]

# THE SYSTEMS DYNAMICS OF THE STRUCTURED PARTICLES

V.M. Somsikov  [<EMAIL_ADDRESS>Laboratory of Physics of the
geoheliocosmic relation, Institute of Ionosphere, Almaty, Kazahstan.

###### Abstract

Dynamics of the structured particles consisting of potentially interacting
material points is considered in the framework of classical mechanics.
Equations of interaction and motion of structured particles have been derived.
The expression for friction force has been obtained. It has been shown that
irreversibility of dynamics of structured particles is caused by increase of
their internal energy due to the energy of motion. It has been shown also that
the dynamics of the structured particles is determined by two types of
symmetry: the symmetry of the space and the internal symmetry of the
structured particles. Possibility of theoretical substantiation of the laws of
thermodynamics has been considered.

nonequilibrium, classical mechanics, thermodynamics

###### pacs:

05.45; 02.30.H, J

††preprint: APS/123-QED

##  I Introduction

All real bodies in nature are the structured particles ($SP$). But the
existing classical mechanics has been developed for material points($MP$) or
hard bodies [1] which does not exist in the nature and which have no internal
structure. Therefore it is desirable to create the mechanics of $SP$. This
mechanics will be more general than the existing mechanics of unstructured
bodies. Indeed, at the $MP$ motion in non-homogeneity space and their
interaction the energy of $MP$ motion changes only, while for the $SP$
internal energy varies also.

As usually the change of $SP$ internal energy is described empirically by the
classical mechanics for $MP$. So a question arises, whether it is possible to
find rigorous mathematical description of $SP$ dynamics within the frames of
the Newtonian mechanics and if possible then how? We found the answer on this
question by studying the motion equation of $SP$ when $SP$ is an equilibrium
system of potentially interacting $MP$.

It turns out that under certain conditions dynamics of such systems is
irreversible [2-4]. These conditions are formulated as follows:

1). The energy of an $SP$ must be presented as a sum of internal energy and
the energy of $SP$ motion as a whole.

2). Each material point in the system must be connected with a certain $SP$
independent of its motion in space.

3). During all the process the subsystems are considered to be equilibrium.

The first condition is necessary to introduce internal energy in the
description of system dynamics as a new key parameter charactering energy
variations of $SP$. The second condition enables not to redefine $SP$ after
mixing of $MP$. The last condition is taken from thermodynamics. It is
equivalent to the condition of weak interactions in the $SP$, which do not
violate $SP$ equilibrium. Moreover, it implies that each $SP$ contains so many
elements that it can be described using the concept of equilibrium system.

In this paper we consider derivation of the motion equation of interacting
$SP$. With the help of this equation it is shown how the mechanism of friction
can be explained in the frame of laws of the classical mechanics. It is shown
also how based on the hypothesis of local equilibrium, which enables to
represent non-equilibrium systems as an ensemble of equilibrium subsystems,
one can generalize the obtained results for two interacting $SP$. It is also
shown how Lagrange, Hamilton and Liouville equations for non-equilibrium
systems are derived from the equation of motion of a set of equilibrium $SP$.
We consider how such equations are different from their canonic prototypes for
the system of $MP$. We consider why the $SP$ dynamics is determined by the two
types of symmetries: the symmetry of space in which the $SP$ motion and
internal symmetry of distributions of elements of $SP$. It is shown how the
main equation of thermodynamics can be derived from the equation of $SP$
interaction and how the concept of entropy arises in classical mechanics.

##  II The motion equation of equilibrium structural particles

It was shown in [4] that for obtaining of the $SP$ motion equation it is
necessary to define the energy of each $SP$ as a sum of internal energy and
energy of its motion. Differentiating energy of system with respect to time
and using a condition of its conservation, the equation for an energy exchange
between $SP$ can be obtained and then with its help the equation of motion of
$SP$ can be found. After that the equation of motion for $SP$ can be obtained
in two stages. At the first stage, based on the condition of energy
conservation, we obtain the equation of motion for the system in the field of
external forces. Then we take a system consisting of $SP$ and obtain their
equations of motion when the external field for one $SP$ is the field of
forces of the other $SP$. Forces acting between the $SP$ can be obtained from
$MP$ potential interaction.

Let us show how the equation of motion for a system of $N$ material points
with weights $m=1$ can be obtained [2-4]. Forces acting between pairs of $MP$
are assumed to be central and potential. The energy of the system $E$ is equal
to the sum of kinetic energies of $MP$. Thus
$T_{N}=\sum\limits_{i=1}^{N}m{v_{i}}^{2}/2$, their potential energy in the
field of external forces, ${U_{N}}^{env}$, and potential energy of their
interaction
${U_{N}}(r_{ij})={\sum\limits_{i=1}^{N-1}}{\sum\limits_{j=i+1}^{N}}U_{ij}(r_{ij})$,
where $r_{ij}=r_{i}-r_{j}$, $r_{i},v_{i}$ are coordinates and velocities of
the $i$-th $MP$. Thus, $E=E_{N}+U^{env}=T_{N}+U_{N}+U^{env}=const$.

By substituting variables we represent the energy of the system as a sum of
the motion energy of the center of mass ($CM$) and the internal energy.
Differentiating this energy with respect to time, we will obtain [3]:

$\displaystyle
V_{N}M_{N}\dot{V}_{N}+{\dot{E}}_{N}^{ins}=-V_{N}F^{env}-\Phi^{env}$ (1)

Here $F^{env}=\sum\limits_{i=1}^{N}F_{i}^{env}(R_{N},\tilde{r}_{i})$,
${\dot{E}}_{N}^{ins}={\dot{T}}_{N}^{ins}(\tilde{v}_{i})+{\dot{U}}_{N}^{ins}(\tilde{r}_{i})$=
$\sum\limits_{i=1}^{N}\tilde{v}_{i}(m\dot{\tilde{v}}_{i}+F(\tilde{r})_{i})$,
$\Phi^{env}=\sum\limits_{i=1}^{N}\tilde{v}_{i}F_{i}^{env}(R_{N},\tilde{r}_{i})$,
$r_{i}=R_{N}+\tilde{r}_{i}$, $M_{N}=mN$, $v_{i}=V_{N}+\tilde{v}_{i}$,
$F_{i}^{env}=\partial{U^{env}}/\partial{\tilde{r}_{i})}$, $\tilde{r}_{i}$,
$\tilde{v}_{i}$ are the coordinates and velocity of $i$-th $MP$ in the $CM$
system, $R_{N},V_{N}$ are the coordinates and velocity of the $CM$ system.

The equation (1) represents the balance of the energy of the system of $MP$ in
the field of external forces.

The first term in the left-hand side of the equation determines the change of
kinetic energy of the system - ${\dot{T}}_{N}^{tr}=V_{N}M_{N}\dot{V}_{N}$. The
second term determines the change of internal energy of the system,
${\dot{E}}_{N}^{ins}$. This energy dependent on coordinates and velocities of
$MP$ relative to the $CM$ of the system.

The right-hand side corresponds to the work of internal forces changing the
energy of the system. The first term changes
${\dot{T}}_{N}^{tr}=V_{N}M_{N}\dot{V}_{N}$. The second term determines the
work of forces changing ${\dot{E}}_{N}^{ins}$.

Let us determine the condition when the work of non-potential forces is not
equal to zero. We must take into account that
$F^{env}=F^{env}(R+\tilde{r}_{i})$ where $R$ is the distance from the source
of force to the $CM$ of the system. Let us assume that $R>>\tilde{r}_{i}$. In
this case the force $F^{env}$ can be expanded with respect to a small
parameter. Leaving in the expansion terms of zero and first order we can
write: $F_{i}^{env}=F_{i}^{env}|_{R}+(\nabla{F_{i}^{env}})|_{R}\tilde{r}_{i}$.
Taking into account that
$\sum\limits_{i=1}^{N}\tilde{v}_{i}=\sum\limits_{i=1}^{N}\tilde{r}_{i}=0$ and
$\sum\limits_{i=1}^{N}F_{i}^{env}|_{R}=NF_{i}^{env}|_{R}=F_{0}^{env}$, we get
from (1):

$\displaystyle
V_{N}(M_{N}\dot{V}_{N})+\sum\limits_{i=1}^{N}m\tilde{v}_{i}(\dot{\tilde{v}}_{i}+F(\tilde{r})_{i})\approx$
$\displaystyle\approx-
V_{N}F_{0}^{env}-({\nabla}F^{env}_{i}|_{R})\sum\limits_{i=1}^{N}\tilde{v}_{i}\tilde{r}_{i}$
(2)

In the right-hand side of equation (2) the force $F_{0}^{env}$ in the first
term depends on $R$. It is a potential force. The second term depending on
coordinates of $MP$ and their velocities relative to the $CM$ of the system
determines changes in the internal energy of the system. It is proportional to
the divergence of the external force. Therefore, in spite of the condition
$R>>\tilde{r}_{i}$ the values of $\tilde{v}_{i}$ may be not small, and the
second term cannot be omitted. Forces corresponding to this term are not
potential forces. So, the change in the internal energy will be not equal to
zero only if the characteristic scale of inhomogeneities of the external field
is commensurable with the system scale.

Thus, inhomogeneity of space leads to the inhomogeneity of time for the
system. It is connected with possibility of increase in internal energy of
system at the expense of energy of its motion and impossibility of returning
of the system’s internal energy into the energy of its motion due to the law
of momentum conservation. But the law of preservation of full energy is
carried out.

Equation (2) confirms assumption of A. Poincare [5] that it is necessary to
take into account structures of interacting bodies at rather small distances
between them.

Dynamics of an individual $MP$ as well as dynamics of a system of $MP$ can be
derived from equation (1). A $MP$ does not have an internal energy, and forces
acting on it are caused by potential forces of interaction with other $MP$ and
the external force. Therefore the motion of a $MP$ is determined by the work
of potential forces transforming the energy of the external field into its
kinetic energy only.

Unlike $MP$, a system has its internal energy. Therefore the work of external
forces over the system causes changes in its $T_{N}^{tr}$ and $E_{N}^{ins}$,
i.e. the external force breaks up into two components. The first component is
a potential force. It changes momentum of the system’s $CM$. The second
component is non-potential. Its work changes $E_{N}^{ins}$. Hence, the motion
of the system is determined by the work of potential and non-potential forces
transforming the external field energy into the energy of $CM$ motion and
internal energy.

Multiplying eq.(1) by $V_{N}$ and dividing by $V_{N}^{2}$ we find the equation
of a system motion [4]:

$\displaystyle M_{N}\dot{V}_{N}=-F^{env}-{\alpha_{N}}V_{N}$ (3)

where $\alpha_{N}=[{\dot{E}}_{N}^{ins}+\Phi^{env}]/V_{N}^{2}$ is a coefficient
determined by the change of internal energy.

The equation (3) is a motion equation for $SP$. The first term in the right-
hand side of the equation determines the system acceleration, and the second
term determines the change of its internal energy. The eq. (3) is reduced to
the Newton equation if it is possible to neglect variation in the internal
energy.

Thus, the system state in the external field is determined by two parameters:
the energy of motion and the internal energy. Each type of energy has its own
force. The change in the motion energy is caused by the potential component of
the force, whereas the change in the internal energy is caused by the non-
potential component.

Let us show how to obtain the equation for interaction two equilibrium $SP$.
For this purpose we take the system consisting of two $ES$-$L$ and $K$. The
$L$ is the number of elements in the $L$-$SP$ and $K$ is the number of
elements in $K$-$SP$, i.e. $L+K=N$. Let $LV_{L}+KV_{K}=0$, where $V_{L}$ and
$V_{K}$ are velocities of $L$ and $K$ equilibrium subsystems relative to the
$CM$ of the system. Differentiating the energy of the system with respect to
time, we obtain:
${\sum\limits_{i=1}^{N}v_{i}{\dot{v}}_{i}}+{\sum\limits_{i=1}^{N-1}}\sum\limits_{j=i+1}^{N}v_{ij}F_{ij}=0$,
where $F_{ij}=U_{ij}=\partial{U}/\partial{r_{ij}}$.

In order to derive the equation for $L$-$SP$, in the left-hand side of the
equation we leave only terms determining change of kinetic and potential
energy of interaction of $L$-$SP$ elements among themselves. All other terms
we displace into the right-hand side of the equation and combine the groups of
terms in such a way that each group contains the terms with identical
velocities. In accordance with Newton equation, the groups which contain terms
with velocities of the elements from $K$-$SP$ are equal to zero. As a result
the right-hand side of the equation will contain only the terms which
determine the interaction of the elements $L$-$SP$ with the elements $K$-$SP$.
Thus we will have:
${\sum\limits_{i_{L}=1}^{L}}v_{i_{L}}{\dot{v}}_{i_{L}}+{\sum\limits_{i_{L}=1}^{L-1}}\sum\limits_{j_{L}=i_{L}+1}^{L}F_{{i_{L}}{j_{L}}}v_{{i_{L}}{j_{L}}}={\sum\limits_{i_{L}=1}^{L}}\sum\limits_{j_{K}=1}^{K}F_{{i_{L}}{j_{K}}}v_{j_{K}}$
where double indexes are introduced to denote that a particle belongs to the
corresponding system. If we make substitution
$v_{i_{L}}=\tilde{v}_{i_{L}}+V_{L}$, where $\tilde{v}_{i_{L}}$ is the velocity
of $i_{L}$ particle relative to the $CM$ of $L$ -$SP$, we obtain the equation
for $L$-$SP$. The equation for $K$-$SP$ can be obtained in the same way. The
equations for two interacting systems can be written as [4]:

$\displaystyle
V_{L}M_{L}\dot{V}_{L}+{\dot{E}_{L}}^{ins}=-{\Phi}_{L}-V_{L}{\Psi}$ (4)
$\displaystyle
V_{K}M_{K}\dot{V}_{K}+{\dot{E}_{K}}^{ins}={\Phi}_{K}+V_{K}{\Psi}$ (5)

Here $M_{L}=mL,M_{K}=mK,\Psi=\sum\limits_{{i_{L}}=1}^{L}F^{K}_{i_{L}}$;
${\Phi}_{L}=\sum\limits_{{i_{L}}=1}^{L}\tilde{v}_{i_{L}}F^{K}_{i_{L}}$;
${\Phi}_{K}=\sum\limits_{{i_{K}}=1}^{K}\tilde{v}_{i_{K}}F^{L}_{i_{K}}$;
$F^{K}_{i_{L}}=\sum\limits_{{j_{K}}=1}^{K}F_{i_{L}j_{K}}$;
$F^{L}_{j_{K}}=\sum\limits_{{i_{L}}=1}^{L}F_{i_{L}j_{K}}$;
${\dot{E}_{L}}^{ins}={\sum\limits_{i_{L}=1}^{L-1}}\sum\limits_{j_{L}=i_{L}+1}^{L}v_{i_{L}j_{L}}[\frac{{m\dot{v}}_{i_{L}j_{L}}}{L}+\\\
+F_{i_{L}j_{L}}]$;
${\dot{E}_{K}}^{ins}={\sum\limits_{i_{K}=1}^{K-1}}\sum\limits_{j_{K}=i_{K}+1}^{K}v_{i_{K}j_{K}}[\frac{{m\dot{v}}_{i_{K}j_{K}}}{K}+\\\
+F_{i_{K}j_{K}}]$.

The equations (4, 5) are equations for interactions two $SP$. They describe
energy exchange between $SP$. Independent variables are macro-parameters and
micro-parameters. Macro-parameters are coordinates and velocities of the
motion of $CM$ of $SP$. Micro-parameters are relative coordinates and
velocities of $MP$.

Therefore the equation of $SP$ interaction binds together two types of
description: on the macrolevel and on the microlevel. The description on the
macrolevel determines dynamics of an $SP$ as a whole and description on the
microlevel determines dynamics of the elements of an $SP$.

The potential force, $\Psi$, determines the motion of an $SP$ as a whole. This
force is the sum of potential forces acting on the elements of one $SP$ from
the other $SP$.

The forces determined by terms ${\Phi}_{L}$ and ${\Phi}_{K}$ transform the
motion energy of $SP$ into their internal energy as a result of chaotic motion
of elements of one $SP$ in the field of forces of the other $SP$. As in the
case of the system in the external field, these terms are not zero only if the
characteristic scale of inhomogeneity of forces of one system is commeasurable
with the scale of the other system. The work of such forces causes violation
of time symmetry for $SP$ dynamics.

The equations for $SP$ motion corresponding to the equations (4,5) can be
written as [4]:

$M_{L}\dot{V}_{L}=-\Psi-{\alpha}_{L}V_{L}$ (6)
$M_{K}\dot{V}_{K}=\Psi+{\alpha}_{K}V_{K}$ (7)

where ${\alpha}_{L}=(\dot{E}^{ins}_{L}+{\Phi}_{L})/V^{2}_{L}$,
${\alpha}_{K}=({\Phi}_{K}-\dot{E}^{ins}_{K})/V^{2}_{K}$,

The equations (6, 7) are motion equations for interacting $SP$. The second
terms in the right-hand side of the equations determine the forces changing
the internal energy of the $SP$. These forces are equivalent to the friction
forces. Their work is a sum of works of forces acting on the $MP$ of one $SP$
from the other $SP$.

The coefficients ”$\alpha_{L}$”, ”$\alpha_{K}$” determine efficiency of
transformation of the energy of $SP$ motion into their internal energy. These
coefficients are friction coefficients. Therefore equations (6, 7) enable to
determine analytical form of non-potential forces in the non-equilibrium
system causing changes in the internal energy of the $SP$.

##  III The generals of Lagrange, Hamilton and Liouville equations for
equilibrium systems

Let us show qualitative difference of Lagrange, Hamilton and Liouville
equations for the systems of $MP$ from similar equations for $SP$.

Using Newton equation one can derive Hamilton principle for $MP$ from
differential D’Alambert principle [6]. For this purpose the time integral of
virtual work done by effective forces is equated to zero. Integration over
time is carried out provided that external forces possess a power function. It
means that the canonical principle of Hamilton is valid only for cases when
$\sum F_{i}\delta R_{i}=-\delta U$, where $i$ is a particle number, and
$F_{i}$ is a force acting on this particle. But for interacting $SP$ the
condition of conservation of forces is not fulfilled because of the presence
of a non-potential component. Therefore Hamiltonian principle for $SP$ as well
as Lagrange, Hamilton and Liouville equations must be derived using eq. (3).

Liouville equation for non-equilibrium system consisting from a set of
equilibrium $SP$ is written as [2, 4]:

$df/dt=-\sum\limits_{L=1}^{R}{\partial}{F_{L}}/{\partial}V_{L}$ (8)

Here $f$ is a distribution function for a set of $SP$, $F_{L}$ is a non-
potential part of collective forces acting on the $SP$, $V_{L}$ is the
velocity of $L$-$SP$.

The right-hand side of the equation is determined by the efficiency of
transformation of the $SP$ motion energy into their internal energy. For non-
equilibrium systems the right-hand side is not equal to zero because of non-
potentiality of forces changing the internal energy.

The state of the system as a set of $SP$ can be defined in the phase space
which consists of $6R-1$ coordinates and momentums of $SP$, where $R$ is the
number of $SP$. Location of each $SP$ is given by three coordinates and their
moments. Let us call this space an $S$-space for $SP$ in order to distinguish
it from the usual phase space for $MP$. Unlike the usual phase space [7,8] the
$S$-space is not conserved. It is caused by transformation of the energy of
$SP$ relative motion into their internal energy. The $SP$ internal energy
cannot be transformed into the $SP$ energy of motion as $SP$ momentum cannot
change due to the motion of its $MP$ [7]. Therefore $S$-space is compressible.

##  IV The dynamics geometry of $SP$

The task of mechanics is definition of trajectories of material bodies in
space with the help of dynamics laws. Therefore the geometry is included
naturally into a formalism of classical mechanics. The interrelation of
geometry and mechanics is carried out through concept of an interval. This
concept lies in bases of the formalism, both classical, and the relativistic
mechanics [6]. We will consider in what difference of an interval for $MP$
from an interval for $SP$.

Let’s consider a point in the configuration space, corresponding to the system
of $MP$. Through an interval time of $dt\longrightarrow 0$ the $MP$ will move
on distance $ds$. The volume of $ds$ is an interval. The interval for a set of
$MP$ is possible to express through the kinetic energies as follows [6]:

$\displaystyle
d\overline{s}^{2}=2T_{N}dt^{2}=\sum^{N}_{i=1}\breve{v}^{2}_{i}dt^{2}=\sum^{N}_{i=1}(d\breve{x}^{2}_{i}+d\breve{y}^{2}_{i}+d\breve{z}^{2}_{i})$
(9)

where ${d\overline{s}}$ is interval displaying infinitesimal distance between
two points of configuration space; ${\breve{x}=\surd{m_{i}x_{i}}}$,
${\breve{y}=\surd{m_{i}y_{i}}}$, ${\breve{z}=\surd{m_{i}z_{i}}}$ are
coordinates of the $i$ element; $m_{i}$ is a mass of the $i$ -element. The
configuration space is $3N$ dimensional Euclidian spaces for $N$ $MP$. In
general case the linear element will be set in the square-law differential
form of corresponding variables:

$\displaystyle
d\overline{s}^{2}=\sum^{n}_{i,k=1}g_{ik}d\breve{x}_{i}d\breve{x}_{k}$ (10)

where $g_{ik}=g_{ki}$ is symmetrical metrics tensor, $n=3N$.

If we have $p$ kinematics restrictions $f_{i}=f_{i}(x_{1},x_{2}...x_{n})$,
$i=1,2...p$, the motion of the system will be in $l=3N-p$ dimensional
hyperspace. In this case we have:
$d\overline{s}^{2}=\sum^{n}_{i,k=1}a_{ik}dq_{i}dq_{k}$, where $a_{ik}$ -is
known function in a new coordinates. If as kinematics conditions are potential
forces then the equation (8) will be equivalent to the motion equation of
$MP$. But for system which is a set of $SP$, the energy part is distributed by
non-potential forces. There is a question what will be an interval in this
case?

Let’s show, that for answer on this question it is necessary to present energy
of system in the form of two parts: energy of motion of the center of mass of
$SP$-$T_{N}^{tr}$, and internal energy of $SP$-$T_{N}^{ins}$. I.e. the
interval corresponding for system $SP$ also should consist of two parts. In
this case the $T_{N}^{tr}$, $T_{N}^{ins}$ expressions (7) can be written down
as:

$\displaystyle
d\overline{s}^{2}=(2T_{N}^{tr}+2T_{N}^{ins})dt^{2}=ds_{tr}^{2}+ds_{ins}^{2}=$
$\displaystyle\
N\breve{V}_{0}^{2}dt^{2}+(\sum^{N-1}_{i=1}\sum^{N}_{j=i+1}\breve{v}_{ij}^{2})dt^{2}/N$
(11)

where $\breve{V}_{0}=(\sum^{N}_{i=1}\breve{v}_{i})/N$,
$\breve{v}_{ij}=\breve{v}_{i}-\breve{v}_{j}$.

Let us transform the energy $T_{N}$ by replacement:
$\breve{v}_{i}=\breve{V}_{0}-\bar{v}_{i}$, where
$\sum^{N}_{i=1}\breve{v}_{i}=N\breve{V}_{0}$, i.e.
$\sum^{N}_{i=1}\bar{v}_{i}=0$. Then we will have:

$\displaystyle
T_{N}=N\breve{V}_{0}^{2}/2+\breve{V}_{0}\sum^{N}_{i=1}\bar{v}_{i}+\sum^{N}_{i=1}\bar{v}_{i}^{2}/2$
(12)

Because $\sum^{N}_{i=1}\bar{v}_{i}=0$, then we have

$\displaystyle\sum^{N}_{i=1}\bar{v}_{i}^{2}/2=1/(2N)(\sum^{N-1}_{i=1}\sum^{N}_{j=i+1}\breve{v}_{ij}^{2})$
(13)

As a result we obtain:

$\displaystyle
d\overline{s}^{2}=(2T_{N}^{tr}+2T_{N}^{ins})dt^{2}=ds_{tr}^{2}+ds_{ins}^{2}=$
$\displaystyle\ N\breve{V}_{0}^{2}dt^{2}+\sum^{N}_{i=1}\bar{v}_{i}^{2}dt^{2}$
(14)

Thus, the square of an interval of non-equilibrium system breaks up to the sum
of squares of two intervals. The first corresponds to the motion energy of
$SP$ center of mass and the second corresponds to the internal energy of
system. It is follows from here that the interval of the non-equilibrium
system which consists of a set of $SP$ breaks up to two independent intervals
characterizing dynamics of system: $ds_{tr}^{2}=N\breve{V}_{0}^{2}dt^{2}$ and
$ds_{ins}^{2}=\sum^{N}_{i=1}\bar{v}_{i}^{2}dt^{2}$. These intervals are
orthogonally and they correspond to adjacent of a triangle for a full interval
of system in configuration space.

The change of the $SP$ center of mass motion energy is caused by work of
potential forces $F^{tr}$. Their work is defined by expression:
$A^{tr}=\int{F^{tr}dR}$, $F^{tr}=\nabla\varphi$, where $\varphi$ is scalar
function, $dR$ is a distance of systems motion.

The forces $F^{ins}$ which change of the internal energy $SP$ are non-
potential. Their work consists from the work on change of $MP$ motion energy
relative to the center of mass, i.e.
$A^{ins}=\sum^{N}_{i=1}\int{F_{i}dr_{i}}$, where $dr_{i}$ -moving of $i$ -th
element of system relative to the center of mass. And because
$\sum^{N}_{i=1}{F_{i}}=0$ then $\int{\sum^{N}_{i=1}F_{i}dR}=0$ for any
possible way of moving of system. I.e. the potential component of the external
force $F^{tr}$ acting on $SP$ changes $s_{tr}$ but does not change $s_{ins}$.
The work of non-potential forces, $F^{ins}$ changes $s_{ins}$ but does not
change $s_{tr}$. The variables defining motion of the center of mass are
macroparameters, and the variables defining change of internal energy are
microparameters.

Thus for the description of dynamics of the non-equilibrium system it is
necessary to present this system as a set of $SP$ and then it is necessary to
represent $SP$’s energy in the form of the sum of two types of energy:
internal energy and energy of $SP$ motion. In the nature we deal with the real
bodies possessing internal energy. At their interaction the part of energy go
to their heating. This energy transforming is realized by the friction force.
So the $SP$ dynamics is determined by the two types of symmetries: the
symmetry of space in which the $SP$ motion and internal symmetry of
distributions of elements of $SP$. Thus the necessity of splitting of the
energy on two parts has under itself a real basis.

##  V The equations of interaction of systems and thermodynamics

Equations (1-8) give relationship between mechanics and thermodynamics [4, 8].
According to the basic equation of thermodynamics the work of external forces
acting on the system splits into two parts. The first part corresponds to
reversible work. In our case it corresponds to the change of the motion energy
of the system as a whole. The second part of energy goes on heating. It
corresponds to the internal energy of the system.

Let us take a motionless non-equilibrium system consisting of ”$R$”
equilibrium subsystems. Each equilibrium subsystem consists of a great number
of elements $N_{L}>>1$, where $L=1,2,3...R,N=\sum\limits_{L=1}^{R}N_{L}$. Let
$dE$ be work done over the system. In thermodynamics energy $E$ is called
internal energy (in our case it is equal to the sum of all energies of
equilibrium subsystems). It is known from thermodynamics that ${dE=dQ-PdY}$
[8]. Here, according to generally accepted terminology, $E$ is the energy of
the system; $Q$ is the thermal energy; $P$ is the pressure; $Y$ is the volume.
The equation of interaction between $SP$ is also a differential of two types
of energy. It means that $dE$ in the $SP$ is redistributed in such a way that
some part of it changes energy of relative motion of the $SP$ and the other
part changes the internal energy. Thus, it follows that entropy may be
introduced into classical mechanics if it is considered as a quantity
characterizing increase in the internal energy of an $SP$ at the expense of
energy of their motion. Then the increase in entropy can be written as [3, 4]:

${{\Delta{S}}={\sum\limits_{L=1}^{R}{\\{{N_{L}}\sum\limits_{k=1}^{N_{L}}\int[{\sum\limits_{s}{{F^{L}_{ks}}v_{k}}/{E^{L}}]{dt}}\\}}}}$
(15)

Here ${E^{L}}$ is the kinetic energy of $L$-$SP$; $N_{L}$ is the number of
elements in $L$-$SP$; $L=1,2,3...R$; ${R}$ is the number of $SP$; ${s}$ is the
number of external elements which interact with ${k}$ element belonging to the
$L$-$SP$; ${F_{ks}^{L}}$ is the force acting on the $k$-element; $v_{k}$ is
the velocity of the $k$\- element.

Based on the generally accepted definition of entropy we can derive expression
for its production and define necessary conditions for stationarity of a
nonequilibrium system [4].

##  VI Conclusion

The classical mechanics collides with insuperable difficulties in attempt to
describe evolution of non-equilibrium systems. The main reason is that the
process of evolution is irreversible but the classical mechanics is reversible
[9, 10]. The reversibility of classical mechanics is defined by the nature of
the second law of Newton. According to this law the acceleration of
unstructured bodies is proportional to the force acting on it. Therefore the
region of application of the second law of Newton is restricted by
unstructured bodies. It means that the second law of Newton is inapplicable
for the description of dynamics of the real bodies possessing a friction.
Hence for removal of the mentioned restrictions of classical mechanics it is
necessary to define friction forces rigorously on the basis of Newton’s second
law.

The analysis of dynamics of a hard-discs system has led to the conclusion that
in order to solve this problem it is necessary to find the motion equation of
$SP$. It has been done for a case when $SP$ represents a system of potentially
interacting $MP$, moving in the field of external forces.

During the process of search of a way which could lead to the $SP$ motions
equation and then as a result of its analysis, the following conclusions were
found out.

The motion and evolution of the system are defined by two types of symmetry:
the symmetry of space in which it is moving and its internal symmetry. In
accordance with these two types of symmetries the energy of system also breaks
up on to two types: the motion energy of system and its internal energy. In
its turn, the change of these types of energy is also defined by two types of
forces. Transformation of energy of $SP$ motion is caused by potential force.
Transformation of internal energy $SP$ is caused by work of non-potential
force. The work of the non-potential force leads to irreversibility of $SP$
dynamics.

The non-equilibrium systems in approach of the local equilibrium can be
presented as a set of the equilibrium subsystems which are in motion relative
to each other. In this case the description of dynamics of system by means of
the $SP$ motion equation can be carried out.

The state of the system as a set of $SP$ can be defined in the phase space
which consists of 6R-1 coordinates and momentums of $SP$, where R is the
number of $SP$. Location of each $SP$ is given by three coordinates and their
momentums. The phase space which is determined by coordinates and velocities
of $SP$ is compressible.

The dynamics of the non-equilibrium system composed of a set of $SP$ is
determined by the Liouville equation for equilibrium $SP$. These systems
acquires an equilibrium state when all energy of $SP$ motion transforms into
its internal energy.

The offered expansion of classical mechanics and the deterministic explanation
of irreversibility open a way to the substantiation of thermodynamics.
According to the motion equation for $SP$ the first law of thermodynamics
follows from the fact that the work of external forces changes both the energy
of particle’s motion and their internal energy. The second law of
thermodynamics follows from irreversible transformation of energy of relative
motion of system’s particles into their internal energy.

The motion equation for $SP$ also states impossibility of existence of
structureless particles in classical mechanics, which is equivalent to
infinite divisibility of matter.

Thus, the replacement of model of system in the form of set $MP$ on a model in
the form of a set of $SP$ leads to essential expansion of classical mechanics.
Such expansion allows, remaining within the frame of laws of Newton’s
mechanics, to offer the deterministic explanation of irreversibility and,
thereby, to enter the concept of entropy and evolution into the classical
mechanics. It is a bright example of that the further development of physics
is impossible without perfection of models on which basis it has been
constructed.

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