## Examples

What is a “**law of physics**”? As a category of being, it is a stable type of relations. However, what is meant by the term relations, and what does stable type actually mean? Depending on the degree of instantiation of these definitions, various laws of Physics can be considered. There is the interpretation of the concepts in the **Theory of Physical Structures** (TPhS), and certain statements have already been obtained on the existence of the “relations”. The Theory of Physical Structures is an algebraic theory of relations among elements of arbitrary nature, and aimed at rethinking of the laws of Physics. It is based on the
**phenomenological symmetry** of the laws of Physics and was formulated by Yu.I. Kulakov in Novosibirsk, in 1966.

### Three-dimensional Euclidean space

Let us move on to some examples, characterizing the heart of the matter. Geometry could be considered as the simplest one (Part 2, Chapter 6, Paragraph 4 of the monograph The theory of physical structures by Yu. I. Kulakov). Indeed, we can see whether the points are on the same line, plane, volume, etc. by making experimental measurements. Let us consider a set $\mathfrak{M}=\{i_1, i_2, \ldots, i_n \},$ that consists of $n$ points that are arbitrarily located in three-dimensional space. Could it be said that despite their arbitrary location, there is THE law of Physics (i.e. the law, validity of which can be established by experiment), which is obeyed by all points of the set $\mathfrak{M}$? To discover it, its necessary to consider all possible pairs of points of the set $\mathfrak{M}$, there will be ${\frac 12}n(n-1)$, by assigning each pair to experimentally measured value characterizing reciprocal positions of the points. We shall take distance as a value measured by experiment using, for example, a scale ruler.

By assigning each pair of points $(i,k)$ to the distance $\ell_{ik}$, we get a set of experimental data that fully characterize the given set $\mathfrak{M}$. The data can be presented in the form of a matrix as follows:

$$ \left.\begin{array}{c|ccccc} {} & i_1 & i_2 & i_3 & \ldots & i_n \\ \hline i_1 & 0 & \ell_{12} & \ell_{13} & \ldots & \ell_{1n} \\ i_2 & \ell_{12} & 0 & \ell_{23} & \ldots & \ell_{2n} \\ i_3 & \ell_{13} & \ell_{23} & 0 & \ldots & \ell_{3n} \\ \ldots &\ldots & \ldots & \ldots & \ldots & \ldots \\ i_n & \ell_{1n} & \ell_{2n} & \ell_{3n} & \ldots & 0 \end{array} \right. $$

It is clear that the reciprocal distances $\ell_{ik}, \ell_{im},\ell_{km}$
among any *three* arbitrary points $i,k,m \in \mathfrak{M}$
cant be functionally related, because when the distances $\ell_{ik}$ and $\ell_{im}$ are fixed, the third one $\ell_{km}$ can take values from $|\ell_{ik}-\ell_{im}|$
to $\ell_{ik} + \ell_{im}$:

$$
\mbox{
\unitlength=1mm
\begin{picture}(60,45)(0,-20)
\thicklines
\put(0,0){\line(6,-1){60}}
\put(0,0){\line(1,1){20}}
\put(20,20){\line(4,-3){40}}
\put(-4,0){$i$}
\put(62,-11){$k$}
\put(17,21){$m$}
\put(27,-10){$\ell_{ik}$}
\put(3.5,11){$\ell_{im}$}
\put(42,5){$\ell_{km}$}
\put(95,8){}
\thinlines
\end{picture}
}
$$
\\
The situation is similar when we take //four// arbitrary points
$i,k,m,n \in \mathfrak{M}$:
$$
\mbox{
\unitlength=1mm
\begin{picture}(60,45)(0,-20)
\thicklines
\put(20,20){\line(4,-3){10.3}}
\put(0,0){\line(6,-1){60}}
\put(0,0){\line(1,1){20}}
\put(0,0){\line(5,2){50}}
\put(20,20){\line(1,0){30}}
\put(50,20){\line(1,-3){10}}
\put(30.5,12){\line(4,-3){29.5}}
\put(-4,0){$i$}
\put(61,-11){$m$}
\put(17,21){$k$}
\put(49,21){$n$}
\end{picture}
}
$$
\\
and consider the relationship among
//six//
reciprocal distances $\ell_{ik}, \ell_{im}, \ell_{in},
\ell_{km},\ell_{kn},\ell_{mn}$.
When the distances $\ell_{ik}, \ell_{im}, \ell_{in},
\ell_{km}, \ell_{kn}$ are fixed, the sixth one $\ell_{mn}$ can take different values from a certain interval.

But if we take *five* arbitrary points $i, k, m, n, p \in \mathfrak{M}$, one of the *ten* reciprocal distances $\ell_{ik},
\ell_{im}, \ell_{in}, \ell_{ip}, \ell_{km}, \ell_{kn}, \ell_{kp}, \ell_{mn},
\ell_{mp}, \ell_{np}$ is a two-valued function of the remaining nine distances.

So, for any five points of the three-dimensional Euclidean space there is the functional relation among their reciprocal distances, and the form of the relation does not depend on the chosen points:

$$
\left|\begin{array}{cccccc}
0 & 1 & 1 & 1 & 1 & 1 \\
1 & 0 & \ell^2_{ik} & \ell^2_{im} & \ell^2_{in} & \ell^2_{ip}\\
1 & \ell^2_{ik} & 0 & \ell^2_{km} & \ell^2_{kn} & \ell^2_{kp}\\
1 & \ell^2_{im} & \ell^2_{km} & 0 & \ell^2_{mn} & \ell^2_{mp}\\
1 & \ell^2_{in} & \ell^2_{kn} & \ell^2_{mn} & 0 & \ell^2_{np}\\
1 & \ell^2_{ip} & \ell^2_{kp} & \ell^2_{mp} & \ell^2_{np} & 0 \\
\end{array}\right|=0.
$$

Full details of this example have been reviewed in the monograph The theory of physical structures by Yu. I. Kulakov.

### Newton's law of motion

To consider more physical examples let us refer to the idea of objects of different nature, as opposed to geometry, where all points are taken from one set. In this case, two points from two different sets assign to a measurement procedure, a kind of an analogue of *distance*.

Let us start with the well-known Newtons second law: $ ma=F.$ First of all, we shall add Latin and Greek indices to the physical quantities: $ m_ia_{\alpha i}=F_\alpha. $

We shall use indices $ i$ and $ k$ to refer to two bodies, indices $ \alpha$ and $ \beta$ to refer to two accelerators (or springs), and double index $ \alpha i$ to refer to the acceleration of a body $i$ under the impact of a spring $ \alpha $.

We shall take two bodies $ i$
and $ k$, two accelerators $ \alpha$ and $ \beta $,
and rewrite the equation in four versions:

$ m_i a_{\alpha i}= F_\alpha$ $ m_k a_{\alpha k}= F_\alpha $

$ m_i a_{\beta i}= F_\beta $ $ m_k a_{\beta k}= F_\beta $

Eliminating two weights $ m_i$ and $ m_k$, two forces $ F_\alpha$ and $ F_\beta $ yields one equation linking four accelerations.
$
a_{\alpha i}a_{\beta k}-a_{\alpha k}a_{\beta k}=0,
$
We shall rewrite it in such a form that the determinant is zero:

$ \left|
\begin{array}{cc}
a_{\alpha i} & a_{\alpha k}

a_{\beta i} & a_{\beta k}
\end{array}\right|\equiv 0
$
We are making a logical jump here! Instead of considering a determinant of the second order, we shall consider its generalization in the form of a numerical function of four variables
$
\Phi(\varphi_{\alpha i},\varphi_{\alpha k},
\varphi_{\beta i},\varphi_{\beta k})\equiv 0
$
, choosing one numeric function of two numeric variables, — $ \xi$ or $ \eta$ and **x** or
**y**, as its argument:

$\varphi_{\alpha i}=\varphi(\xi,x)\quad \varphi_{\alpha k}=\varphi(\xi,y)$

$\varphi_{\beta i}=\varphi(\eta,x)\quad \varphi_{\beta k}=\varphi(\eta,y)
$

As a result, we have previously unknown functional equation with respect to two unknown numeric functions $ \Phi$
and $\varphi$

$\Phi\bigl(\varphi(\xi,x),\varphi(\xi,y),\varphi(\eta,x),\varphi(\eta,y)\bigr)\equiv
0$ that is true for arbitrary $ \xi,\eta,x,y\in\mathbb{R}.$

Full details of this example have been reviewed in the monograph The theory of physical structures by Yu. I. Kulakov.

### Ohms law

Let us consider the example from Yu.I. Kulakovs book.
If we take three arbitrary conductors $i, k, m \in\mathfrak M$ and two arbitrary current sources $\alpha, \beta \in \mathfrak N$
and measure electric current *six* times ${\cal J}_{i\alpha}, {\cal J}_{i\beta}, {\cal
J}_{k\alpha}, {\cal J}_{k\beta}, {\cal J}_{m\alpha}, {\cal J}_{m\beta}$
using ammeter, according to the following pattern:

The following relation holds:
$$
\left|\begin{array}{ccc}
{\cal J}_{i\alpha} {\cal J}_{i\beta} & {\cal J}_{i\alpha} & {\cal
J}_{i\beta}\\
{\cal J}_{k\alpha} {\cal J}_{k\beta} & {\cal J}_{k\alpha} & {\cal
J}_{k\beta}\\
{\cal J}_{m\alpha} {\cal J}_{m\beta} & {\cal J}_{m\alpha} & {\cal
J}_{m\beta}\\
\end{array}\right|=0,
$$
From which, using the reference points $k, m \in \mathfrak M, \beta \in \mathfrak N$
, well-known Ohm's law for a complete circuit can be obtained.
$$
{\cal J}_{i\alpha}=\frac{{\cal E_{\alpha}}}{R_i+r_{\alpha}},
$$
${\cal E_{\alpha}}$ – the electromotive force of the current source,

$R_i$ – the resistances of the conductors,

$r_{\alpha }$ – the internal resistance of the current source.

Full details of this example have been reviewed in the monograph The theory of physical structures by Yu. I. Kulakov

### Thermodynamics

We shall consider the example from the monograph by G.G. Mikhailichenko (Introduction).

Let us consider the set of states of some thermodynamic system. We shall assign to each pair of states $\langle ij\rangle $
two numbers equal to two quantities of heat
$Q^{TS}_{ij}$ and $Q^{ST}_{ij}$, which the system gives away to other bodies in the course of the transition from the state $i$ to the state $j$, first along the isotherm $(T=const)$, then along the adiabat $(S=const)$. There is the process **TS**, going first along the adiabat, then along the isotherm in the first case and the process **ST** in the second one, where **T** is the temperature and **S** is the entropy of the system.

A two-component numeric function $Q_{ij}=(Q_{ij}^{TS},Q_{ij}^{ST})$
sets *two-dimensional geometry* on the plane $(S,T)$ of states of the thermodynamic system, and this function in this geometry is a kind of an analogue of *distance* between points $i$ and $j$.
Let us take *three* arbitrary states
$\langle ijk\rangle$ on the plane $(S,T)$, and
the order of the states is determined by the entry of the triple. Then, in addition to the *distance* $Q_{ij}$, it is possible to write two ones
$Q_{ik}, Q_{jk}$ for the pairs of state $\langle ik\rangle$ and
$\langle jk\rangle$.
All three two-component *distances* turn out to be tied by the two following equations:
$$
\left|
\begin{array}{ccc}
0 & -Q^{ST}_{ij} & -Q^{ST}_{ik} \\
Q^{TS}_{ij} & 0 & -Q^{ST}_{jk} \\
Q^{TS}_{ik} & Q^{TS}_{jk} & 0
\end{array}
\right| =0,
\hspace{1 cm}
\left|
\begin{array}{ccc}
Q^{TS}_{ij} & Q^{TS}_{jk} & -Q^{ST}_{ik} \\
Q^{TS}_{ik} & 0 & -Q^{ST}_{ik} \\
Q^{TS}_{ik} & -Q^{ST}_{ij} & -Q^{ST}_{jk}
\end{array}
\right| =0,
$$

Which are true for any triple of states $\langle ijk\rangle$.

### Examples by Yu.I. Kulakov

Examples from the monograph The theory of physical structures by Yu. I. Kulakov.

### Additional examples

These are several examples of physical structures proposed by Yu.I. Kulakov:

1. Analytic geometry,

2. Analytic thermodynamics,

3. Time as a physical structure.