamiltonian systems with the holonomic constraints. The second type of non-linearity consists of nonlinearities, which determines the dynamics of non-Hamiltonian systems with nonholonomic constraints. This type of non-linearity belongs to the dissipative systems [

The main purpose of the work is to analyze the class of the nonlinearities, responsible for the irreversible dynamical systems of the classical mechanics. For this purpose, we determine the nature of the class nonlinearities which response for irreversibility. The connection between the nonlinearities and irreversibility of the systems dynamic will be analyzed also. The characteristics of these nonlinearities for nonequilibrium systems (NS), taking into account their hierarchical structure, will be studied.

The possibility of integrating the dynamics equations of the classical mechanics is determined by the properties of the system itself and by the nature of the restriction that are imposed on the system. The restrictions are divided on the holonomic and nonholonomic.

Holonomic constraints are such restrictions, which can be expressed through the full differential of the spatial variables. For systems with holonomic constraints, the task is reduced to the solution of independent equations for the generalized independent variables. Hypothesis about holonomicity constraints is used for obtain the Lagrange and Hamilton canonical equations [

For Hamiltonian systems, the energy of each element is invariant. The nonlinearities, existing in Hamiltonian systems, causes fragmentation of scale that characterizes the dynamics of each individual element [

Non-holonomic constraints cannot be expressed through the total differential of a function of space variables [

Thus, the nonlinearities responsible for the evolution of the NS to the equilibrium or for the interaction systems are inherent only for the systems with non- holonomic constraints. Since these nonlinearities responsible for the evolution of the system, we will call them “evolutionary nonlinearities”.

We will construct this work based on the mechanics of structured particles (SP), where SP is equilibrium system from a great enough number of the potentially interaction MP [

The generality of the analysis of dynamics of systems of nonlinear equations of classical mechanics is that any non-equilibrium systems in the local thermodynamics equilibrium approach with sufficient accuracy can be represented by a set of interacting SP [

The motion of MP system is determined by the influence of two independent types of internal and external of forces. The internal forces are determined by the interaction of MP. The sum of these forces is equal to zero. External forces determine the motion of the system in space. The external forces may be different for different MP due to the inhomogeneity of the external field. In this case, the external forces will change the internal energy of the system. The internal motions of the MP in the system relative to the CM, do not contribute into the motion of the system in space.

Since the internal energy and the motion energy are independent, the energy of the system should be presented as the sum of the energy of motion of the system and of the internal energy. We can do this presentation of the energy with the help of independents macro- and micro-variables. The energy of the system that provided in these variables has the form [

where

where

Let’s assume that

and

We see that the second term on the right hand side of Equation (5) depends on the micro- and macro-variables. It is different from zero only when the existence of a gradient of external forces. It is because a change of internal energy of the bodies is possible only due to the difference of the external forces acting on the different elements of the system. It is not just a bilinear but bisymmetrical as it depends on the variables of two different groups of symmetry. The motion equation for the system can be obtained from Equation (2) [

where

The first term in the right hand side of Equation (4) is a potential force, which applied to the CM of the system. The second term is determine the change of the SP’s internal energy and depends on both micro- and macro-variables. This nonlinear term is appeared due to taking into account of the internal energy of the system.

Equation (4) generalizes equation Aristotle and Newton. As Aristotle equation says that the speed of a body is proportional to the force. According to Equation (4) is true when the frictional force becomes equal to the driving force.

Newton’s equation states that the acceleration is proportional to force. This case is realized when external forces are uniform, or when the internal forces are large, so that the system can be considered a solid.

Let us compare the mechanics of SP [

Nature of the “evolutionary nonlinearity” can be easily understood by the ex- ample of the oscillator in a non-uniform field of the force. To understand this nature the energy of oscillator should be expressed in terms of micro- and macro-variables as the sum of the motion energy and the internal energy. In the non-uniform external forces when the characteristics scales of irregularities of the forces are commensurate with the characteristics scale of oscillator, both types of energy are related. Depending on the initial phase, the oscillator is able to pass through the potential barrier, even when the energy of motion is less than the height of the barrier. This transition is carried out due to transformation of the internal energy into the motion energy [

We start from the condition that the symmetry breaking always connected with the violation of the invariance of the measure corresponding to this symmetry group. A measure of the translational symmetry in classical mechanics is the phase volume, or motion energy of the system. The violation of the invariance of the motion energy is a breaking of the time symmetry. In our case, a breaking of the time symmetry is due to transformation of the motion energy into internal energy. Such a transformation is determined by the second bilinear term of Equation (5), which is simultaneously dependent from the micro- and macro-variables.

Let us consider the dynamics of the nonequilibrium systems. In the approach of the local thermodynamic equilibrium, nonequilibrium system can be submitted as a set of the equilibrium subsystems SP, which in the relative motion to each other (

Let us consider nonequilibrium system consists from _{p} number of MP, i.e. l = 1, 2, 3 … L_{p}; _{p} is coordinates of CM for p-SP relative to the CM of nonequilibrium system;

Here

The first term in Equation (5) is motion energy of nonequilibrium system. The second term is a sum of internal kinetic energy of SP. The third term defines of the kinetic component of the internal energy of each MP. The fourth term is the sum of the internal potential energy of all SP. It is determined by summing the indices corresponding for SP. The fifth term is the potential energy of interaction of SP. The term

As in the case of SP, the motion equation of the nonequilibrium system can be obtained from the energy Equation (5) which represented by the hierarchy of the micro- and macro-variables. The internal energy of the nonequilibrium system is divided into the sum of the SP motion energies and their internal energy. This means that the energy of the external field go on the change the motion energy of nonequilibrium system, on the change of the energy of SP relative motion and on the change of the internal energy of SP. The last two types of energy is the internal energy of the nonequilibrium system, i.e. in the nonequilibrium system, in contrast to the SP, there is an additional hierarchical level. It leads to a hierarchy of energy and entropy of the nonequilibrium system.

The phase space of nonequilibrium system is determined by coordinates and velocity of CM for SP. It was called S-space [

The physical nature of the irreversibility of SP dynamics in an inhomogeneous field of external forces was enough to set out in detail in [

According to the statistical physics, the closed NS tend to equilibrium, which corresponds to zero relative velocities of all SP. That is, the kinetic energy of the relative motions of the SP,

Now let us show how the equilibration of the system follows from the evolutionary nonlinearity. For this purpose we consider the motion SP in an inhomogeneous field of forces. According to Equations (3) and (4) the motion energy of the SP can be converted into the internal energy. Suppose that the value of this transformation is equal to

If the field of external forces is small, the equilibrium violation for SP can be ignored. This means that the energy of its motion is irreversible converted into the internal energy. But with sufficient quantities of forces and their gradients, equilibrium of the SP can be broken. In this case, it can represent as a set of equilibrium subsystems having nonzero relative velocity. Then the increase in the internal energy of the system

Only the energy of the relative motions of equilibrium subsystems can go back into the energy of motion of the full system because the internal energy of the equilibrium system cannot be going into the motion energy of this system due to momentum conservation. Let the return energy is equal to

Let us explain why the increase of the internal energy becomes irreversible only for the big enough systems.

According to the law of statistical physics, the fluctuations in the value of the energy are proportional to the

It is necessary to emphasize that the only condition that was used for proving of violations of equality forward and reverse energy flow in a sufficiently large system, moving in a non-uniform force field, was the condition of inability to transform the internal energy of an equilibrium system in its kinetic energy. It is equivalently to the statement of inability transformation of the thermal energy into the body’s energy of motion due to law of momentum conservation. This provided us an opportunity to introduce the concept of entropy in classical mechanics. Let us briefly explain how this was done.

The entropy in classical mechanics was called as D-entropy. D-entropy is defined as the ratio of the increment of internal energy of the system at the expense of the energy of motion to the value of the internal energy. Expression of D-entropy has the form [

The amount of D-entropy is determined by summing over the part of the work of external forces that is goes to the change internal energy of the NS. D- entropy is a deterministic because its follows from the motion equations of the system.

Determination of D-entropy is acceptable for small systems. However, for them the D-entropy, in contrast from the entropy of Clausius, may be both positive and negative. The oscillator is example of small systems. The energy of the relative motion of the elements of oscillator, which is the internal energy of the oscillator, in a non-uniform field of force can goes into its motion energy [

D-entropy can be used to determine the limits of applicability of the thermodynamic description of the system, the limits of applicability of the laws of statistical probability. This is confirmed by the numerical calculations of the change of the internal energy of the system depending from the number of MP when the system is moving in the inhomogeneous field of external forces.

The possibility of transformation of the motion energy of the NS into the internal energy depends on the characteristic scale of the inhomogeneity of the external field of forces

Let the scale of the NS is

When

If

It is follows from the mechanics of the SP that according to the laws of classical mechanics, the matter is infinitely divisible [

The nonlinear terms defining the process of transformation of the body’s motion energy into their internal energy are functions of the variables hierarchical levels. Therefore, D-entropy, and different types of energies are defined, in general, by the variables for the all hierarchical levels. The dynamics of the body is determined by the dualism of energy on any hierarchical level, because the work of external forces for any hierarchical element going both on its motion and on the change of the internal energy.

Evolution of the bodies is connected with the non-linear transformation of the flows of external energy into internal energy of its structural elements arranged in the hierarchy. Therefore, the description of the evolution is a self-consistent nonlinear dynamic task. That is, the field of forces created by the structure the nested hierarchical elements of the system, defines the structure of the system, as well as the structure of the system is determined by the hierarchy of the field of forces.

The conditions of stationary of the body structures, requires a balance of incoming and out-going energy flows at all hierarchical levels. These flows are hierarchically chain of non-linear transformations of the body’s motion energy. Therefore, for stationary the compensation of dissipative processes for the all hierarchical levels is necessary. It is only known the electromagnetic radiation for the body, which can to compensate the energy inflow into the bodies [

A key role in the dynamics of systems plays symmetry. The nature of the transformation of energy at all hierarchical levels is determined by the external symmetries and symmetries of these levels. The nature of spontaneous symmetry breaking in the microcosm [

In the framework of classical mechanics, all systems divided into two categories― 1) systems with holonomic constraints and 2) systems with nonholonomic constraints. This fact defines an obvious criterion to classify the types of nonlinearity.

So, there are fundamentally different types of nonlinearities that characterize the dynamics of the classical systems. The first type is nonlinearities that belong to the systems with holonomic constraints. The nonlinearities of the Hamiltonian systems belong to the first type of nonlinearities, since the requirement of holonomicity of the constraints is used in the derivation of Lagrange equation. Hamiltonian systems are reversible. For them, the volume of the phase space is preserved. The nonlinearities inherent to the Hamiltonian systems do not lead to evolution, because evolution is impossible without the dissipative processes.

Another type of nonlinearity, which we have called “evolutionary nonlinearities”, is only possible for systems with nonholonomic constraints. A distinctive feature of “evolutionary nonlinearity” is that the corresponding nonlinear terms depend on variables of different symmetry groups, which is connected with the different hierarchical levels of the body. In the classical mechanics, the “evolutionary nonlinearities” have a place when the system of MP is moving in a nonhomogeneous field of force. The presence of the gradient of field of force leads to engagement of the micro- and macro-variables defining the system dynamics and the dynamics of its internal structure. This leads to the transformation of the system motion energy into the internal energy. Thus, the “evolutionary nonlinearity” is inherent for the dissipative processes. They lead to the formation and development of structures accompanied by production of D-entropy in all hierarchical levels of the system. They determine the irreversibility of the system dynamics and violation of time symmetry.

The irreversibility of the dynamics of systems occurs when a sufficiently large number of MP in a system have a place. This is due to the fact that direct flow of energy into the system is determined by the non-linear terms of the second order, and the reverse flow of internal energy into kinetic energy of the system is determined by the terms of more than fourth order of smallness. With an increasing number of MP, the fluctuations of these flows decrease. Therefore, for sufficiently large numbers of MP, the value of the fluctuation of reverse flow energy not more than the value of the direct flow of energy

The condition of the infinite divisibility of matter implies that the bodies are represented by a number of embedded systems. Their dynamics are determined by the “evolutionary nonlinearities”, and by the hierarchical order, which corresponds to the degree of heterogeneity of the field of forces.

Dynamics systems are determined by the principle of dualism of symmetry at all hierarchical levels, as the work for any hierarchical level and on the change of its internal energy.

Somsikov, V.M. (2017) Non-Linearity of Dynamics of the Non-Equilibrium Systems. World Journal of Mechanics, 7, 11-23. https://doi.org/10.4236/wjm.2017.72002