# INCLUSION OF THE TENSOR ANALYSIS OF NETWORKS INTO DIFFERENTIAL GEOMETRY

Olivier Maurice

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Olivier Maurice. INCLUSION OF THE TENSOR ANALYSIS OF NETWORKS INTO DIFFERENTIAL GEOMETRY. 2023. [hal-04286165]

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# INCLUSION OF THE TENSOR ANALYSIS OF NETWORKS INTO DIFFERENTIAL GEOMETRY

Olivier MAURICE, AFSCET

19 Octobre 2023

## Abstract :

Kron's tensor analysis of networks was developed partly based on differential geometry, but with few accurate links between both formalisms. This paper tries to submit some complements for enriching these links. This work shows new points of view and writings as new perspectives for the use of this formalism in the electromagnetic compatibility (EMC) context then in all physics. In a first paragraph we locate the space of configuration in a polyhedral (polytopic) representation using some cellular topology description. This gives us the context for defining a dual space and other concepts that belong to the differential geometry. Once this environment acquired we can rewrite Kron's equations and defined various physical interactions. In particular we can exchange on the meaning of least action and geodesics, and submit a global electronic space view.

The whole document is written in a way usable by engineers. But it can give ideas and perhaps new approaches for researchers in particular in the case of multiphysics modelling. Some symbolism is subject also for identifying the various electromagnetic interactions in graphs.

This article is a tentative for synthesizing and sharing my work on Kron's tensor analysis of networks realized during more than thirty three years.

In all the article we use Einstein's mute notation and we suppose that the readers know tensor algebra basics.

## 1. THE UNDERLYING SPACE

Any physical problem is projected onto a polyhedra which is itself projected onto a space and wavelength grid for a given time axis value. In our proposal the space - wavelength hypersurface ($x^\alpha, \lambda/c$) supports an electronic circuit behavior for one particular value of time $\tau$. The volume ($x^\sigma, \tau$) contains all the system working points, which is locally described at a given space point under Laplace's formalism using $k = j2\pi\lambda/c$. The figure 1 illustrates this concept (we will detail the concept of layers further).

The polyhedra is a cellular topology made of nodes (set $\mathbb{N}$), edges (set $\mathbb{B}$), nodes pairs (set $\mathbb{P}$) and meshes (set $\mathbb{M}$). A network is made of many circuits, each circuit being a connex graph. When we hover the polyhedra layer, we perceive both local definitions of the circuits, branches and meshes, and a larger view of curvilinear currents that can involve various circuits. The three visions constitute the bases of two systems of spaces. One are local edges description
<figure><img src="image_2.png" /><figcaption>FIGURE 1. SpaceTime layer subnetwork</figcaption></figure>

associated with the base vectors $\vec{b}_{\alpha}$, $\alpha \in \mathbb{B}$. The other is a local curvilinear description of mesh
circulations with its covectors base (cobase) $\mathcal{C}^{\sigma}$, $\sigma \in \mathbb{M}$. This description beginning remains quite
difficult to understand. We imagine flows of particles linked with closed circulations. Each of
these circulations is a sort of function associated with classical vectors. Each vector belongs to
a branch which is a part of the underlying curvilinear grid. The set of vectors $\{\vec{b}_{\alpha}\}$ makes the
edge space $\mathbb{B}$. They are also connected with $\mathbb{R}^3$ through the connectivity $\Lambda/\vec{b}_{\alpha} = \Lambda_{\alpha}^{\sigma}\vec{x}_{\sigma}$.

The definition of the functional $\varphi$ is not so trivial. In fact what we want is that the branches
vector $\vec{b}_k$ are the base vector of the local tangent space $TpS$. As there are obtained from the
functional derivation $\partial\varphi/\partial\vec{b}_j$, we must define this functional. For doing that we consider the
curvilinear axis of the underlying space and their connections with meshes (see figure 2). The
curvilinear base $\vec{u}_k$ being available, we can write the dependencies between the mesh (on the
draw $a_i$) and these directions. This defines a function $\varphi$ with $\varphi(a_i) \in \{b^k\}$, $\varphi = b_i^k$ if a side of $a_i$
follows the direction $\vec{u}_k$, $-b_i^k$ if it follows the inverse direction and 0 otherwise. The functional
$\varphi$ is a set of equations linking the meshes $a_i$ (which are scalars) and the directions $\vec{u}_k$. For
<figure><img src="image_3.png" /><figcaption>FIGURE 2. Space connection</figcaption></figure>

example:

$$ (1) \qquad \varphi : \left\{ \begin{array}{l} a_1 = b_1^1 \vec{u}_1 + b_1^2 \vec{u}_2 + b_1^3 \vec{u}_3 + \dots \\ a_2 = b_2^1 \vec{u}_1 + b_2^2 \vec{u}_2 + b_2^3 \vec{u}_3 + \dots \equiv a_j = \varphi_j^k \vec{u}_k = b_j^k \vec{u}_k \\ a_3 = \dots \end{array} \right. $$

The branch vectors are obtained writing:

$$ (2) \qquad \vec{b}_k = \frac{\partial \varphi}{\partial \vec{u}_k} = b_k^j \vec{u}_j, \quad \vec{s} = s^k \vec{b}_k = s^k b_k^l \vec{u}_j, \quad b_i^k \in \{-1, 0, 1\} $$

Then computing $g_{ij} = \langle \vec{b}_i, \vec{b}_j \rangle$, we obtain N coefficients of the metric g. But with this operation some of g lines doesn't exist in $\varphi_j^k$, they appear like ghost connections to unexisting meshes (M being arbitrary defined with Euler-Poincaré's characteristic as single constraint). Supressing these lines, we define $g = \varphi_j^k$ as the metric giving the measurements of the numered mesh $a_i$ and the curvilinear space directions. As a consequence, the branch vectors coordinates are obtained using the inverse metric: $s^k = g^{kj} a_j$, they are a measure of the mesh lengths in each space direction possibly shared between more than one mesh. They define the branch vector $\vec{s} = s^k \vec{b}_k$. Meshes are local concretization of the underlying space and branches give the measures of each mesh side.
We can now formally write the fundamental relation between the currents of branches co-ordinates $s^\sigma$ (which points out the current vector $\vec{b}$ in the branch space $\mathbb{B}$) and the currents of meshes coordinates $a_\alpha$ (which points out the covector of the currents $\check{a}$ in the mesh space $\mathbb{M}: \check{a} = a_i \check{c}^i$):

$$ (3) \qquad s^\alpha = g^{\alpha\sigma} a_\sigma, \quad g^{\alpha\sigma} = (g_{\sigma\alpha})^{-1} $$

In practice, we start from this last equation for defining the relations between the branch currents and the mesh ones. When a branch is shared between two meshes for example, the coefficients $g^{bm}$ is +1 between one mesh and the branch current, and -1 between the same branch current but with the other mesh, if the two circulations $\check{c}^\alpha$ are of opposite way in the branch. With this formalism contrarily to Kron's one, the current variance depends on the space. $s^\alpha$ concerns the branche space while $a_\alpha$ concerns the mesh one. For potential differences and electromotive forces (emf), $e_\alpha$ belongs to $\mathbb{B}$ and $E^\alpha$ belongs to $\mathbb{M}$. These results help also for clarifying the writings.

For Gabriel KRON[1], $g$ in the equation 3 was a connectivity, comparable to the incidence matrix $A$ between the nodes space and the branch space. At the contrary, the impedance tensor was seen as a sort of metric. In my opinion and for having used this approach, while the function $\varphi$ gives a physical measurement of the mesh, based on the space of branches (well defined on the underlying space), the impedance tensor is too far from a simple Riemannian concept of distance for being seen one way or another as a metric.

In our proposal, the metric is well defined mathematically, and all the objects involved in the definitions find their place without any ambiguity. More, we will see that this new approach give answers for other equations interpretations.

## 2. FROM THE ENERGY TO LAGRANGE'S EQUATIONS

One of the most beautiful result in physics obtain these last centuries comes from Lagrange's least action principle analysis. The action $a$ is the product of the energy $T$ by the time:

$$ (4) \qquad a = \int_t dt T $$

Our purpose is to determine the conditions in which the term $T$ remains an extremum when the temporal trajectory changes. Many demonstrations of this principle was made, so our purpose is not to make them once more, but to understand its major mechanism. We pose that the energy function $T$ depends on time, space and speed, i.e.: $T(t, x, \dot{x})$. We imagine that two operators exist, $\alpha, \beta$, applied on space and speed and depending on time such a way that $T = \alpha \cdot \dot{x} + \beta \cdot x$. We want that between two trajectories $x, \dot{x}$ and $x + \epsilon, \dot{x} + \dot{\epsilon}$, the difference tends to zero if these trajectories are near the physical trajectory.

Lagrange's genius[2] was to differentiate the derivatives on $T$ depending on $x$ or $\dot{x}$ from the difference $\delta a$ between the two trajectories. For determining the speed and location of the particle from its energies and compare the trajectories we first apply the space and speed
derivatives. Then we can compare the trajectories using the differential $\delta$. The idea is so to compute first for example $\partial T_i/\partial x$, then to compute $\delta(\partial T_i/\partial x)$, $T_i$ meaning both trajectories, $T_1$ and $T_2$.

Making these operations we obtain on this first tentative (neglecting $\epsilon^2 = 0$ and $\dot{\epsilon}^2 = 0$):

$$ (5) \qquad \delta \frac{\partial T}{\partial \dot{x}} = 2\alpha \cdot \dot{\epsilon} \quad \delta \frac{\partial T}{\partial x} = 2\beta \cdot \epsilon $$

We know that Newton's law is a good solution of the problem. So we should retrieve some- where its components that do not appear in this first tentative. It's evident that one operation must be modified to conduct:

$$ (6) \qquad \delta \frac{d}{dt} \frac{\partial T}{\partial \dot{x}} = 2\alpha \cdot \ddot{\epsilon} \quad \delta \frac{\partial T}{\partial x} = 2\beta \cdot \epsilon $$

Our objective for reaching $\delta a = 0$ becomes Lagrange's equations:

$$ (7) \qquad \frac{d}{dt} \frac{\partial T}{\partial \dot{x}} - \frac{\partial T}{\partial x} = 0 $$

whose are another expression of Newton's law.

Usually [3], this demonstration uses integration by parts. By rewriting the neighborhood of $a$ with a function $\eta\epsilon$ where this time $\epsilon$ doesn't depends on $t$:

$$ (8) \qquad a(\epsilon) = \int_{t_1}^{r_2} dt T(t, x + \epsilon\eta, \dot{x} + \epsilon\dot{\eta}) $$

Looking for an extremum with $\epsilon = 0$, this leads to:

$$ (9) \qquad \frac{\delta a}{\delta \epsilon} (\epsilon = 0) = \int_{t_1}^{t_2} dt \left( \frac{\partial T}{\partial x} \eta + \frac{\partial T}{\partial \dot{x}} \dot{\eta} \right) = 0 $$

This can be written:

$$ (10) \qquad \int_{t_1}^{t_2} dt \frac{\partial T}{\partial x} \eta + \left[ \frac{\partial T}{\partial \dot{x}} \eta \right]_{t_1}^{t_2} - \int_{t_1}^{t_2} dt \eta \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{x}} \right) = \int_{t_1}^{t_2} dt \eta \left( \frac{\partial T}{\partial x} - \frac{d}{dt} \frac{\partial T}{\partial \dot{x}} \right) = 0 $$

As this must remain true whatever $\eta$, it comes:

$$ (11) \qquad \frac{\partial T}{\partial x} - \frac{d}{dt} \frac{\partial T}{\partial \dot{x}} = 0 $$

Which is our previous result. The equation can be easily generalized, using the coordinates $x^k$. Note that in this case, we already work in a 4-dimension space-time, $x^k$ covering $\mathbb{R}^3$ and the time derivative $dt$ pointing out the time axis:

$$ (12) \qquad \frac{\partial T}{\partial x^k} - \frac{d}{dt} \frac{\partial T}{\partial \dot{x}^k} = 0 $$

Lagrange's equations wear intrinsically a tensor product. The energy sends a tensorial product involving speeds (for the kinetic one) to a scalar $T$. Under Grassmann's algebra [4] this may be written:

$$ (13) \qquad T = \frac{1}{2} t_{\mu\nu} \dot{x}^{\mu} \wedge \dot{x}^{\nu} $$
As a time derivation doesn't change a tensor variance, the underlying space being the fundamental base $(\vec{x}_1, \vec{x}_2, \vec{x}_3) \in \mathbb{R}^3$, a point coordinates are given by $x^k$ and the tensor $t_{\mu\nu}$ is twice covariant. But the base $\vec{x}_k$ is a local one. Depending on where the point is located in the whole space, their properties can change. The underlying space is a parametrized hypersurface $\mathcal{A}$ of parameters $u_\alpha$ so that $x^k = \mathcal{A}(u_\alpha)$.

The fundamental base is by the fact defined through:

$$ (14) \qquad \vec{x}_{\alpha} = \delta_{\alpha\beta} \frac{\partial \mathcal{A}}{\partial u_{\beta}} $$

and the fundamental metric of the underlying space by:

$$ (15) \qquad G_{\mu\nu} = \langle \vec{x}_{\mu}, \vec{x}_{\nu} \rangle $$

In practice, working directly in the underlying space is too heavy. The experience has shown the extremely powerful use of macro models. That's the polyhedra projection that supports the cellular topology in which we describe our electronical circuits and other physical circuits.

To do that we can imagine an outline $\check{c}^k$ defined on the fundamental base $\check{c}^k = f^{kj}\vec{x}_j$. As $\vec{b}_k$ is supported by $\vec{x}_k$, we define a connectivity between both spaces writing:

$$ (16) \qquad \vec{b}_k = \Lambda_k^j \vec{x}_j \Rightarrow \vec{x}_j = V_j^k \vec{b}_k $$

This connection give us the way to describe the outline as a set of physical branches $b^k$ by writing:

$$ (17) \qquad \check{c}^{\alpha} = f^{\alpha\beta} V_{\beta}^{\sigma} \vec{b}_{\sigma} $$

Defining $\varphi^{\alpha\sigma} = f^{\alpha\beta}V_{\beta}^{\sigma}$, we finally retrieve a dual of our first relation between the curvilinear space $\check{c}$ and the local macro models of circuit $\vec{b}$ through:

$$ (18) \qquad \check{c}^{\alpha} = \varphi^{\alpha\sigma} \vec{b}_{\sigma} $$

Somewhere this is clearer with the coordinates where the local circuit depends on a parametrized space attached with the meshes:

$$ (19) \qquad s^{\alpha} = g^{\alpha\sigma} a_{\sigma} $$

The mesh space appears clearly here as defined by the curvilinear space which parameters are the underlying space directions, and the local tangent space is the one of the branches. The mesh space is the fundamental circuits space. We show this now, being inspired by a previous demonstration given by Gabillard[7].

In mechanics, for a frictionless system, a mass $m$ in equilibrium has for kinetic energy $T = 1/2m\dot{x}^2$ and for potential energy $U = 1/2kx^2$. From these relations we obtain:

$$ (20) \qquad \frac{d}{dt} \frac{\partial T}{\partial \dot{x}} = m \frac{d\dot{x}}{dt} = m\gamma = F_1 - \frac{\partial U}{\partial x} = -kx = -F_2 $$

$F_1$ is the inertia force and $F_2$ the stiffness force. Lagrange's equation only says that the summation of these two forces must be equal to zero: $F_1 + F_2 = 0$. If the system has dissipation forces
$F_f$ and is constrained by an external force $F_e$, Lagrange's equations says that the summation of all the internal forces must be equal to the applied external forces, i.e. $F_1 + F_2 + F_f = F_e$.

In the case of an electrical circuit, the external electromotive force applied $e(t)$ must be equal to the voltage developed across the resistors $v_r$, plus the voltages developed across the capacitors $v_C$ and plus the voltages developed across the inductances $v_L$:

$$ (21) \qquad e(t) = v_r + v_C + v_L $$

We see that the mesh equation is nothing else than the electrical expression of Lagrange's equations.

### 3. FROM KIRCHHOFF'S LAW TO KRON'S ONE

The most elementary circuit we can do is a single branch. If this branch is power supplied by both an external current source $s^y$, with an external emf $e_x$, a self impedance $z$, the difference of potentials across the current source being $V_x$ we can write:

$$ (22) \qquad e_x - V_x = z_{xy}s^y \Leftrightarrow e_x = V_x + z_{xy}s^y $$

Using our metric we can now write:

$$ (23) \qquad e_x = V_x + z_{xy}g^{y\alpha}a_\alpha $$

and multiplying both members by $g^{\sigma x}$:

$$ (24) \qquad g^{\sigma x}e_x = g^{\sigma x}V_x + g^{\sigma x}z_{xy}g^{y\alpha}a_\alpha $$

$g^{\sigma x}V_x$ on a closed circulation is a rotational of gradients, and so equal to zero: $g^{\sigma x}V_x = 0$. $g^{\sigma x}e_x$ is the mesh emf $E^\sigma$, and $\zeta^{\sigma\alpha} = g^{\sigma x}z_{xy}g^{y\alpha}$ the fundamental tensor in the mesh space. The circuit equation becomes Kron's equation in the mesh space but with our new writing:

$$ (25) \qquad E^\sigma = \zeta^{\sigma\alpha}a_\alpha $$

Once the mesh currents are known, we can know the branches ones as a consequence of the metric, as it is $g$ which allows to measure the branch currents through $s^k = g^{k\alpha}a_\alpha$. The dimension of the mesh space is the number of mesh $M$ and is also the number of holes of the manifold equivalent to the circuit studied [8] (another way is to say that $M$ has a link with Euler's characteristic). But there's one axis that is not underlying the base $\check{c}^\alpha$: the time axis. As we want to use Laplace's transform for our computations on electronics, all vectors or covectors should be associated with the wavelength $\lambda$ (or the frequency, or ...) defining Laplace's operator as $k = j2\pi c/\lambda$. And to obtain a direct link with the time scale we choose to use $\lambda/c$ rather than $\lambda$. Our equation becomes:

$$ (26) \qquad \left( E^\sigma, \frac{\lambda}{c} \right) = \zeta^{\sigma\alpha} \left( a_\alpha, \frac{\lambda}{c} \right) $$

The Laplace's expression of the vectors or covectors are valid for one location on the time scale $\tau$, which is the self-time of the circuit in its referential. The twice contravariant tensor $\zeta$ for
example is parametrized by $\lambda$ but depends on $\tau$. By a similar way, the circuit movements are described inside $(\mathbb{R}^3, \tau)$. And for generalizing our approach, taking into account relative movements between referentials, it can exists various $\tau$. The underlying space in that case is of the form $(x^k, \tau^k)$. A first representation of Kron's generalized equation becomes with these concepts:

$$ (27) \qquad \left. \left( E^{\sigma}, \frac{\lambda}{c} \right) \right|_{(x^k, \tau^k)} = \zeta^{\sigma\alpha} \left. \left( a_{\alpha}, \frac{\lambda}{c} \right) \right|_{(x^k, \tau^k)} $$

## 4. THE RADIATIVE MEANING OF THE UNDERLYING SPACE

($\mathbb{R}^3, t$) wears the field. The gravitational field is the geometry, and the electromagnetic field is the interactions between particles (chemical part). The electromagnetic field that we cannot access directly, follows the lines defined by the gravitation.

If we take a magnifying glass for amplifying our vision of the underlying layer, we find a disturbed space where the light goes everywhere[9]. But while going away and enlarging the view, we just discern the mains ways used by the light which are the most probabilistic ones: an application of the least action principle at the quantum level. Quantum mechanics is just a magnified view of the underlying space, or of the circuit one. Interactions act as strings[5] between the underlying space and the circuit space, exchanging couplings between the charges particles in the circuit, and the electromagnetic field on the underlying layer.

These strings doesn't change the topology characteristics. They translate interactions between various graph elements, symbolizing reports of emf issued from a mesh current on another mesh without modifying Poincaré's characteristic of the network.

We detail now these various interactions.

### 4.1. Coulomb's interaction.
It is very interesting to model the field using Coulomb's gauge. In this gauge, the photonic and delayed vector potential field $A$ is transverse to the light line (it's the far electric field that can be associated with photons). The near and longitudinal electric field is obtained from the scalar potential $\psi$, not delayed. The magnetic field is similar to the one defined in the Lorentz's gauge ($B = \nabla \times A$). Coulomb's interaction concerns the scalar potential which is the support for this interaction:

$$ (28) \qquad \psi(2) = \frac{q(1)}{4\pi\epsilon R(1, 2)} $$

This formula must be interpreted following: the potential on the node (2) created by the load on the node (1) is given by equation 28. This load is obtained through the capacitor that exists between the node and the reference node $C^{11}$. The potential difference constructed across this branch $V_1$ leads to the load $q(1) = C^{11}V_1$. The interaction is then defined by:

$$ (29) \qquad \psi_2 = \frac{C^{11}V_1}{4\pi\epsilon R^{12}} $$
The Gauss's equation of Maxwell's group is:

$$ (30) \qquad \iint_S d\vec{s} \cdot \nabla \vec{\psi}_2 = \frac{q(2)}{\epsilon} $$

The potential $\psi_2$ creates between the node (2) and the reference node a current source<fnref n="1" />:

$$ (31) \qquad a_2 = k\epsilon \iint_S d\vec{s} \cdot \nabla \vec{\psi}_2 $$

Coulomb's interaction is translated through an admittance between two nodes of two capacitance branches referenced to a common and virtual reference node of potential and defined by:

$$ (32) \qquad y_2^1 = \frac{a_2}{V_1} = k\epsilon \iint_S d\vec{s} \cdot \left( \vec{\nabla} \frac{C^{11}}{4\pi\epsilon R^{12}} \right) $$

As this interaction is an application going from the branch space to the mesh one: $y : \mathbb{B} \to \mathbb{M}$, it is not a fundamental tensor neither a metric. It's not a tensor in fact and has only the property of a matrix. A consequence of its definition, is that the graph contains capacitors between the nodes involved in the interaction and a common potential reference. This last property is in coherence with the absence of delay for $\psi$. The fact to share the same potential reference implies to be in the same circuit inside which no time delays are needed. The question may be "how the distance acts on Coulomb's interaction?". What's the threshold up to which this interaction disappears?

We consider only the dynamic case, not the static one for which no distance can be defined, but for which the absence of time is not a problem as we work with a null frequency. We know, and it's a well known property of Fourier's transform, that the continuous component can be added separately to the dynamic spectrum.

In a dynamic case, oscillations exist between various parts of two separate circuits with a Coulomb's coupling between them. We can imagine for example two monopoles of height $h_m$ referenced to the same ground plane, separated by a distance $d$, and coupled through a capacitance $\gamma^{21}$ defined by:

$$ (33) \qquad \gamma^{21} = \delta^{22} \left( \frac{y_2^1}{k} \right) $$

The two monopoles are perpendicular to the ground and parallel one with respect to the other. Their images on the other side of the ground makes that for wavelength much higher than their heights, they wear a constant load along their conductors, and their images an opposite load. We match each monopole with a resistance $R$, then we apply a power supply $V$ on one monopole, defined by $V(k) = 1/k$. After a short time equal to $6R\gamma$, the voltage across $\gamma$ is $V$. But each monopole has a capacitance between its foot and the ground. When increasing

<footnote n="1">¹current sources are virtual meshes in $\mathbb{M}$ for which the mesh currents $a_i$ are fixed and the potentials difference across the branch receiving $a_i$ unknown[6]</footnote>
<figure><img src="image_4.png" /><figcaption>FIGURE 3. Coulomb's coupling between two monopoles</figcaption></figure>

the distance between the monopoles, these capacitances will play a role more important than $\gamma$, and $\gamma$ will even disappear beyond a given value of $d$.

If we look to the figure 3 we see the electrostatic field lines going back to the ground around the antennas feet, and higher, lines going directly from one monopole to the other. These direct lines are the electrostatic flux associated with $\gamma$, while the other lines make another capacitance in parallel to the load $R: C_m$.

Just seeing this drawing, we understand that for distances higher than $h_m$, $\gamma$ vanishes to zero. The exact computation is more difficult, but the principle is here. We mean that beyond a given distance, the $\gamma$ coupling disappears. If we reduce the wavelength, we go through the resonance for which $\lambda/4 = h_m$ where the field line remain similar to the previous ones. Then the distribution of loads all along the monopole will alternate positive and negative ones, making the Coulomb's interaction meaningless. In resume, when propagation appears, Coulomb's coupling disappears. That why there are no contradiction in its definition where the delays are not present. The symbol for this interaction is a line between the two nodes concerned, with two little bars perpendicular on this line, like a capacitance.

4.2. **Neumann's interaction.** Neumann's interaction concerns the near magnetic field interaction. There is a technique by the past a little forgotten, which uses susceptances. At this time it was usual to construct complex magnetic systems, including amplifiers, etc. For the conception of transformers, the susceptances model is without any doubt the most efficient. Somewhere it's just the inverse of the mutual inductance. By definition the reluctance is given by:

$$ (34) \qquad \mathcal{R} = \frac{1}{\mu} \left( \frac{l}{As} \right) $$
* $\mu$ is the magnetic permeability,

* $As$ is the magnetic flux section,

* $l$ is the magnetic flux length.

Having the susceptances tensor, we can employ the fundamental relation for the magnetic circuits, including the Neumann's interaction which is the near magnetic field interaction:

$$ (35) \qquad \mathcal{F} = \mathcal{R}\phi $$

$\mathcal{F}$ is the magnetomotive force obtained by the product of the current in a loop of circulation equal to $\partial As$ by the number of turns the loop contains, and $\phi$ is the magnetic flux defined by:

$$ (36) \qquad \phi = \oint_S d\vec{S} \cdot \vec{B} $$

As $\mu$ is a tensor, $\mathcal{R}$ is a tensor. Generalizing our previous relation leads to:

$$ (37) \qquad \mathcal{F}_{\alpha} = \mathcal{R}_{\alpha\beta}\phi^{\beta} $$

Note that $\partial As$ is the border operator applied to the surface $As$: it defines the closed circulation of the mesh of current $\mathcal{F}$ surrounding the surface $As$. The fact that in the relation 37 the magnetomotive force is covariant is coherent with our dual space definition.

The relation with the mutual inductance is immediate: the electromotive force comes from the flux through $e^{\beta} = -k\phi^{\beta}$. As $-kM^{\beta\sigma} = e^{\beta}/\mathcal{F}_{\sigma}$ it implies that:

$$ (38) \qquad -kM^{\beta\sigma} = \frac{-k\phi^{\beta}}{\mathcal{F}_{\sigma}} = -k(\mathcal{R}_{\sigma\beta})^{-1} \Rightarrow M^{\beta\sigma} = (\mathcal{R}_{\sigma\beta})^{-1} $$

With materials of high permeability, it's easy to compute $\mathcal{R}$. For more complex situation, it's always possible to use Neumann's formula:

$$ (39) \qquad M^{\alpha\sigma} = \frac{\mu^{\alpha\sigma}}{4\pi} \oint_{\vec{c}^{\alpha}} \oint_{\vec{c}^{\sigma}} \frac{dc_{\alpha} \cdot dc_{\sigma}}{R^{\sigma\alpha}} $$

The symbolism used for this interaction is a line joining both magnetomotive force and receiving circuit wearing a little circle in its center.

4.3. **Fraunhofer's interaction.** Fraunhofer's interaction points out the radiative process associated with far field, photons and light. Used under a classical approach, the field is clearly defined with Coulomb's gauge using the vector potential. From the quantum point of view, the field uses various modes for going from a space location to another. These modes are seen as field states and the photon must be seen like the probability function saying how many photons may populate each state[12].

The vector potential field definition is (formally it comes from the 4-speed $(c, v)$). Multiplying by $\rho$, $(\rho c, \rho v)$ with $\rho v = Jx$ and finally by $\mu_0/(4\pi R^2)$ with $\mu_0\epsilon_0c^2 = 1$; it gives the 4-field $(\psi/c, \mathbf{A}))$:
$$ (40) \qquad \mathcal{A}^{\alpha} = \oint_{C_{\sigma}} da_{\sigma} G^{\sigma\alpha}, \quad G^{\sigma\alpha} = \frac{1}{r_{\alpha\sigma}} \exp\left(-k \frac{r_{\alpha\sigma}}{c_{\sigma\alpha}}\right) $$

$r_{\alpha\sigma}$ is the distance between the points $\alpha$ and $\sigma$ and $c_{\sigma\alpha}$ the light celerity between the same points.

The mutual impedance interaction based on this field is given by:

$$ (41) \qquad \chi^{\sigma\omega} = \frac{\mu^{\sigma\omega}}{4\pi} \oint_{C_{\sigma}} \delta_{\sigma\beta} d\check{c}^{\sigma} \mathcal{A}^{\beta}(a_{\omega}) $$

We may want as in the static case 39 to find an expression field-less, as it was discussed previously by Feynman[13]. But as also says by the same author, this remains very difficult. The difficulty appears when we try to obtain the equation of the interaction impedance. We beed to recall from where comes the field due to the fact that it is a delayed one. This delay cut the causal relation between the field source and the target mesh. By the fact, we need to keep the field nature information through its origin, which is intrinsically included in its expression.

The field traveling can be followed using the admittance $y_{ii}$ which comes from the ratio between an electromotive force $e^i$ induced on a mesh $i$ and the mesh current $a_i$ giving:

$$ (42) \qquad a_i = y_{ii} e^i $$

On a specific travel, the field can go from mesh 1 to mesh 2: $e^2 = \chi^{21}a_1$. This determines the mesh current $a_2 = y_{22}e^2$. This current radiates a field creating an electromotive force on the mesh 3 by $e^3 = \chi^{32}a_2 = \chi^{32}y_{22}\chi^{21}a_1$, etc. The final complete interaction between meshes 3 and 1 is defined by$^2$:

$$ (43) \qquad \mathcal{H}^{31} = \chi^{32} y_{22} \chi^{21} $$

This can be generalized for any field traveling:

$$ (44) \qquad \mathcal{H}^{A1} = \chi^{AB} y_{BB} \chi^{BC} y_{CC} \chi^{CD} y_{DD} \dots \chi^{S1} $$

This particular way is a particular field mode and can be associated with a field state $|ABCD \dots 1\rangle$. The final current induced on the target mesh $a_1$ can be analyzed separating each possible traveling:

$$ (45) \qquad a_1 = a_1(\mathcal{H}^{A1}) + a_1(\mathcal{H}^{B1}) + \dots $$

The transfer function:

$$ (46) \qquad \mathcal{H}^{A1} \leftrightarrow P(A, 1) $$

is linked with the probability that a photon uses the traveling $\mathcal{H}^{A1}$ to go from the point A to the point 1. This is the meaningfull concept of photons. The probability function is not a simple one. The total electromotive force on the target mesh (the observation point) is $e = \mathcal{H}^{A1} + \mathcal{H}^{B1} + \dots$. And so the module of $e$ is not the simple summation of the module of the

<footnote>$^2$As each mesh is matched, Huygens's principle implies that the light does not go back[10]</footnote>
terms $\mathcal{H}^{i1}$. What we can do is to evaluate the contribution of each $\mathcal{H}^{i1}$ on the final amplitude of $a(e)$, normalizing $a_1$. The probability of number of photons to populate one possible way is equal to the contribution of this way to the total amplitude of energy transmitted[9]:

$$ (47) \qquad P(\mathcal{H}^{i1}) = \left| \frac{\Delta a_i (+\mathcal{H}^{i1})}{a_i \left( \sum_{j\infty} \mathcal{H}^{j1} \right)} \right| $$

We obtain the probability through a integration: the density is not directly obtainable.

By describing the field travelings in a quantum way, we describe the underlying porosity. At very little scales, these multiple distribution of photons appears while taking height makes appear a more concive view of the classical field, giving the most used way i.e. the main classical field traveling. It's exactly the same feeling when looking to a photography. If you use a magnifying glass you will see pixels that seems in disoder, without any image information. But taking ahead higher, the image appears and the pixels give a coherent structure. This distribution is distributed over a flat or curved space, depending on the gravity. We don't need to quantified the geometry in this model where gravitation is analyzed separately from the electromagnetic forces, even if they influence it at second order[11]. The symbol for this interaction is a continuous line having an arrow on both its start and end.

### 4.4. Heaviside's interaction.
Heaviside's interaction concerns all energies guided through longitudinal structures capable for wearing electromagnetic (or other) energy. It can be waveguides, but also cavities seen as waveguides short-circuited on their ends. What propagates is a wave plan where stationary waves are developed. The propagation speed depends on these stationary waves and of the medium. The various travelings that the field can borrow inside the waveguide correspond to the various modes that the field shown.

For modeling this physical process, we plaintly use the underlying space in its space of waves form. On each mesh that belongs to the physical layer, waves can power supply energy or at the contrary, energy can leave a mesh under a wave form. The physical process operates in three steps:

(1) the mesh space creates a potential source for a wave vector in the wave space;

(2) this wave vector propagates between the wave ports in the wave space;

(3) some meshes take some of the waves as electromotive forces induced on them through a metric $h$.

The technique is deeply inspired from the *mobile waves technique* initiated by Bergeron for pressure waves[14] and taken back by Vabre[15] for electromagnetic waves. The figure 4 illustrates the mechanism.

In this example we have a wave guide of length $x$. The waves propagate with a speed $v$ and the waveguide medium impedance of propagation is $n_c$. This case has two wave planes: one on the left, $A$; and another on the right, $B$. In the waves space, these two wave planes having eachother two ports, the wave vector is of dimension 4. It contains the complex waves
<figure><img src="image_5.png" /><figcaption>FIGURE 4. Mobile waves principle</figcaption></figure>

on the ports $p_1, r_1, p_2, r_2$. The p-ports are associated with traveling waves while the r-ports are associated with the back-scattered waves.

The mesh space creates on the mesh associated with the waveguide input a potential $V_0$ that depends on the power supply impedance $z_i$ and of the waveguide propagation impedance $n_c$. At $t = 0$, if $E_0(k) = E_0$, the input wave amplitude transmitted to the waveguide is simply:

$$ (48) \qquad V_0 = \left( \frac{n_c}{n_c + z_i} \right) E_0 $$
At the same time, the wave vector in the waves space is:

$$ (49) \qquad \mathcal{V}_{\alpha} = \begin{bmatrix} V_0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \Leftrightarrow \begin{bmatrix} p_1 \\ r_1 \\ p_2 \\ r_2 \end{bmatrix} $$

Through this mechanism, the mesh space feeds the waves one defining the terms of the wave vector at time 0.

Knowing the waveguide and the limit conditions on the wave planes A and B, we can compute the reflexion coefficients $G_a$ and $G_b$ defined by:

$$ (50) \qquad G_a = \left( \frac{z_i - n_c}{z_i + n_c} \right), \quad G_b = \left( \frac{z_o - n_c}{z_o + n_c} \right) $$

Between the waves on the ports we have the following relations :

$$ (51) \qquad \left\{ \begin{array}{l} p_1 = G_a r_1 \\ r_2 = G_b p_2 \\ p_2 = p_1 \exp(-k_v x) \\ r_1 = r_2 \exp(-k_v x) \end{array} \right. $$

The multiple back and forth of the waves are weared by an operator $\gamma$ that modifies the wave vector components gradually as the time increases. For example after one traveling, the wave vector becomes

$$ (52) \qquad \mathcal{V}_{\alpha} = \begin{bmatrix} 0 \\ 0 \\ V_0 e^{-kx/v} \\ 0 \end{bmatrix} \Leftrightarrow \begin{bmatrix} p_1 \\ r_1 \\ p_2 \\ r_2 \end{bmatrix} $$

We can show that a matrix can give all the evolutions of the wave vector components depending on time[16][17]. If $G_{ab}$ is Green's function defined by:

$$ (53) \qquad G_{ab} = e^{-kx/v} $$
The *gamma matrix* is:

$$
(54) \qquad \gamma = \begin{bmatrix} 0 & G_a & 0 & 0 \\ 0 & 0 & 0 & G_{ab} \\ G_{ab} & 0 & 0 & 0 \\ 0 & 0 & G_b & 0 \end{bmatrix}
$$

Once can verify that each time you apply $\gamma$ on $\mathcal{V}_\alpha$, you make appear the wave vector compo-
nent when the time is running: $\gamma\mathcal{V}_\alpha$, $\gamma\gamma\mathcal{V}_\alpha$, .... All the back and forth of the waves are taken
into account if we make a sufficient number of products to reach the equilibrium state. Writing
that in general we have:

$$
(55) \qquad \mathcal{V}_{\alpha} (t \to +\infty) = (1 + \gamma + \gamma\gamma + \gamma\gamma\gamma + \dots) \mathcal{V}_{\alpha} (t = 0)
$$

In general $2n + 1$ products are sufficient where $n$ is the wave planes number.

Noting (1 + γ + γγ + γγγ + ...) = γ̄, the wave vector is known through V_α(t → +∞) = γ̄V_α(t = 0), and it gives all the wave amplitudes on all the ports over the waves space W. We need now to report these wave onto the mesh space as electromotive forces. The electromotive forces induced by W on M are defined by:

$$
(56) \qquad e^{\alpha} = 2h^{\alpha\sigma} \tilde{\gamma} \mathcal{V}_{\sigma}
$$

For example the wave amplitude reported on the wave planes A and B are the wave vector
amplitudes on r₁ (for A) and p₂ (for B) and are obtain through:

$$
(57) \qquad \begin{bmatrix} e^A \\ e^B \end{bmatrix} = 2 \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \tilde{\gamma} \mathcal{V}_\sigma
$$

A waveguide works through a quite simple mechanism, more simply than often presented[18].
Once the cutoff wave length is exceeded, the field found configurations to propagates along the
waveguide. The first action consists in finding a field distribution in the wave plane. Let's
consider such a distribution for a rectangular waveguide of height h (direction z), length in the
direction x and width a in the direction y (Az can be the scalar potential or the electric or
magnetic field for exemple):

$$
(58) \qquad A_z = A_0 \cos \left( \pi \frac{y}{a} \right) \cos (\omega t)
$$

The field must respect Helmholtz's equation defined by:

$$
(59) \qquad \frac{\partial^2 A_z}{\partial x^2} + \frac{\partial^2 A_z}{\partial y^2} + \frac{\partial^2 A_z}{\partial z^2} = \frac{1}{c^2} \frac{\partial^2 A_z}{\partial t^2}
$$
Here we have $\partial^2 A_z/\partial z^2 = 0$ and with 58:

$$ (60) \qquad \frac{\partial^2 A_z}{\partial y^2} = -\frac{\pi^2}{a^2} A_z, \quad \frac{\partial^2 A_z}{\partial t^2} = -\omega^2 A_z $$

Then Helmholtz's equation becomes:

$$ (61) \qquad \frac{\partial^2 A_z}{\partial x^2} = \left( -\frac{\omega^2}{c^2} + \frac{\pi^2}{a^2} \right) A_z $$

It's a propagation equation but with a speed $v$ different from the celerity $c$. A propagation under $v$ may be:

$$ (62) \qquad \frac{\partial^2 A_z}{\partial x^2} = - \left( \frac{\omega^2}{v^2} \right) A_z $$

It implies that:

$$ (63) \qquad \frac{\omega^2}{v^2} = \frac{\omega^2}{c^2} - \frac{\pi^2}{a^2} \Rightarrow v = c \left[ \sqrt{1 - \left( \frac{\pi c}{a\omega} \right)^2} \right]^{-1} $$

but $\omega = 2\pi c/\lambda$, the free space propagation and $\lambda_c = 2a$, the cutoff wavelength. This leads to:

$$ (64) \qquad v = c \left[ \sqrt{1 - \left( \frac{\lambda}{\lambda_c} \right)^2} \right]^{-1} $$

$v$ is the phase speed that can be higher than the celerity, but the group speed is given by $c^2/v$. Knowing the wave propagation speed and the field distribution in the wave plane give us the information for computing the delays and couplings between emitters and receivers in interaction with the waveguide. But it lacks the propagation impedance $n_c$ to compute the reflexion coefficients and the losses to take into account the wave decreasing while propagating in the waveguide.

The wave impedance is defined by $\mu v$. Between the wave impedance and the propagation one, the difference comes from the integration of the field in potentials. If $h$ is the height on which the electric field is constant (it evolves depending on sinusoidal functions, but it can be made equivalent to a constant function like for a rms value) for a given mode and $W$ the equivalent width for the magnetic field, we have:

$$ (65) \qquad n_c = \mu v \left( \frac{h}{W} \right) $$

The losses can be estimated considering the metallic structure used by the currents in the waveguide wall. It often simplifies estimations when we can replace functions $q$ changing in sinus form by a constant one $Q$. This means that:

$$ (66) \qquad \int_0^X dx q \sin \left( \pi \frac{x}{X} \right) = \int_0^X dx Q \Rightarrow Q = \frac{2}{\pi} q $$
For example, in case of the first mode TE10 of the wave plane in a rectangular waveguide of width a, the current in the propagation direction uses $2a/\pi$ of equivalent width for propagating. The corresponding resistance per meter is:

$$ (67) \qquad R = 2 \frac{\rho}{2a/\pi} \sqrt{\pi f \mu \sigma} $$

and the attenuation per meter in propagation:

$$ (68) \qquad \alpha \approx \frac{1}{2} \frac{R}{n_c} $$

So Green's function of propagation taking into account the losses in the waveguide becomes:

$$ (69) \qquad G_{ab} = \exp\left(-k\frac{x}{v}\right) \exp(-\alpha x) $$

### 4.4.1. Couplings with the waveguide, waveguide to receiver direction.
For computing the couplings between electronics and a waveguide we work with three steps:

(1) we determine the phasor giving the location in interaction in the waveguide.

(2) We determine the scalar product which belongs to [0, 1] between the field direction in the waveguide and the receivers or the emitters.

(3) We determine the potentials induced in the receiver or from the emitter to the waveguide.

This approach can be used only in the cases where the interaction between the circuits and the waveguide are weak ones. It means that the propagation impedance $n_c$ is nearly the same in the presence of the circuits than without them. If this condition is not respected, the waveguide must be interrupted by a circuit including the ones in interaction, then the waveguide starts again at the output of this circuit.

In the case of an interaction with a receiver, the final objective is to find an electromotive force of coupling between some circuit and the waveguide extremities. The reasoning in general is conducted using the field potentials rather than the field forces and in the direction waveguide - receiver.

In the waves space we know the propagating and backscattered waves through the ports $p_1$ and $r_2$ coming from the meshes at the waveguide extremities. For computing the coupling at another location in the waveguide, we define a phasor that gives the wave transport from the extremities to this location. This phasor is given by:

$$ (70) \qquad \Theta = \begin{bmatrix} \exp(-k\Delta x/v) & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \exp(-k(X - \Delta x)/v) \end{bmatrix} $$
The operator $\bar{\gamma}$ becomes $\overline{\Theta\gamma}$ in this new waves plane at the abscissa $x$ for a waveguide length $X$.

The postponement of both electromotive forces coming from ports $p_1$ and $r_2$ pass through a metric $h$ which takes into account the normalized scalar product between the field direction in the waves plane and the receiver circuit. If $\theta$ is the angle between the waves plane field and the circuit branch coupled with the waveguide, $h$ has the form:

$$ (71) \qquad h = \begin{bmatrix} h_1(\theta) & 0 & 0 & h_2(\theta) \end{bmatrix} $$

The total electromotive force $e^c$ coming from the waveguide and induced on the receiving circuit is classically obtained through:

$$ (72) \qquad e^c = 2 [h(\theta)]^{\alpha} \overline{\Theta\gamma} \mathcal{V}_{\alpha} $$

In most cases, the circuit receiving the waveguide energy is an interface in the waveguide structure (an opening for example) having its own equivalent circuit. It is connected after with the rest of the electronics, cavity or anything else. It acts as a frontier.

### 4.4.2. Couplings with the waveguide, emitter to waveguide direction.
In this way, a circuit power supplies the waveguide defining the original wave vector amplitude on one or more ports. If there is a weak interaction between a circuit and the waveguide, it implies that somewhere, an opening exists in the waveguide structure on which a circuit is connected. In the plane shared between the circuit and the waveguide, we must determine the scalar product between the normalized current trajectories in the circuit and the waves plane in the waveguide. If $f(\theta)$ is this scalar product, then the potential developed across the shared branch is transmitted to the waveguide waves plane like in a classical dividing bridge $\eta$ computation. If $E_0$ is the original electromotive forces of the emitter having a self impedance $z_0$, then the transmitted potential $\phi_t$ is defined by:

$$ (73) \qquad \phi_t = f(\theta)\eta E_0, \quad \eta = \frac{z_0}{z_0 + n_c} $$

It remains to translate this induction into the waveguide extremities. To do that we use a dephasor $\Theta(x)$ and the wave vector becomes power supplied following:

$$ (74) \qquad \mathcal{V}_{\alpha} = \begin{bmatrix} 0 & \Theta(x)f(\theta)\eta E_0 & \Theta(X-x)f(\theta)\eta E_0 & 0 \end{bmatrix} $$

Finally, the wave vector is projected into the mesh space sources covector using the relation 56. The symbol for Heaviside's interaction is a heavy line.

## 5. THE GLOBAL CONFIGURATION SPACE WITH ITS LAYERS AND ITS SCALES

We consider three fundamentals layers:

(1) the gravitational and geometric layer: Einstein's layer;

(2) the electromagnetic fields layer (including thermal flux): Maxwell's layer;

(3) the mechanical and circuits layer: Lagrange's layer.
We have associated famous scientist names to these layers for refering to their fundamental contributions in physics. Each layer is used with the implicit assumption always associated with any modeling: the model dimension must be in accordance with the wavelengths domain of its configuration space.

We see further that a fourth layer can be defined for studying very complex systems. It's an upper layer named Freud's layer $\mathcal{LF}$, involving game theory for modeling human factor.

**5.1. Einstein's layer: $\mathcal{LE}$.** Einstein's layer defines the geometry. All the objects and circuits dimensions, locations, etc. are defined thanks to the geometry defined in this layer. The geometry is mainly determined by the gravitational field. For recall, Galilée's main relation on movement was to give the equation of an object speed $v$ in a referential having a speed $V$ compared to an observer referential. This observer sees the object with a speed $v'$ such as:

$$ (75) \qquad v' = v + V \Rightarrow x' = x + V\tau $$

By analogy this leads to:

$$ (76) \qquad ct' = ct + V\tau, \quad \tau = \frac{x}{c} \Rightarrow t' = t + \frac{Vx}{c^2} $$

In relativity with $\gamma = \sqrt{1 - V^2/c^2}^{-1}$ this becomes:

$$ (77) \qquad t' = \gamma \left( t + \frac{Vx}{c^2} \right) $$

With $\gamma t = \sigma$ and $a = V/t$ we obtain:

$$ (78) \qquad t' = \sigma \left( 1 + \frac{ax}{c^2} \right) $$

Accepting the equivalence between the inertial mass and the weighted one: $a = g$. As $g = \text{grad}\phi \Rightarrow ax/2 = \phi$ in average and:

$$ (79) \qquad t' = \sigma \left( 1 + 2\frac{\phi}{c^2} \right) $$

This marvellous relation which was Einstein's genius shows that the gravitational potential determines space and time[19].

Starting from the idea that the gravitational potential determines the space geometry, we can imagine a sphere, so a curvilinear space on which we may live, and on its center there is a big mass fixing the sphere curvature. Our problem means to write the movement equation on this sphere[20]. To do that, we define a function of two parameters $(u, v)$ and a parametrized
surface S depending on (u, v), locally seen as a cartesian space (x, y, z):

$$ (80) \qquad S: \left\{ \begin{array}{l} x = x(u, v) = R\cos(u)\cos(v) \\ y = y(u, v) = R\cos(u)\sin(v) \\ z = z(u, v) = R\sin(u) \end{array} \right. $$

Note that we can define by the same way any function, the sphere is only one example. On our curved space we can define a local base[21]:

$$ (81) \qquad \vec{b}_1 = \frac{\partial S}{\partial u}, \quad \vec{b}_2 = \frac{\partial S}{\partial v} $$

A local normal vector can be defined using:

$$ (82) \qquad \vec{n}_3 = \frac{\vec{b}_1 \times \vec{b}_2}{||\vec{b}_1 \times \vec{b}_2||} $$

Once these vectors are known, two fundamental metrics are defined:

$$ (83) \qquad g_{ij} = \langle \vec{b}_i, \vec{b}_j \rangle, \quad h_{ij} = \langle \vec{b}_{ij}, \vec{n}_3 \rangle, \quad \vec{b}_{ij} = \frac{\partial \vec{b}_i}{\partial u_j} $$

Note that $det(g) = g = g_{11}g_{22} - g_{12}^2 = ||\vec{b}_1 \times \vec{b}_2||^2$ with $g_{21} = g_{12}$. A curve $\gamma(t)$ traced on S is a geodesic if its tangential acceleration $\vec{a}_t$ is equal to zero. The total acceleration is given by $\vec{a} = \vec{a}_t + \vec{a}_S$. By the fact, the derivative of the local matched referential gives all the information on geometry. Let's define this derivatives:

$$ (84) \qquad \vec{b}_{ij} = \Gamma_{ij}^{1} \vec{b}_{1} + \Gamma_{ij}^{2} \vec{b}_{2} + h_{ij} \vec{n}_{3} $$

$\Gamma$ are Christoffel's coefficients, $h$ the second fundamental form. We have:

$$ (85) \qquad \Gamma_{ij,m} = \langle \vec{\gamma}_{ij}, \vec{b}_m \rangle $$

and:

$$ (86) \qquad \Gamma_{ij}^{k} = g^{k1} \Gamma_{ij,1} + g^{k2} \Gamma_{ij,2} $$

If we can write the acceleration relation on the curvilinear space, we should be able to express the movement equation. The curve $\gamma$ is defined by $S(u, v)$. So $\dot{\gamma} = \dot{u}\vec{b}_1 + \dot{v}\vec{b}_2$. This because:

$$ \dot{x}_i = \frac{dx_i}{dt} = \frac{\partial x_i}{\partial u_i} \frac{du_i}{dt} = \vec{b}_i \dot{u}_i $$

We can now express the acceleration:

$$ (87) \qquad \ddot{\gamma} = \ddot{u}\vec{b}_1 + \ddot{v}\vec{b}_2 + \dot{u}\dot{\vec{b}}_1 + \dot{v}\dot{\vec{b}}_2 $$
The base vector can be defined depending on their derivatives:

$$ (88) \qquad \vec{b}_1 = \frac{\partial \vec{b}_1}{\partial u} \frac{du}{dt} + \frac{\partial \vec{b}_1}{\partial v} \frac{dv}{dt} = \vec{b}_{11} \dot{u}_1 + \vec{b}_{12} \dot{v} $$

by replacement we obtain:

$$ (89) \qquad \ddot{\gamma} = \ddot{u}\vec{b}_1 + \ddot{v}\vec{b}_2 + (\dot{u})^2\vec{b}_{11} + 2\ddot{u}\ddot{v}\vec{b}_{12} + (\dot{v})^2\vec{b}_{22} $$

With:

$$ (90) \qquad \left\{ \begin{array}{l} \vec{b}_{11} = \Gamma_{11}^1 \vec{b}_1 + \Gamma_{11}^2 \vec{b}_2 + h_{11} \vec{n}_3 \\ \vec{b}_{12} = \vec{b}_{21} = \Gamma_{12}^1 \vec{b}_1 + \Gamma_{12}^2 \vec{b}_2 + h_{12} \vec{n}_3 \\ \vec{b}_{22} = \Gamma_{22}^1 \vec{b}_1 + \Gamma_{22}^2 \vec{b}_2 + h_{22} \vec{n}_3 \end{array} \right. $$

After replacement this leads to the three important equations:

$$ (91) \qquad \left\{ \begin{array}{l} \ddot{\gamma}_S(\vec{b}_1) = \ddot{u} + \Gamma_{11}^1 \dot{u}^2 + 2\Gamma_{12}^1 \dot{u}\dot{v} + \Gamma_{22}^1 \dot{v}^2 = 0 \\ \ddot{\gamma}_S(\vec{b}_2) = \ddot{v} + \Gamma_{11}^2 \dot{u}^2 + 2\Gamma_{12}^2 \dot{u}\dot{v} + \Gamma_{22}^2 \dot{v}^2 \\ \ddot{\gamma}_t(\vec{n}_3) = \dot{u}^2 h_{11} + 2h_{12} \dot{u}\dot{v} + h_{22} \dot{v}^2 \end{array} \right. $$

These three equations can be synthesized in the next two:

$$ (92) \qquad \ddot{\gamma}_S(\vec{b}_\alpha) = \frac{d\dot{u}^\alpha}{dt} + \Gamma_{\omega\sigma}^\alpha \dot{u}^\omega \dot{u}^\sigma, \quad \ddot{\gamma}_t(\vec{n}_3) = h_{\alpha\sigma} \dot{u}^\alpha \dot{u}^\sigma $$

Without any external constraints, the first equation is equal to zero and is the geodesics equation. Making the correspondence with the classical derivative noted $D\dot{u}^\alpha/dt$, the first equation also gives the covariant derivative definition:

$$ (93) \qquad D\dot{u}^\alpha = d\dot{u}^\alpha + \Gamma_{\omega\sigma}^\alpha \dot{u}^\omega \dot{u}^\sigma dt $$

The equations 91 give the acceleration, and so the forces seen in a referential of an observer. In presence of an external force, a term is added on the right equal to this force divided by the mass $m_2$ of the receiver, in the observer referential [22] (see figure 5).

Let $S^\alpha$ being the source term associated with a force. We can be inspired by Maxwell's equation $Rot\vec{B} = dE/c^2dt + \mu_0J$ and apply similar relation to the gravitational fields, i.e. the acceleration. Computing $D_\beta\ddot{\gamma}^\alpha - D_\alpha\ddot{\gamma}^\beta$ and the same operation on $S^\alpha$ and an added term coming from the time component of the equivalent current $\rho v$ of $J$, we obtain for the first operation (the calculation is a little tedious, see [23]) a tensor $R_\sigma^\alpha$ called Riemann's tensor and on the right, the equation $8\pi G/c^4(T_\sigma^\alpha - \delta_\sigma^\alpha T)$ where $T$ is the energy-impulse tensor of the form $\rho v^\alpha v_\sigma$. The final result is Einstein's equation:

$$ (94) \qquad R_\sigma^\alpha = \frac{8\pi G}{c^4} (T_\sigma^\alpha - \delta_\sigma^\alpha T) $$
<figure><img src="image_6.png" /><figcaption>FIGURE 5. Mobile particle in a gravitational field</figcaption></figure>

Einstein's equations define the gravitational field structure, and so the geometry structure. As a consequence, this defines our geometrical layer structure. Note that we make measures on Einstein's layer.

5.2. **Maxwell's layer: $\mathcal{L}M$.** Maxwell's layer, is unreachable. Photons or any other particles can be observed only undirectly through macro quantities like currents, voltages or heat. Equations 40 and 41 gives the keys for going from the physical layer $\mathcal{L}L$ to the field one $\mathcal{L}M$, and from the field layer $\mathcal{L}M$ to the physical one $\mathcal{L}L$. These relations show that the field $A^\alpha$ cannot be accessed directly. The field $A^\alpha$ is basically a 4-potential vector of the form $(\psi, A^\alpha)$. Teh scalar potential do not belongs to $\mathcal{L}M$: it's a near field interaction modeled in $\mathcal{L}L$ using capacitances. It means that $\mathcal{L}M$ includes only photonic field, and intrinsically, photons are not locals, and they cannot be directly observed. The field between two antennas for example can be studied theoretically with a large view authorizing to concentrate the photon population and to draw a perceptible field radiation.

Note that all the dimensions used for defining $\mathcal{L}M$ come from $\mathcal{L}E$. Finally $\mathcal{L}M$ can be associated with the light, but the light wears all our world information...

5.3. **Lagrange's layer: $\mathcal{L}L$.** Lagrange's layer is the layer of our perceptible world. It's the layer of the energy: a macroscopic marvellous world and whatever we can imagine on fields, what count in final it's our world and its evolvings. It's the world of our circuits and some bridges go to and come back from the other layers, intervening in the circuits models equations.

In any problem, the properties of the three layers are involved implicitly, and they come to complete the equations. It's important to have conscious of these layers, all considerations to end in $\mathcal{L}L$ which is the only accessible layer. For example, we may be estonished that EMC engineers often speak of fields even for immunity, while the field is not the constraint!
5.4. **From Newton to Lagrange.** Forces $F^i$ are perceived in Lagrange's layer, as the weight for example. Newton's law can be written in a tensor way[24]: $ma^i = F^i$. Knowing what we have discussed before, this equation can be written:

$$ (95) \qquad m \left( \frac{d^2 u^i}{dt^2} + \Gamma^i_{kh} \frac{du^h}{dt} \frac{du^k}{dt} \right) = F^i $$

or using the covariant components:

$$ (96) \qquad ma_i = m \left( g_{ij} \frac{d^2 u^j}{dt^2} + \Gamma_{kh,i} \frac{du^h}{dt} \frac{du^k}{dt} \right) = F_i \Leftrightarrow m \left( g_{ij} \frac{d\dot{u}^j}{dt} + \Gamma_{kh,i} \dot{u}^k \dot{u}^h \right) = F_i $$

using:

$$ (97) \qquad \Gamma_{kh,i} = \frac{1}{2} \left( \frac{\partial g_{ki}}{\partial u^h} + \frac{\partial g_{ih}}{\partial u^k} - \frac{\partial g_{kh}}{\partial u^i} \right) $$

Replacing this relation in 96 and organizing the terms we obtain:

$$ (98) \qquad m \left[ g_{ij} \frac{d\dot{u}^j}{dt} + \frac{1}{2} \left( \frac{\partial g_{ik}}{\partial u^h} + \frac{\partial g_{ih}}{\partial u^k} \right) \dot{u}^k - \frac{1}{2} \frac{\partial g_{hk}}{\partial u^i} \dot{u}^h \right] = F_i $$

We can use indices properties:

$$ (99) \qquad \left( \frac{\partial g_{ik}}{\partial u^h} + \frac{\partial g_{ih}}{\partial u^k} \right) \dot{u}^h \dot{u}^k = 2 \frac{\partial g_{ih}}{\partial u^k} \dot{u}^h \dot{u}^k $$

We finally obtain:

$$ (100) \qquad m \left[ g_{ij} \frac{d\dot{u}^j}{dt} + \frac{\partial g_{ih}}{\partial u^k} \dot{u}^h \dot{u}^k - \frac{1}{2} \frac{\partial g_{hk}}{\partial u^i} \dot{u}^h \dot{u}^k \right] = F_i $$

Let's compute now the kinetic energy:

$$ (101) \qquad T = \frac{1}{2} mg_{ij} \dot{u}^i \dot{u}^j \Rightarrow \frac{\partial T}{\partial \dot{u}^i} = mg_{ij} \dot{u}^j $$

and so:

$$ (102) \qquad \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{u}^i} \right) = m \left( g_{ij} \frac{d\dot{u}^j}{dt} + \frac{\partial g_{ih}}{\partial u^k} \dot{u}^h \dot{u}^k \right) $$

On another side:

$$ (103) \qquad \frac{\partial T}{\partial u^i} = \frac{1}{2} m \frac{\partial g_{hk}}{\partial u^i} \dot{u}^h \dot{u}^k $$

Using 102 and 103 in 100, it leads to:

$$ (104) \qquad \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{u}^i} \right) - \frac{\partial T}{\partial u^i} = F_i $$
We retrieve our Lagrange's equations in the other way. In the case where the force derives from a potential V:

$$ (105) \qquad F_i = -\frac{\partial V}{\partial u^i} $$

Lagrange's equations become:

$$ (106) \qquad \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{u}^i} \right) - \frac{\partial T}{\partial u^i} + \frac{\partial V}{\partial u^i} = 0 $$

5.5. A particle obliged for moving on a surface defined on $\mathcal{L}E$. We make the assumption that a particle is obliged to move on a surface $S$ perfectly smooth. We choose a coordinate system $x^\alpha = x^\alpha(u^1, u^2, u^3)$ such a way that we obtain the choosen surface by fixing $u^3$ as a constant value. The surface $S$ constitutes a Riemann's space of two dimensions with a metric defined by:

$$ (107) \qquad ds^2 = g_{ij} du^i du^j $$

The fundamental tensor $g$ is obtained through:

$$ (108) \qquad g_{ij} = \delta_{\alpha\beta} \frac{\partial x^\alpha}{\partial u^i} \frac{\partial x^\beta}{\partial u^j} $$

The particle speed is:

$$ (109) \qquad v^i = \frac{du^i}{dt} $$

We obtain the covariant components with $v_k = g_{ki}v^i$, and the acceleration:

$$ (110) \qquad a^i = \frac{Dv^i}{dt} = \frac{dv^i}{dt} + \Gamma^i_{kh} v^h v^k $$

We find the same processes as before. Noting $L = T - V$ Lagrange's function, knowing that $V$ doesn't depend on $u^i$, Lagrange's equations can be written:

$$ (111) \qquad \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{u}^i} \right) - \frac{\partial L}{\partial u^i} = 0 $$

Christoffel's symbols belong to $\mathcal{L}E$ while the observed forces belong to $\mathcal{L}L$.

As the lagrangian is linked with the energy, the action depends on $L$. For an electrical particle, the energy is $qV$. It means that we can define basically the action $\Theta$ through:

$$ (112) \qquad \Theta = \frac{1}{c} \int_{\lambda} d\lambda |qV| $$

As the load is the current integral, we can write for a trajectory $\gamma$ made of a succession of branches of indices $\alpha$: see figure 6 ($V_\alpha$ is the potential difference across the branch $\alpha$). The
<figure><img src="image_7.png" /><figcaption>FIGURE 6. Example of trajectory</figcaption></figure>

process is equivalent to link the succession of potential differences, including those of sources of currents, as a continuous summation of them.):

$$ (113) \qquad \Theta(\alpha \in \gamma) = \frac{1}{c} \int_{\lambda} d\lambda \left| \frac{s^{*\alpha}}{k} V_{\alpha} \right| $$

but the branch currents $s^\alpha$ are defined over the meshes ones $s^\alpha = g^{\alpha\sigma}a_\sigma$ which leads to:

$$ (114) \qquad \Theta(\alpha \in \gamma) = \frac{1}{2c} \int_{\lambda} d\lambda \left( \frac{g^{\alpha\sigma}}{k} a_{\sigma}^{*} V_{\alpha} \right) $$

We can wonder how the action can be computed when the trajectory involves an interaction belonging to $\mathcal{L}M$? Let's imagine two meshes of impedacnes $Z1$ and $Z2$ connected by a radiative interaction through a $\chi^{\alpha\sigma}$ coupling. The fundamental impedance tensor $\zeta$ is given by:

$$ (115) \qquad \zeta = \begin{bmatrix} Z1 & -\chi^{12} \\ -\chi^{21} & Z2 \end{bmatrix} $$

If we verify $\chi^{12} = \chi^{21}$, we can replace the free space coupling by a shared branch of the same impedance. If we transform each mesh self impedance by $Z1 \rightarrow Z1 - \chi^{21}$ and $Z2 \rightarrow Z2 - \chi^{21}$,
the fundamental tensor of this new circuit is (keeping the same orientation for both meshes):

$$ (116) \qquad \zeta' = \begin{bmatrix} Z1 - \chi^{21} + \chi^{21} & -\chi^{12} \\ -\chi^{21} & Z2 - \chi^{21} + \chi^{21} \end{bmatrix} $$

We verify that $\zeta' = \zeta$. This topological transformation gives us a way for easily including free space interaction in an action computing.

The least action principle says us that between two possible trajectories $\gamma$ and $\gamma'$, $\delta\Theta = |\Theta(\alpha \in \gamma') - \Theta(\alpha \in \gamma)|$, this difference should tend towards zero if $\gamma$ is an extremum: $\delta\Theta \to 0$.

## 6. THE CASE OF ELECTRICAL MACHINES

Electrical machines are a first example of electro-mechanical coupling. Basically these couplings consist in the report of the mechanical effort on the electrical budget ($\phi\Omega$), and the report of the electrical forces on the mechanical torque ($\phi J$). It involves:

* the self and mutual inductances of and between coils $\mathcal{L}$;

* the cutted flux induction $G$;

* the electrical resistance of wires $r$;

* the currents in the coils $a$;

* the applied electromotive forces $E$;

* the angular speed $\Omega$;

* the magnetic flux $\phi$;

* the rotor inertia moment $Q$;

* the frictions of the rotor axis $\rho$;

* and the machine torque $\mathcal{T}$.

Basics equations obtained with these quantities are:

$$ (117) \qquad \begin{cases} E^{\mu} = \mathcal{L}^{\mu\nu}ka_{\nu} + r^{\mu\nu}a_{\nu} + G^{\mu\nu}\Omega a_{\nu} \\ \mathcal{T} = Qk\Omega + \rho\Omega - G^{\mu\nu}a_{\mu}a_{\nu} \end{cases} $$

The mutual inductance translates the coupling through reluctance between aligned coils, while the cutted flux coupling acts between coils in quadrature.

Let's imagine an electrical machine made of four coils, two that belong to the stator and two that belong to the rotor. When the machines turn of an angle $\theta$ ($\dot{\theta} = \Omega$), as each current is projected on a space vector: $a_i/\tilde{a} = a_i\tilde{c}^i$, a change of referential must be operated between the static referential elaborated on a virtual static machine ($\tilde{c}^\mu$) and the dynamic one coming from the running machine. We note the base vectors attached to the running machine ($\tilde{\delta}^\mu$). We can define a change of base $\Lambda(t)$ saying that:

$$ (118) \qquad \tilde{c}^{\alpha} = \Lambda_{\sigma}^{\alpha}(t)\tilde{\delta}^{\sigma} \Rightarrow \Lambda_{\sigma}^{\alpha} = \frac{\partial \tilde{c}^{\alpha}}{\partial \tilde{\delta}^{\sigma}} $$
A typical form for $\Lambda$ is:

$$ (119) \qquad \Lambda(t) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\theta(t) & -\sin\theta(t) & 0 \\ 0 & \sin\theta(t) & \cos\theta(t) & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$

All the terms in 117 like $ka_ν$ becomes after the change of referentials $k\Lambda_ν^μp_μ$, the current covector being in this referential $\check{p} = p_μ\delta^μ$, and:

$$ (120) \qquad k\Lambda_{\nu}^{\mu} p_{\mu} = (k\Lambda_{\nu}^{\mu}) p_{\mu} + \Lambda_{\nu}^{\mu} (kp_{\mu}) $$

and:

$$ (121) \qquad (k\Lambda_{\nu}^{\mu}) p_{\mu} = \frac{\partial \Lambda_{\nu}^{\mu}}{\partial \theta} \Omega p_{\mu} $$

so:

$$ (122) \qquad ka_{\nu} = k\Lambda_{\nu}^{\mu}p_{\mu} = \frac{\partial\Lambda_{\nu}^{\mu}}{\partial\theta}\Omega p_{\mu} + \Lambda_{\nu}^{\mu}(kp_{\mu}) $$

Using these relations in the first equation of 117 we obtain:

$$ (123) \qquad E^{\sigma} = \mathcal{L}^{\sigma\nu} \frac{\partial \Lambda_{\nu}^{\mu}}{\partial \theta} \Omega p_{\mu} + \mathcal{L}^{\sigma\nu} \Lambda_{\nu}^{\mu} (kp_{\mu}) + r^{\sigma\nu} \Lambda_{\nu}^{\mu} p_{\mu} + G^{\sigma\nu} \Omega \Lambda_{\nu}^{\mu} p_{\mu} $$

We can now multiply both members by the inverse transformation $V_\sigma^\beta = [\Lambda_\beta^\sigma]^{-1}$:

$$ (124) \qquad V_{\sigma}^{\beta} E^{\sigma} = V_{\sigma}^{\beta} \mathcal{L}^{\sigma\nu} \frac{\partial \Lambda_{\nu}^{\mu}}{\partial \theta} \Omega p_{\mu} + V_{\sigma}^{\beta} \mathcal{L}^{\sigma\nu} \Lambda_{\nu}^{\mu} (kp_{\mu}) + V_{\sigma}^{\beta} r^{\sigma\nu} \Lambda_{\nu}^{\mu} p_{\mu} + V_{\sigma}^{\beta} G^{\sigma\nu} \Omega \Lambda_{\nu}^{\mu} p_{\mu} $$

Let's write:

$$ (125) \qquad \left\{ \begin{array}{l} R^{\beta\mu} = V_{\sigma}^{\beta} r^{\sigma\nu} \Lambda_{\nu}^{\mu} \\ L^{\beta\mu} = V_{\sigma}^{\beta} \mathcal{L}^{\sigma\nu} \Lambda_{\nu}^{\mu} \\ H^{\beta\mu} = V_{\sigma}^{\beta} G^{\sigma\nu} \Lambda_{\nu}^{\mu} \\ \Gamma_{\theta}^{\beta\mu} = V_{\sigma}^{\beta} E_{\theta}^{\sigma\mu} = V_{\sigma}^{\beta} \mathcal{L}^{\sigma\nu} \partial_{\theta} \Lambda_{\nu}^{\mu} \\ N^{\beta} = V_{\sigma}^{\beta} E^{\sigma} \end{array} \right. $$

Our equation becomes:

$$ (126) \qquad N^{\beta} = \Gamma_{\theta}^{\beta\mu} \Omega^{\theta} p_{\mu} + k L^{\beta\mu} p_{\mu} + R^{\beta\mu} p_{\mu} + H^{\beta\mu} \Omega p_{\mu} $$
and

$$ (127) \qquad k\Lambda(t) = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & -\sin\theta(t) & -\cos\theta(t) & 0 \\ 0 & \cos\theta(t) & -\sin\theta(t) & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} $$

and the torque is now defined by:

$$ (128) \qquad \mathcal{T} = Qk\Omega + \rho\Omega - H^{\mu\nu}p_{\mu}p_{\nu} $$

## 7. THE MULTIPHYSICS COUPLING MECHANISMS

As for the electrical machine, the simplest way for coupling various equations coming from various physics is to report in each equations system a term coming from another physic.

Rigorously, if we look for example a resistance depending on temperature at first order, Ohm's law becomes $R(1 + \alpha T_0)a_0 \Leftrightarrow Ra_0 + Ra(a_0 T_0)$: the observables are $a_0$ and $a_0 T_0$. These observables belong to a space resulting of the tensor product of both spaces $M$ and $T$: $M \otimes T$. The hybrid covector pointing out the measurements comes from $d\check{c}^\alpha \wedge d\check{t}^\sigma$, with $\check{t}^\sigma$ the cobase of $T$. In practice, the increasing of the configuration space makes the problem solving very difficult and unintelligible.

Rather than this first rigorous method, we choose a less mathematical one but easier to employ in physics. It consists in reporting observables coming from one physic in the equations of another physic. Then, by solving each physics separately, next values change in the equations where they are reported on the next time step. The approximation error depends directly on the time step choice.

Graphs can be employed for any modellings and any physics[25]. The approach consists in giving basic equations for each kind of physic, then to couple between them the group of equations system attached with the group of physics involved.

### 7.1. Mechanical equations.
In mechanics, the inertia given by $Ldi/dt$ in electronics can be written $md\dot{x}/dt$ following first Newton's law[7]. The stiffness is similar to a capacitor and defined by:

$$ F_r = K \int_t dt \dot{x} $$

And the losses if proportional to the speed are $R = f\dot{x}$. Mechanical vibrations can be modelled using equivalent electrical circuits and their mechanical impedances ($m$, $K$, $f$). Out of thick domains, $f$ and $k$ depends on the temperature. The movement can be created by an external force added to an electrical solicitation. A typical application of these kinds of couplings are the
electromechanical mechanisms. The electrical current must be replaced by the speed $\dot{x} \to j^\alpha$. The topology works in $\mathbb{B}$ and $\mathbb{N}$. A typical electromechanical equations system may be:

$$ (129) \qquad \left\{ \begin{array}{l} kmj^1 + fj^1 + \frac{K}{k}j^1 = F(k) + lBs^1 \\ kLs^1 + Rs^1 + \frac{1}{kC}s^1 = e(k) - lBj^1 \end{array} \right. $$

Both equations can be projected in $\mathbb{M}$ on two meshes without considering the sources. Then each source is a mesh source and the coupling between the two physics appears as a string between the two meshes and so the two physics. The impedance coupling is $lB$ (here $l$ is a length and $B$ the magnetic field, but it's not important for our purpose). For our example, the fundamental tensor may be:

$$ (130) \qquad \zeta = \begin{bmatrix} km + f + \frac{K}{k} & -lB \\ lB & kL + R + \frac{1}{kC} \end{bmatrix} $$

the currents are $a_1 = g_{11}j^1$, $a_2 = g_{21}s^1$.

It's a general method to transform in couplings the shared forces between two elementary systems. Another example may be two masses sharing the same spring. Looking the situation in each mass referential: each mass is a force of inertia, plus its own spring reaction. This gives two meshes:

$$ (131) \qquad m_1kv^1 + \frac{K_1}{k}v^1 = 0, \quad m_2kv^2 + \frac{K_2}{k}v^2 $$

Now the first mass sees the resistance of the spring attached to the second mass, adding a term $-Kv^2$ with a string and the same for the second mass. The fundamental tensor is:

$$ (132) \qquad \zeta = \begin{bmatrix} m_1k + \frac{K}{k} & -\frac{K}{k} \\ -\frac{K}{k} & m_2k + \frac{K}{k} \end{bmatrix} $$

the vector being $v = [v^1, v^2]$ and no source being applied.

## 7.2. Thermal equations.
The observable is the temperature variation $\delta T$ compared to a reference value, often the ambient temperature. The basic constant are influenced by the temperature through the first order equations:

$$ (133) \qquad \left\{ \begin{array}{l} \epsilon(T) = \epsilon(1 + \alpha_{\epsilon}T) \\ \mu(T) = \mu(1 + \alpha_{\mu}T) \\ \rho(T) = \rho(1 + \alpha_{\rho}T) \\ K(T) = K(1 + \alpha_{K}T) \end{array} \right. $$
The thermal graph uses as inputs the power coming from the Joule energy of the electrical circuits. Two kinds of impedances exist: the thermal resistance $Rth$ and the thermal capacity $1/(km_C)$. The Joule power coming from an electrical circuit is defined by:

$$ (134) \qquad P_J (d \in \mathbb{B}) = \frac{1}{4} [u_d s^{*d} + u_d^* s^d] \rightarrow Q_d $$

$Q_d$ becomes a current source for a mesh in the heat space $\mathbb{T}$. Note that our approach is completely coherent as the source $Q_\alpha$ appears covariant and is effectively associated with a 1-form, being intrinsically a scalar, while the target quantity $\delta T$ can be associated with a gradient, i.e. a natural vector.

### 7.2.1. Thermal diffusion.
Diffusion (conduction and convection) through solids or gases can be modeled using the thermal resistance with $\delta T^u = Rth^{uv}Q_v$. Some thermal inertia can complete this relation giving:

$$ (135) \qquad \delta T^{\sigma} = \left( Rth^{\sigma\alpha} + \frac{1}{km_{C\alpha\alpha}} \right) Q_{\alpha} $$

$Rth = q^{-1}l/S$ where $q$ is a coefficient in $[W][^\circ K.m]^{-1}$ or $[W][^\circ C.m]^{-1}$ (for example in the case of copper, $q = 387 [W][^\circ C.m]^{-1}$). Sometimes $[kCal]$ is also an unit used. In that case $q(kCal) = q(^\circ C)/1,16$. The electronics heat when running and create Joule's power. This power creates temperature that modifies the electronics following 133.

### 7.2.2. Thermal radiation.
Stefan Boltzmann's law says us that a body at a temperature $T$ radiates a total power defined by:

$$ (136) \qquad Rad = \sigma T^4 \Rightarrow Rad(\text{grey body}) = \epsilon \sigma T^4 $$

$\epsilon = 5,67.10^{-8} [W][m^2.K^4]$. This relation can also be translated using the luminance notion. With this observable:

$$ (137) \qquad Lrad = \frac{\sigma}{\pi} T^4 \Rightarrow Lrad(\text{grey body}) = \epsilon \frac{\sigma}{\pi} T^4 $$

Let's considering the physical exchange of heat between two bodies of surfaces $S1$ and $S2$ represented figure 7 (this example was developed by Pr. D.Venot - SUPELEC at this time).

The energy flux radiates by $dS1$ to $dS2$ is defined by Bouguer's law:

$$ (138) \qquad d\phi = \frac{L_1}{r^2} (dS_1 \cos(i_1) dS_2 \cos(i_2)) $$

$L_1$ being first body luminance which emits the heat. If $L_1$ is associated with a grey body:

$$ (139) \qquad d\phi = \epsilon_1 \frac{\sigma}{\pi} (T_1)^4 \frac{1}{r^2} (dS_1 \cos(i_1) dS_2 \cos(i_2)) $$
<figure><img src="image_8.png" /><figcaption>FIGURE 7. Radiated interaction</figcaption></figure>

$T_1$ being S1 temperature. So the total incident flux $\phi_{21}^i$ radiates by S1 to S2 is obtained through:

$$ (140) \qquad \phi_{21}^i = \epsilon_1 \frac{\sigma}{\pi} (T_1)^4 \iint_{S1,S2} dS_1 dS_2 \frac{\cos(i_1)\cos(i_2)}{r^2} $$

The double integral that depends only of the bodies geometry is called the form factor $\varphi_{21}$:

$$ (141) \qquad \varphi_{21} = \iint_{S1,S2} dS_1 dS_2 \frac{\cos(i_1)\cos(i_2)}{r^2} $$

and $\phi_{21}^i = (\sigma/\pi)\epsilon_1 T_1^4 \varphi_{21}$. If we design by $\alpha(T_1, T_2)$ the absorbtion factor of the surface S2, the absorbed flux by S2, $\phi_2^a$ is given by:

$$ (142) \qquad \phi_2^a = \frac{\sigma}{\pi} \epsilon_1 \alpha(T_1, T_2) T_1^4 \varphi_{21} $$

If $\alpha(T_1, T_2)$ doesn't depend on temperature at first order, then $\alpha(T_1, T_2) = \epsilon_2$ and:

$$ (143) \qquad \phi_2^a = \frac{\sigma}{\pi} \epsilon_1 \epsilon_2 T_1^4 \varphi_{21} $$

If the ambient temperature is an offset common to the whole system studied, all the bodies are initially at the same temperature $T_a$. Under this assumption $T_1 \to \delta T^1$ and:

$$ (144) \qquad \phi_2^a = \frac{\sigma}{\pi} \epsilon_1 \epsilon_2 \varphi_{21} (\delta T^1)^4 \Leftrightarrow \phi_2^a = \tilde{R}_{21} (\delta T^1) $$

with the radiated heat operator defined by:

$$ (145) \qquad \tilde{R}_{21}(\bullet) = \frac{\sigma}{\pi} \epsilon_1 \epsilon_2 \varphi_{21}(\bullet)^4 $$
The complete thermal equation 135 becoming:

$$ (146) \qquad \delta T^{\sigma} - \tilde{\mathcal{R}}_{\beta}^{\sigma} (\delta T^{\beta}) = \left( Rth^{\sigma\alpha} + \frac{1}{km_{C_{\alpha\alpha}}} \right) Q_{\alpha} $$

As said before, $\delta T^{\sigma}$ depends on $Q_{\alpha}$ and $Q_{\alpha}$ depends on $\delta T^{\sigma}$ through the relations 133. While we have expressed all the relations under Laplace's formalism, these cycles of exchanges between physics are distributed along the time axis. The time axis allows to identify steps in states, and plays a fundamental role in the interactions between moving referentials, in the quantum changes of states as for non linearities as we see after.

$Rth$ and $m_C$ belong to $LL$ while $\tilde{\mathcal{R}}_{\beta}^{\sigma}$ belongs to $LM$.

### 7.3. Fluids equation.
Fluids equations group three main equations[24]:

(1) the continuity equation;

(2) the perfect fluid equation and fluids with vortices;

(3) the viscous fluid equation.

On any point of $LE$ the fluid is animated by a speed $v^{\alpha}(x^{\alpha}, t) = \dot{x}^{\alpha}$. The fluid acceleration at the same point is given by:

$$ (147) \qquad \gamma^{\alpha} = \frac{dv^{\alpha}}{dt} = \frac{\partial v^{\alpha}}{\partial t} + \frac{\partial v^{\alpha}}{\partial x^{\beta}} \frac{dx^{\beta}}{dt} = \frac{\partial v^{\alpha}}{\partial t} + \frac{\partial v^{\alpha}}{\partial x^{\beta}} v^{\beta} $$

If we use curvilinear coordinates $y^i$ this becomes:

$$ (148) \qquad v^{\alpha} = \frac{dy^{\alpha}}{dt} $$

and

$$ (149) \qquad \gamma^{\alpha} = \frac{\partial v^{\alpha}}{\partial t} + D_{\beta}v^{\alpha} \cdot v^{\beta} \Leftrightarrow \gamma^{\alpha} = \frac{\partial v^{\alpha}}{\partial t} + \frac{\partial v^{\alpha}}{\partial x^{\beta}} v^{\beta} + \Gamma^{\alpha}_{\beta\sigma}v^{\beta}v^{\sigma} $$

### 7.4. The continuity equation.
Being inspired by Maxwell's continuity equation, it comes naturally a similar equation for fluids (remembering Maxwell self inspiration by fluid mechanics) with $\rho v \equiv J$:

$$ (150) \qquad \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x^{\beta}} (\rho v^{\beta}) = 0 \quad \xrightarrow{cur} \quad \frac{\partial \rho}{\partial t} + D_{\beta} (\rho v^{\beta}) = 0 $$

This is the continuity equation for fluids.

### 7.5. The perfect fluids equation.
Two kinds of forces act on a fluid portion: the massive force $\rho X^{\alpha} dV$ where $X^{\alpha}$ is the force vector and the pressure $p = \partial_{\alpha} p dV$. Equalling previous acceleration to these constraints leads to:

$$ (151) \qquad \frac{\partial v^{\alpha}}{\partial t} + \frac{\partial v^{\alpha}}{\partial x^{\beta}} v^{\beta} = X^{\alpha} - \delta^{\alpha}_{\alpha} \frac{1}{\rho} \frac{\partial p}{\partial x^{\alpha}} $$
and in curvilinear coordinates:

$$ (152) \qquad \frac{\partial v^\alpha}{\partial t} + D_\beta v^\beta \cdot v^\alpha = X^\alpha - g^{\alpha\alpha} \frac{1}{\rho} \frac{\partial p}{\partial y^\alpha} $$

7.6. **Fluids with vortices.** The tensor $T_{ij} = D_j v^i - D_i v^j$ is the rotational of the fluid speed. It is equal to:

$$ (153) \qquad T_{\alpha\beta} = \frac{\partial v^\alpha}{\partial y^\beta} - \frac{\partial v^\beta}{\partial y^\alpha} $$

When vortices appear in the fluid transportation Euler's equation complete our previous one giving:

$$ (154) \qquad \gamma^\alpha = \frac{\partial v^\alpha}{\partial t} + D_\beta v^\alpha \cdot v^\beta + \delta_\sigma^{\alpha\omega\beta} (T_{\omega\beta} \times v^\sigma) = X^\alpha - g^{\alpha\alpha} \frac{1}{\rho} \frac{\partial p}{\partial y^\alpha} $$

7.7. **The viscous fluid equation.** Navier Stokes's equation concerns fluids having viscosity in their behavior. If $\mu$ is the viscosity coefficient, it adds two terms to the perfect fluid equation 151.

A first term comes from the gradient of the speed divergence. It translates the fluid dispersion evolving from a point to another:

$$ (155) \qquad \nabla^2 v \rightarrow \frac{\mu}{\rho} \delta^{\beta\sigma} \frac{\partial^2 v^\alpha}{\partial x^\beta \partial x^\sigma} $$

A second term is the divergence of the speed gradient. This terms explains how the speed variation diverge locally. It is expressed by:

$$ (156) \qquad \nabla (\nabla v) \rightarrow \frac{\mu}{3\rho} \delta^{\alpha\beta} \frac{\partial}{\partial x^\beta} \left( \frac{\partial v^\omega}{\partial x^\omega} \right) $$

The complete Navier-Stokes's equation is:

$$ (157) \qquad \frac{\partial v^\alpha}{\partial t} + \frac{\partial v^\alpha}{\partial x^\beta} v^\beta - \frac{\mu}{\rho} \delta^{\beta\sigma} \frac{\partial^2 v^\alpha}{\partial x^\beta \partial x^\sigma} - \frac{\mu}{3\rho} \delta^{\alpha\beta} \frac{\partial}{\partial x^\beta} \left( \frac{\partial v^\omega}{\partial x^\omega} \right) = X^\alpha - \frac{1}{\rho} \frac{\partial p}{\partial x^\alpha} $$

This equation is associated with the continuity equation, and if the fluid is compressible, it is necessary to use a third equation describing the fluid state, of the form $\rho = f(p, T)$. For example the Mariotte - Gay-Lussac's equation is:

$$ (158) \qquad \rho = \frac{p}{(1 + T/273)} $$

In curvilinear coordinates, Navier-Stokes's equation becomes:

$$ (159) \qquad \frac{\partial v^\alpha}{\partial t} + D_\beta v^\beta \cdot v^\alpha - \frac{\mu}{\rho} g^{\sigma\beta} D_\sigma D_\beta v^\sigma - \frac{\mu}{3\rho} g^{\alpha\beta} D_\beta (D_\sigma v^\sigma) = X^\alpha - \frac{1}{\rho} g^{\alpha\sigma} \frac{\partial p}{\partial x^\sigma} $$
<figure><img src="image_9.png" /><figcaption>FIGURE 8. Exponential order function</figcaption></figure>

## 8. NON LINEARITIES AND DOMAINS

A function $F(t)$ is said to be piecewise continuous in an interval $\alpha < t < \beta$ if this interval can be divided in a finite number of intervals on which the function is continuous and has finited limits on right and left [26] (see figure 8).

If it exists $M > 0$ and $\gamma$ so that:

$$ (160) \qquad \forall t > N, \quad |e^{-\gamma t} F(t)| < M $$

we say that $F(t)$ is of exponential order $\gamma$ when $t \to +\infty$.

By definition if $F(t)$ is piecewise continuous on any finite interval $0 < t < N$ and if $F(t)$ is of exponential order $\gamma$ then its Laplace's transform exists for all $s > \gamma$, $s$ being here the Laplace's operator defined on the pulsation $\omega$.

Non linear functions are functions piecewise continuous and can be studied using Laplace's transform under the assumption that they are of exponential order. The change of states in mechanics or electronics (including thermal analysis) arrives at some moments that can be identified on the time axis. These connections are accompanied by two consequences:

(1) some added terms can come from this state changes;

(2) final conditions in energy must be reported as initial conditions on the next state.
At the end of a state $t_1$ (the time is seen as a time interval), some potential energy can be reached across a capacitor C: $U_f(t_1)$. This potential must be reported as initial condition on the next state $t_2$ adding on this state a potential differences in series with C, $U_f/k$. This value $U_f/k$ must complete the source covector of the circuit for $t_2$. In the case of magnetic energy storing, the final condition on $t_1$ is a current in a coil $I_f(t_1)$. As initial condition on $t_2$, the inductance becomes in factor of $(ks^\alpha - I_f(t_1))$. In that case, the source covector on $t_2$ must be completed with the term $LI_f(t_1)$.

A non linearity is a set of n virtual branches in parallel where only one is connected on a single time interval $t_i$. On $t_i$ the whole circuit is solved under Laplace's formalism with the operator $k = j2\pi c/\lambda + \alpha$. Let's take a simple example with a circuit having three branches and two meshes $\{\vec{c}^1, \vec{c}^2\}$. A shared branch has a non linear behavior depending on the voltage across it. The set of branches is $\mathbb{B} = \{\vec{b}_1, \vec{b}_2, \vec{b}_3\}$. A non linear set of branches exists also with $\mathbb{B}^{NL} = \{\vec{l}_1, \vec{l}_2, \vec{l}_3, \vec{l}_4\}$, $\vec{l}_2$ and $\vec{l}_3$ supporting the two non linear properties of one electronic component modeled with $\vec{b}_2$. This is translited by associating one impedance of one non linear state to each virtual branch $\vec{l}^i$. As only the real branch $s^{(i)}$ exists for each time interval, this branch has an impedance that depends on a parameter $p$, replacing $s^{(i)}$ by one of the possible $\vec{l}_\alpha$ depending on $p$.

Various mathematical functions can be used for defining the components of $z(\vec{l}_\alpha)$. One quite efficient is a combination of sigmoid functions:

$$ (161) \qquad z(\vec{l}_{\alpha}) \rightarrow \mathcal{D}_{[\pi, \pi')}^{p} = [1 + \exp(-a(q - \pi))] [1 - 1 + \exp(-a(q - \pi))] $$

$a$ in the sigmoid function defines the domain stiffness on its limits.

The parameter $p$ (here a voltage) belongs to the interval $[\pi, \pi']$. If $p \in [\pi, \pi']$, $\mathcal{D}_{[\pi, \pi')}^{p} = 1$. If not, $\mathcal{D}_{[\pi, \pi')}^{p} = 0$. With this function, $z(\vec{l}_\alpha)$ becomes defined by:

$$ (162) \qquad z(\vec{l}_{\alpha}) = \mathcal{D}_{[0, V]} z_n(\vec{l}_{\alpha}) + \mathcal{D}_{[V, U]} z_m(\vec{l}_{\beta}) $$

With this non linear branch, the real branch (i) develops a potential difference given by:

$$ (163) \qquad z_{ab} s^{(i)} = \left\{ \mathcal{D}_{[0,V]} z_{ab}^{1} + \mathcal{D}_{[V,U]} z_{ab}^{2} \right\} s^{(i)}, \quad z_{ab}^{1} = z(\vec{l}_{\alpha}), \quad z_{ab}^{2} = z(\vec{l}_{\beta}) $$

But this impedance changing can have an impact when the time derivatives are involved and when the changing covers more than one time interval. As the inductances belong to $\mathbb{M}$ and are driven mainly by the free energy enclosed in each mesh, they can be first order affected by the change of a branch impedance. By the fact when replacing $s^\alpha$ by $g^{\alpha\beta}a_\beta$ this leads to:

$$ (164) \qquad g^{\omega\alpha} \frac{\delta}{\delta t} (L_{\alpha\beta} g^{\beta\sigma} a_{\sigma}) = g^{\omega\alpha} \frac{\delta L_{\alpha\beta}}{\delta t} g^{\beta\sigma} a_{\sigma} + (g^{\omega\alpha} L_{\alpha\beta} g^{\beta\sigma}) \frac{\delta a_{\sigma}}{\delta t} $$
The second term gives a classical component of $\zeta^{\omega\sigma}$. The first term must be developed:

$$ (165) \qquad g^{\omega\alpha} \frac{\delta L_{\alpha\beta}}{\delta t} g^{\beta\sigma} = g^{\omega\alpha} \Xi_{\alpha\beta,t} g^{\beta\sigma} = \Gamma_t^{\omega\sigma} $$

Finally the generalized network equation becomes:

$$ (166) \qquad E^{\alpha} \left( t \subset x^k, \frac{\lambda}{c} \right) = \left( \zeta^{\alpha\sigma} + \Gamma_t^{\omega\sigma} \right) a_{\sigma} $$

Non linearities are often driven by some static component also called polarization. If we consider a signal $s(t) = D+f(t)exp(-\alpha t)$ where D is its static component and f(t) its dynamic one, its Laplace's transform is:

$$ (167) \qquad \mathcal{L}[s(t)] = \int_{0}^{+\infty} dt (D + f(t)exp(-\alpha t)) e^{-kt} $$

The static part Laplace's transform is:

$$ (168) \qquad \int_{0}^{+\infty} dt De^{-kt} = -\frac{D}{k} [e^{-kt}]_{0}^{+\infty} = \frac{D}{k} $$

$s(t)$ being of exponential order. The final value is defined by $lims(t)(t \to +\infty) = lim [ks(k)] (k \to 0)$. So for the static part, this final value is obviously D.

when programming this kind of problems, we have two nested loops. The external one unfolding the time, and the internal one exploring k. The polarization value is extracted from some observable in $\mathbb{B}$ and translated as a parameter value p involved in the domains states.

## 9. COUPLINGS BETWEEN SCALES

Scales are mainly defined by the wavelength values. Any model must be firstly defined versus its wavelength dependencies. If a physical phenomenon covers various scales it can be interesting to define various models connected by domains through functions like 162. Another solution appearing in couplings with chemical equations for example, is to have one equation per scale, then to couple equations through shared observable as in multiphysics. For example a chemical oxydation is represented by:

$$ Zn(s) \to Zn^{2+}(aq) + 2e $$

Then the load stored during the time step $\Delta t$ creates the current source $2e/\Delta t$, involved in a coupled electronical circuit. Another example is the photosensitivity $S_{ph}$ of a photodiode ($[lux][A]^{-1}$) defined by:

$$ (169) \qquad \Delta s^{(i)} S_{ph} = \Delta E_{\nu} $$

$\Delta E_{\nu}$ being the luminous illuminance (in lux) and $\Delta s^{(i)}$ the current gain (in ampere). It gives the scale connection between photons and currents.

In general, measurement sensors give the way to translate the interaction between micro and macro scales or bridge models between $\mathcal{L}M$ and $\mathcal{L}L$. Through these sensors, both worlds can
coexist with one equation dedicated to each scale and the measurement giving the common relation shared between both equations[27].

## 10. A LINK WITH GAME THEORY

I couldn't make a synthesis without speaking a lot of my thesis subject: the introduction of a game theory inside network analysis[28].

When human mind must be introduced in the physical problem to solve, game theory constitutes a possible model for taking it into account. Once the game rules are defined, a Kuhn's tree can be drawn for modeling the game and its issues. On a vertical sequences axis where each gamer plays in turn, gamer's choices $C_k$ are written horizontally with the associated gains $G_k$. The probabilities for the gamer to go from one choice $c_i$ at one stage of play to another at the next stage of play, is a bayesian probability of the form:

$$ (170) \qquad P(s_1 \rightarrow s_2) = \{c_i | (c_{i-1}, c_{i2}, \dots, C_k, \dots), (G_k, G_l, \dots), \text{Psy}\} $$

Psy being a psychological gamer profile.

These transitions are components of a markovian process matrix $M$ giving gamer's choices probabilities between each stage of play, the rules determining gamer's gains at each stage of play.

The inputs of the process come from some observables in the network describing the gamer physical layer $LL$. The outputs are gamer's gains acting like activators of sources in the same $LL$. This mechanism for taking into account human factor can be seen as an upper layer $LF$: a Freud's layer.

We can imagine a simple case to illustrate the process, represented figure 9.

We have two gamers. Gamer one begins (a given game rule is defined) with an initial condition materialized by the mesh current $a_1$. At each play stage the gamers have two possible choices. On time $\tau_1$, the gamer one begins and can play the transfer function $H^{10}$ or the transfer function $H^{20}$ with bayesian probabilities $p_1^0$ and $p_2^0$. The new possible consequences of these choices are $c^1$ and $c^2$ leading to the gains (new initial conditions) $a_1$ or $a_2$ through the impedances (translations of the choices in gains) $z^{11}$ and $z^{22}$. Now the gamer two plays on time $\tau_2$. He can play $H^{31}$ or $H^{41}$ from $a_1$, or $H^{52}$, $H^{62}$ from $a_2$. Corresponding probabilities are $p_3^1, p_4^1, p_5^2, p_6^2$. The new available consequences through the $z^{ii}$ are $a_3, a_4, a_5, a_6$. All these stages can be computed automatically defining a gamer gain $a_k$ and choice probability $P^k$:

$$ (171) \qquad a_k = [a_0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 0], \quad P^k = [p_0 \ 0 \ 0 \ 0 \ 0 \ 0 \ 0] $$

and consequence:

$$ (172) \qquad c^k(\tau_1) = [0 \ c^1 \ c^2 \ 0 \ 0 \ 0 \ 0] $$
<figure><img src="image_10.png" /><figcaption>FIGURE 9. Kuhn's tree</figcaption></figure>

a transfer function matrix process:

$$ (173) \qquad H = G \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ H^{10} & 0 & 0 & 0 & 0 & 0 & 0 \\ H^{20} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & H^{31} & 0 & 0 & 0 & 0 & 0 \\ 0 & H^{41} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & H^{52} & 0 & 0 & 0 & 0 \\ 0 & 0 & H^{62} & 0 & 0 & 0 & 0 \end{bmatrix} $$
and the markovian matrix:

$$ (174) \qquad M = G \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ p_1^0 & 0 & 0 & 0 & 0 & 0 & 0 \\ p_2^0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & p_3^1 & 0 & 0 & 0 & 0 & 0 \\ 0 & p_4^1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & p_5^2 & 0 & 0 & 0 & 0 \\ 0 & 0 & p_6^2 & 0 & 0 & 0 & 0 \end{bmatrix} $$

The last equation being $a_i = y_{ii}c^i$. At each stage we obtain the consequence by $c^i = H^{ik}a_k$. At $\tau_1$ this gives $c^{1,2} = H^{(1,2)0}a_0$. At $\tau_2$ this becomes :

$$ (175) \quad c^{3,4,5,6} = H^{(3,4,5,6),(1,2)} a_{(1,2)} = H^{(3,4,5,6),(1,2)} y_{(1,2),(1,2)} c^{1,2} = H^{(3,4,5,6),(1,2)} y_{(1,2),(1,2)} H^{(1,2)0} a_0 $$

Using a similar process (markovian one) we obtain the probabilities by: $P_i(\tau_1) = M \cdot p_0$, $P_i(\tau_2) = M \cdot M \cdot p_0$, etc. $H$ can be called the game actions. The effective gain $EG$ of a player is obtained through:

$$ (176) \qquad EG = a_k P^k $$

In a system graph, the game interaction is introduced using strings in the Freud's layer $LF$. All details for modeling games through game theory is given in [29].

For recall, a complex system has three major properties[30]:

(1) this system has a large number of interactions ($\zeta$ tensor with many extra-diagonal components);

(2) it has stochastic behaviors;

(3) it presents emergences.

An emergence is a pure systemic concept. It exists when observable phenomenons cannot be explained at lower scales. For example, an inductance $L$ associated with a loop made of two branches, doesn't exist at the branch level: $L \notin z(\mathbb{B})$.

# 11. UNCERTAINTIES AND DETERMINISTIC CHAOS

As we spoke of measurements, we can conclude this synthesis speaking of uncertainties. There are two simple ways for an engineer for evaluating uncertainties. First one consists in replacing variables $u$ in an equation by the same variable increased by its uncertainty: $u \pm \delta u$. Neglecting second order terms in the equation development makes appear the original expression plus an added term containing the equation uncertainty. The second approach starts from the
variation of some quantity and by development, to make appear the uncertainties. For example
we compute $\delta(ab) = \delta a.b + a\delta b$, then we develop each variation $\delta a$ and $\delta b$.

Taking into account uncertainties is fundamental and often too neglected. In particular,
various system trajectories can be explore by a deterministic approach. But the fact that the
system follows one trajectory rather than another depends on initial conditions on which it
may exist large uncertainties. This is more particularly true in electromagnetic compatibility
(EMC) for example, where systems are often associated with equations having a large number
of eigenmodes and poles.

12. CONCLUSION

In this synthesis, I've tried to compile main notions of thirty three years of developements
in EMC under Kron's formalism, and to replace them in a differential geometry context. This
may give clearer writing and mathematical expressions, making appear in particular a clearer
coherence and description of the various spaces involved.

It confirms also that tensor analysis of networks is a good candidate for theoretical multi-
physic modeling.

More than ever today when sometimes young scientists think that anything can be solved
using numerical finite element methods or equivalent, an analytical approach giving a tool for
demonstrating chaotic behavior, possible integration of a problem, etc.; is an indispensable
reasoning before to consider any numerical simulation. We must always keep in mind that
these simulations are only virtual experiments, and do not constitute demonstrations.
GLOSSARY

<table><thead><tr><th>Variable: meaning</th><th>Variable: meaning</th></tr></thead><tbody><tr><td>x<sup>α</sup>: 4-space-time coordinates</td><td>λ: wavelength</td></tr><tr><td>N: nodes space</td><td>B: branches space</td></tr><tr><td>M: meshes space</td><td>b̂<sub>α</sub>: branch space base</td></tr><tr><td>ĉ<sup>α</sup>: mesh space base</td><td>k: Laplace's operator</td></tr><tr><td>C, Λ: branch space to ℝ<sup>3</sup> space connectivity</td><td>φ: branch versus mesh function</td></tr><tr><td>g<sub>ij</sub>: metric</td><td>δ<sub>ij</sub>: cartesian metric</td></tr><tr><td>s<sup>u</sup>: branch current coordinates and vectors</td><td>a<sub>u</sub>: mesh currents and covectors</td></tr><tr><td>a, Θ: action</td><td>t: time</td></tr><tr><td>T: energy</td><td>t<sub>μν</sub>: energy tensor</td></tr><tr><td>A: parametrized hypersurface</td><td>u<sub>β</sub>: hypersurface parameter</td></tr><tr><td>G<sub>μν</sub>: underlying space metric</td><td>k: stiffness</td></tr><tr><td>e, v<sub>X</sub>: electromotive forces (emf) and voltages</td><td>V: potentials difference</td></tr><tr><td>z<sub>ab</sub>: impedance tensor ∈ B</td><td>E<sup>a</sup> mesh emf</td></tr><tr><td>ζ<sup>ab</sup>: mesh space impedance tensor</td><td>(E<sup>α</sup>, λ/c): 4-space emf</td></tr><tr><td>τ: time particular value</td><td>A: field vector potential</td></tr><tr><td>ψ: field scalar potential</td><td>B: magnetic field</td></tr><tr><td>C<sup>ij</sup>, γ<sup>ab</sup>: capacitances</td><td>q: electric load</td></tr><tr><td>R<sup>ab</sup>: electrical resistance</td><td>y<sup>12</sup>: electrostatic interaction</td></tr><tr><td>R: magnetic reluctance</td><td>As: magnetic flux section</td></tr><tr><td>μ: magnetic permeability</td><td>l: magnetic flux length</td></tr><tr><td>F: magnetomotive force</td><td>φ: magnetic flux</td></tr><tr><td>M<sup>ab</sup>: mutual inductance</td><td>R<sup>ab</sup>: distance between a and b</td></tr><tr><td>(ψ/c, A): 4-field</td><td>χ<sup>μν</sup>: Fraunhofer's interaction</td></tr><tr><td>μ<sup>ab</sup>: permeability tensor</td><td>y<sub>ij</sub>: admittance</td></tr><tr><td>H<sup>ab</sup>: photonic interaction</td><td>P(1, 2): probability of photon traveling between points 1 and 2</td></tr><tr><td>n<sub>c</sub>: waveguide characteristic impedance</td><td>V<sub>α</sub>: wave vector</td></tr><tr><td>G<sub>a</sub>: reflexion coefficient</td><td>p<sub>l</sub>, r<sub>i</sub>: wave ports identifications</td></tr><tr><td>G<sub>ab</sub>: Green's function</td><td>γ: partial wave propagator</td></tr><tr><td>γ̄: complete wave propagator</td><td>h<sup>μν</sup>: wave to mesh metric</td></tr><tr><td>c:celerity</td><td>v: wave phase speed</td></tr><tr><td>h, W: waveguide dimensions</td><td>α: attenuation factor</td></tr><tr><td>Θ: phasor</td><td>σ: self time</td></tr></tbody></table>
<table><thead><tr><th>Variable: meaning</th><th>Variable: meaning</th></tr></thead><tbody><tr><td>φ: gravitational field</td><td>n̄<sup>3</sup>: normal vector</td></tr><tr><td>Γ<sup>q</sup><sub>ij</sub>, Γ<sub>ij,q</sub>, Γ<sup>μν</sup><sub>t</sub>: Christoffel's coefficients or similars</td><td>γ, ˙γ, ˙˙γ: curve and its speed and acceleration on A</td></tr><tr><td>m, M: masses</td><td>h<sub>μν</sub>: second fundamental form metric</td></tr><tr><td>D(•): covariant derivative</td><td>G: gravitational constant</td></tr><tr><td>T<sup>a</sup><sub>b</sub>: energy tensor</td><td>R<sup>a</sup><sub>b</sub>: Riemann's tensor</td></tr><tr><td>LL: Lagrange's layer</td><td>LM: Maxwell's layer</td></tr><tr><td>LE: Einstein's layer</td><td>F<sup>i</sup>: force vector</td></tr><tr><td>V: potential energy</td><td>ds: elementary 4-distance</td></tr><tr><td>L: lagrangian</td><td>L: self and mutual inductances</td></tr><tr><td>G: cutted flux induction</td><td>r: wires resistance</td></tr><tr><td>Ω: angular speed</td><td>Q: inertia moment</td></tr><tr><td>ρ: rotor frictions</td><td>T: machine torque</td></tr><tr><td>p<sub>ν</sub>: mesh current</td><td>N<sup>β</sup>: mesh emf</td></tr><tr><td>K, K<sub>i</sub>: mechanical stiffness</td><td>j<sup>a</sup>: mechanical distance</td></tr><tr><td>T: temperature</td><td>Q<sub>d</sub>: mesh current</td></tr><tr><td>R<sub>th</sub>: thermal resistance</td><td>m<sub>C</sub>: thermal inertia</td></tr><tr><td>δT<sup>u</sup>: thermal variable vector</td><td>Rad: thermal radiation</td></tr><tr><td>L<sub>rad</sub>: luminance</td><td>dφ: radiated thermal flux</td></tr><tr><td>φ<sub>21</sub>: form factor</td><td>σ: thermal radiation coefficient</td></tr><tr><td>φ<sup>a</sup><sub>2</sub>: absorbed flux</td><td>R̃<sub>21</sub>: radiated flux operator</td></tr><tr><td>γ<sup>a</sup>: fluid acceleration</td><td>v<sup>a</sup>: fluid speed</td></tr><tr><td>ρ: fluid density</td><td>T<sub>αβ</sub>: rotational fluid tensor</td></tr><tr><td>μ: viscosity coefficient</td><td>p: pressure</td></tr><tr><td>D<sup>p</sup><sub>I</sub>: parametrized temporal domain</td><td>→: chemical relation</td></tr><tr><td>C<sub>k</sub>, c<sub>k</sub>: gamer's choices</td><td>G<sub>k</sub>: gamer's gain</td></tr><tr><td>P(s<sub>1</sub> → s<sub>2</sub>): bayesian probability</td><td>M: Markov's matrix</td></tr><tr><td>LF: Freud's layer</td><td>H, c: game actions and consequences</td></tr></tbody></table>
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