IEEE Circuits and Systems Magazine - Q4 2022 - 55

derived from these laws also apply in the other domains.
Remarkable properties of electrical networks
were derived from Tellegen's theorem. This is also the
case for analogous laws in other domains. For example,
a theorem such as Tellegen's theorem can be derived for
quantum wave functions [1, p. 111], [22].
Kirchhoff's laws and Tellegen's theorem can therefore
be applied in many sciences. The fact that we can derive
these generalized theorems without an appeal upon physics
and that they can be applied in different disciplines indicates
a form of 'universality' [35]. This finding opens the
door to new insights and application possibilities.12
As mentioned before, the duality of the currents
through and the voltages across the branches of an electrical
network is a remarkable property [3]. The analog
'through variables' and 'over variables' in other science
domains are also dual. It can be proved that these dual
variables are orthogonal. [1, pp. 19-21], [4]. Orthogonality
appears to be a 'universal' property.
Kirchhoff's laws and Tellegen's theorem lead to a
remarkable finding. Even if information about the components
and topology of a network is missing, it is still
possible to say important things about the network that
may be useful. This is for instance the case for abstract
energy concepts such as 'content' and 'cocontent' [1, pp.
39-41], [10], [27]. Both concepts lead to 'mini-max' theorems
which can be applied for optimization purposes.
They are mainly used in structural engineering, where
they play an important part in modelling the elements
for the Finite Element Method (FEM).
Symmetries already play an important role in quantum
mechanics and fundamental physics [28]. Their possibilities
for deriving models are still insufficiently recognized
in other scientific domains. The fact that this is an abstract
and generalizing concept, together with the fear for the
mathematical formulation, apparently forms an obstacle.
Roger Penrose points in the foreword of the book [29] to
the beauty of the symmetries in nature and culture to
make the concept of symmetry more accessible.
It is observed that in the engineering sciences too
little attention is paid to invariances and symmetries.
They are not only important for unifying the sciences.
Invariances and symmetries can also give the mathematical
models from system theory more physical
content and bring them closer to reality. In engineering
sciences, mathematical models are used that do not necessarily
represent the physical reality correctly. They
are experimentally verified and appear to be useful.
With invariances and symmetries, it is possible to better
determine the formulation of the models. This is also
the case for the derivation of models with dimensional
12We can consider this finding as a kind of black box approach.
FOURTH QUARTER 2022
analysis [30]. In order to get an effective grip on reality,
all models should be validated whether invariances and
symmetries for objective knowledge are respected.
VII. Deeper Nature of Reality
Laws are recurring patterns in space and time. They allow
descriptions of the reality that require less information,
and offer the opportunity of information reduction,
compression, and economy. To be considered as a law, a
mathematical model identified with a measurement must
be invariant of the position and the time of day. The invariance
for the scale of the measuring instrument is also
required. These symmetries make it possible to choose
mathematical models that physically correctly represent
reality. The mathematical models in system theory do
not necessarily have a physical content. However, they
must be able to be applied successfully. Laws point also
to the existence of simplicity in the complex reality.
We have also shown in this paper how a general form
of Kirchhoff's laws can be derived from invariances and
the corresponding symmetries. It is in fact a generalization
of Kirchhoff's laws for all measurable quantities.
Kirchhoff's laws can be generalized in a simple way assuming
invariances in sets of real numbers. It is remarkable
that no appeal is made to the physics.
Only three simple assumptions are made (1) that we
can measure voltages and currents, (2) that invariant
functions of these variables exist and (3) that the mathematical
properties of real numbers hold. By abstracting
from the physical content, one obtains generalized
Kirchhoff's laws that can be applied to networks, structures,
and systems in various branches of science.13
Note also that Kirchhoff's laws and their analogous
and generalized forms are identities that hold for arbitrary
values of measurable variables. The possibility of
generalizing Kirchhoff's laws by introducing invariances
and symmetries implies more than deriving these laws
without the laws of physics. There is further distance
from physics. We are basing ourselves on what can be
called 'the deeper nature of reality'. These are the fundamental
properties that underpin the physical laws.
The invariants (6) and (7) and the symmetry (8) turn
out to be 'universally' valid and can be regarded as axioms
for generalized network and structure theories.14
They seem, as it were, designed to make these theories
possible. It is surprising that similar properties can be
13The generalization of Kirchhoff's laws implies an additional form of
information economy.
14For electrical networks, the topology of the network is taken into account.
For mechanical structures, geometry must also be considered.
See e.g., for the calculation of the shear force, bending moment, deflection,
and slope of a cantilever beam [36]. The generalization for mechanical
structures requires the introduction of a generalized continuity
principle.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
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