# On Energy Conversion in Port-Hamiltonian Systems

Arjan van der Schaft and Dimitri Jeltsema A.J. van der Schaft is with the
Bernoulli Institute for Mathematics, Computer Science and AI, University of
Groningen, P.O. Box 407, 9700 AK, The Netherlands. E–mail:
<EMAIL_ADDRESS>D. Jeltsema is with the School of Engineering and
Automotive, HAN University of Applied Sciences, P.O. Box 2217, 6802 CE Arnhem,
The Netherlands. E–mail<EMAIL_ADDRESS>
###### Abstract

We study port-Hamiltonian systems with two external ports, and the strategies
and limitations for conversion of energy from one port into the other. It
turns out that, apart from the cyclo-passivity of port-Hamiltonian systems,
this is related to the internal connection structure of port-Hamiltonian
systems. A source of motivation for energy conversion is provided by
thermodynamics, in particular the Carnot theory of conversion of thermal into
mechanical energy. This is extended to general port-Hamiltonian systems which
are satisfying structural conditions on their topology; thus generalizing the
Carnot-Clausius theory of heat engines. In particular, the operation of Carnot
cycles is extended, which is illustrated by the example of a precursor to the
Stirling engine, as well as an electromagnetic actuator. Furthermore,
alternative energy conversion control schemes such as energy-routers are
discussed.

##  I INTRODUCTION

Energy conversion and harvesting are among the most important current
engineering problems. This motivates a number of control questions such as the
development of a structural theory of (optimal) energy conversion and
efficiency, and the development of effective control strategies for achieving
this. This paper aims at addressing some of these questions by making use of
port-Hamiltonian systems theory (see e.g. [9, 11]), which offers a systematic
framework for modeling and control of multiphysics systems. This line of
research was already initiated in [12] by developing the notion of one-port
cyclo-passivity, which identifies topological conditions which limit the
energy transfer of one port of the system to the other. In the present paper
this is extended by defining and analyzing Carnot cycles for one-port cyclo-
passive systems, and applying this to a gas-piston system and an electro-
mechanical actuator. Furthermore, initial steps are made towards obviating the
conditions for one-port cyclo-passivity by establishing direct topological
connections by the use of feedback; thus relating to the Duindam-Stramigioli
energy router [1] as also discussed in [8]. The paper closes by formulating a
number of open research problems for the theory of energy conversion for port-
Hamiltonian systems.

##  II CYCLO-PASSIVITY

Before moving on to port-Hamiltonian systems, let us first recall passivity
theory, and especially the less well-known theory of cyclo-passivity. Consider
a standard input-state-output system

$\Sigma:\left\\{\,\begin{aligned} \dot{x}&=f(x)+G(x)u,\\\ y&=h(x),\quad
x\in\mathcal{X},\ u,y\in\mathbb{R}^{m},\end{aligned}\right.$ (1)

on an $n$-dimensional state space manifold $\mathcal{X}$. Cyclo-passivity was
coined in [15] generalizing [14], and was further developed in [3, 4] with
recent extensions in [10].

###### Definition 1

$\Sigma$ is cyclo-passive if

$\int\limits_{0}^{\tau}y^{\top}(t)u(t)dt\geq 0$ (2)

for all $\tau\geq 0$ and all $u:[0,\tau]\to\mathbb{R}^{m}$ such that
$x(\tau)=x(0)$. Furthermore, $\Sigma$ is cyclo-passive with respect to $x^{*}$
if (2) holds for all $\tau\geq 0$ and all $u:[0,\tau]\to\mathbb{R}^{m}$ such
that $x(\tau)=x(0)=x^{*}$. In case (2) holds with equality, we speak about
cyclo-losslessness (with respect to $x^{*}$).

Cyclo-passivity thus means that cyclic motions always require a net
nonnegative amount of externally supplied energy. While energy might be
released during some sub-interval of the cyclic motion, over the whole time-
interval this needs to be compensated by an amount of supplied energy which is
at least as large. Clearly, this is a fundamental property of any physical
system, where $y^{\top}u$ is the externally supplied power. The
characterization of the set of possible energy storage functions of a cyclo-
passive system is done via the dissipation inequality [14].

###### Definition 2

A (possibly extended) function
$S:\mathcal{X}\to-\infty\cup\mathbb{R}\cup\infty$ satisfies the dissipation
inequality for system $\Sigma$ if

$S(x(t_{2}))\leq S(x(t_{1}))+\int\limits_{t_{1}}^{t_{2}}y^{\top}(t)u(t)dt$ (3)

holds for all $t_{1}\leq t_{2}$, all input functions
$u:[t_{1},t_{2}]\to\mathbb{R}^{m}$, and all initial conditions $x(t_{1})$,
where $y(t)=h(x(t))$, with $x(t)$ denoting the solution of
$\dot{x}=f(x)+g(x)u$ for initial condition $x(t_{1})$ and input function
$u:[t_{1},t_{2}]\to\mathbb{R}^{m}$. If $S$ is an ordinary function
$S:\mathcal{X}\to-\infty\cup\mathbb{R}$ satisfying (3), then $S$ is called a
storage function.

By assuming differentiability of $S$, the dissipation inequality (3) is easily
seen [9] to be equivalent to the differential dissipation inequality

$\left[\frac{\partial S}{\partial
x}(x)\right]^{\top}\\!\\!\big{(}f(x)+G(x)u\big{)}\leq h^{\top}(x)u,$

or equivalently,

$h(x)=G^{\top}(x)\frac{\partial S}{\partial x}(x)\ \text{and}\
\left[\frac{\partial S}{\partial x}(x)\right]^{\top}\\!\\!\\!f(x)\leq 0.$

Obviously, if there exists a storage function for $\Sigma$, then by
substituting $x(\tau)=x(0)$ in (3), it follows that $\Sigma$ is cyclo-passive.
A converse is given as follows. Assuming reachability from a ground-state
$x^{*}$ and controllability to this same state $x^{*}$, define the (possibly
extended) functions $S_{ac}:\mathcal{X}\to\mathbb{R}\cup\infty$ and
$S_{rc}:\mathcal{X}\to-\infty\cup\mathbb{R}$ as

$\displaystyle S_{ac}(x)$ $\displaystyle:=\\!\\!\\!\\!\mathop{\sup_{u,\tau\geq
0}}_{x(0)=x,\,x(\tau)=x^{*}}-\int\limits_{0}^{\tau}y^{\top}(t)u(t)dt,$ (4)
$\displaystyle S_{rc}(x)$
$\displaystyle:=\\!\\!\\!\\!\\!\\!\\!\mathop{\inf_{u,\tau\geq
0}}_{x(-\tau)=x^{*},\,x(0)=x}\ \ \int\limits_{-\tau}^{0}y^{\top}(t)u(t)dt,$

which have an obvious interpretation in terms of energy. Indeed, the function
$S_{ac}(x)$ denotes the maximal (in fact, supremal) energy that can be
recovered from the system at state $x$ while returning the system to its
ground-state $x^{*}$, and the function $S_{rc}(x)$ denotes the minimal (in
fact, infimal) energy that is needed in order to transfer the system from the
ground-state $x^{*}$ to $x$.

It directly follows [14, 9, 10] from the properties of ‘supremum’ and
‘infimum’ that $S_{ac}$ and $S_{rc}$ satisfy the dissipation inequality.
Hence, if either $S_{ac}$ or $S_{rc}$ are ordinary functions (i.e., not taking
values $\pm\infty$), then the system is cyclo-passive. The following basic
theorem was obtained in [10], extending the results of [3, 4].

###### Theorem 1

Assume $\Sigma$ is reachable from $x^{*}$ and controllable to $x^{*}$. Then
$\Sigma$ is cyclo-passive with respect to $x^{*}$ if and only if

$S_{ac}(x)\leq S_{rc}(x),\quad\mbox{ for all }x\in\mathcal{X}.$ (5)

Furthermore, if $\Sigma$ is cyclo-passive with respect to $x^{*}$ then both
$S_{ac}$ and $S_{rc}$ are storage functions for $\Sigma$, implying that
$\Sigma$ is cyclo-passive. Moreover, if $\Sigma$ is cyclo-passive with respect
to $x^{*}$, then $S_{ac}(x^{*})=S_{rc}(x^{*})=0$, and any other storage
function $S$ for $\Sigma$ satisfies

$S_{ac}(x)\leq S(x)-S(x^{*})\leq S_{rc}(x).$ (6)

If the system is cyclo-lossless from $x^{*}$ then $S_{ac}=S_{rc}$, and the
storage function is unique up to a constant. More generally, $\Sigma$ is
cyclo-passive with unique (up to a constant) storage function if and only if
for every $x\in\mathcal{X}$

$\mathop{\inf_{u,\tau\geq
0}}_{x(0)=x(\tau)=x^{*}}\,\oint\limits_{0}^{\tau}y^{\top}(t)u(t)dt=0.$ (7)

Note that the choice of the ground-state $x^{*}$ in the above theorem is
arbitrary. Indeed, if $\Sigma$ is reachable from and controllable to $x^{*}$,
then so it is from any other ground-state (and, equivalently, for any two
states $x_{1},x_{2}$ there exists a trajectory from $x_{1}$ to $x_{2}$).
Furthermore, by the above theorem, cyclo-passivity with respect to $x^{*}$
implies cyclo-passivity and thus cyclo-passivity with respect to any other
ground state.

###### Example 1

A simple example of a passive system which is not cyclo-lossless, but still
has unique storage function is the mass-spring-damper system

$\displaystyle\begin{bmatrix}\dot{q}\\\ \dot{p}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}0&\frac{1}{m}\\\
-k&-\frac{d}{m}\end{bmatrix}\begin{bmatrix}q\\\
p\end{bmatrix}+\begin{bmatrix}0\\\ 1\end{bmatrix}u,\quad(u=\text{force})$
$\displaystyle y$
$\displaystyle=\begin{bmatrix}0&\frac{1}{m}\end{bmatrix}\begin{bmatrix}q\\\
p\end{bmatrix}\quad(=\text{velocity)}$

Here, $q$ is the displacement of a linear spring with stiffness $k$, $p$ is
the momentum of mass $m$, and ${d>0}$ is the damping coefficient. Let
$x=\text{col}(q,p)$. The dissipation inequality for quadratic storage
functions $S(x)=\frac{1}{2}x^{\top}Qx$ with

$Q=\begin{bmatrix}q_{11}&q_{12}\\\ q_{12}&q_{22}\end{bmatrix}$

reduces to the linear matrix inequality (LMI)

$\begin{array}[]{c}\begin{bmatrix}0&-k\\\
\frac{1}{m}&-\frac{d}{m}\end{bmatrix}\\!\begin{bmatrix}q_{11}&q_{12}\\\
q_{12}&q_{22}\end{bmatrix}+\begin{bmatrix}q_{11}&q_{12}\\\
q_{12}&q_{22}\end{bmatrix}\\!\begin{bmatrix}0&\frac{1}{m}\\\
-k&-\frac{d}{m}\end{bmatrix}\preceq 0,\\\\[14.22636pt]
\begin{bmatrix}0&1\end{bmatrix}\\!\begin{bmatrix}q_{11}&q_{12}\\\
q_{12}&q_{22}\end{bmatrix}=\begin{bmatrix}0&\frac{1}{m}\end{bmatrix}.\end{array}$

The equality implies $q_{12}=0$ and $q_{22}=\frac{1}{m}$. Substitution in the
inequality yields the unique solution $q_{11}=k$, corresponding to a unique
quadratic storage function $S$ which equals the physical energy
$H(q,p)=\frac{1}{2m}p^{2}+\frac{1}{2}kq^{2}$. Thus, the mass-spring-damper
system is such that for all $x$ we have

$\mathop{\inf_{u,\tau\geq
0}}_{x(0)=x(\tau)=0}\,\oint\limits_{0}^{\tau}\frac{p(t)}{m}u(t)dt=0.$

Indeed, although all possible trajectories involve non-zero energy-dissipation
due to the damper $d>0$, leading to

$\oint\limits_{0}^{\tau}\frac{p(t)}{m}u(t)dt>0,$

this dissipation can be made arbitrarily small.

The stronger notion of passivity corresponds to the existence of a nonnegative
storage function. One considers instead of $S_{ac}$ the (possibly extended)
function

$S_{a}(x):=\mathop{\sup_{u,\tau\geq
0}}_{x(0)=x}-\\!\int\limits_{0}^{\tau}y^{\top}(t)u(t)dt,$ (8)

which is the supremal energy that can be extracted from the system at state
$x$; whence the name available storage for $S_{a}$. Obviously, $S_{a}$ is
nonnegative, and, again by the property of ‘supremum’, satisfies the
dissipation inequality.

It follows [14, 9] that $\Sigma$ is passive if and only if $S_{a}$ is an
ordinary function, i.e., $S_{a}(x)<\infty$ for every $x\in\mathcal{X}$.
Furthermore, if $\Sigma$ is passive then $S_{a}$ is a nonnegative storage
function satisfying $\inf_{x}S_{a}(x)=0$, and all other nonnegative storage
functions $S$ satisfy

$S_{a}(x)\leq S(x)-\inf_{x}S(x),\quad x\in\mathcal{X}.$ (9)

Obviously, $S_{ac}(x)\leq S_{a}(x)$ for any $x\in\mathcal{X}$. If
$S_{a}(x^{*})=0$, then it directly follows from the dissipation inequality (3)
that passivity implies (and in case $\Sigma$ is reachable from $x^{*}$, is
equivalent to)

$\int\limits_{0}^{\tau}y^{\top}(t)u(t)dt\geq 0,$ (10)

for all $u:[0,\tau]\to\mathbb{R}^{m},\tau\geq 0$, where $y(t)$ is the
corresponding output for initial condition $x^{*}$. This is sometimes taken as
the definition of passivity, -especially in the linear case with $x^{*}=0$\- ,
and implies that the supplied energy is nonnegative for any motion from the
state of minimal energy. However, in a nonlinear context there is often no
natural ground state $x^{*}$ with $S_{a}(x^{*})=0$, and there may even not
exist such a ground state, as demonstrated by the following example [10].

###### Example 2

Consider the scalar system

$\Sigma:\left\\{\,\begin{aligned} \dot{x}&=u,\\\
y&=e^{x}.\end{aligned}\right.$

Then, the available storage (8) takes the form

$S_{a}(x)=\mathop{\sup_{u,\tau\geq
0}}_{x(0)=x}\left\\{e^{x(0)}-e^{x(\tau)}\right\\}=e^{x},$

and $\Sigma$ is lossless with nonnegative storage function $e^{x}$, which is
unique up to a constant. Note that $\inf S_{a}(x)=0$, but there does not exist
a finite $x^{*}$ such that $S_{a}(x^{*})=0$. Obviously, $\Sigma$ is reachable
from and controllable to, e.g., $x^{*}=0$, with resulting
$S_{ac}(x)=S_{rc}(x)=e^{x}-1$.

##  III ENERGY CONVERSION BY CARNOT-LIKE CYCLES

While the net total energy supplied to a cyclo-passive system during a cyclic
motion is always non-negative, energy conversion is concerned with the
transfer of energy from one port of the system to the other. As we will see,
an important aspect in energy conversion is the internal topology of the
system. This topology is captured in the port-Hamiltonian formulation of
physical systems. A general input-state-output port-Hamiltonian system is
given as [9, 11]

$\displaystyle\dot{x}$ $\displaystyle=J(x)e-\mathcal{R}(x,e)+G(x)u,\quad
e=\frac{\partial H}{\partial x}(x),$ (11) $\displaystyle y$
$\displaystyle=G^{\top}(x)e,\quad x\in\mathcal{X},$

with $n$-dimensional state space $\mathcal{X}$, Hamiltonian
$H:\mathcal{X}\to\mathbb{R}$, skew-symmetric matrix $J(x)=-J^{\top}(x)$, and
dissipation mapping $R$ satisfying $e^{\top}\mathcal{R}(x,e)\geq 0$ for all
$x,e$. By the properties of $J(x)$ and $\mathcal{R}(x,e)$, any port-
Hamiltonian system (11) satisfies the dissipation inequality

$\frac{d}{dt}H(x)=e^{\top}J(x)e-e^{\top}\mathcal{R}(x,e)+e^{\top}G(x)u\leq
y^{\top}u,$

and thus is cyclo-passive, with storage function $H$. Conversely, almost any
cyclo-passive system can be formulated as a port-Hamiltonian system for some
$J$ and $\mathcal{R}$ as above.111However in systematic systems modeling one
would start from a port-Hamiltonian description with $J$ and $\mathcal{R}$
dictated by the physics of the system. Thus, by cyclo-passivity theory

$S_{ac}(x)\leq H(x)-H(x^{*})\leq S_{rc}(x).$

Now, for purposes of energy conversion, consider port-Hamiltonian systems (11)
with two ports $(u_{1},y_{1})$ and $(u_{2},y_{2})$ as in Fig. 1. Consequently

$\frac{d}{dt}H(x)\leq y_{1}^{\top}u_{1}+y_{2}^{\top}u_{2}.$ (12)

The central question addressed in this paper is how we can convert energy
which is flowing into the system at port $1$ into energy which is flowing out
of the system at port $2$, and to identify the possible obstructions for doing
so. Indeed, it turns out that the feasible strategies for energy conversion
depend on the structure of the interconnection matrix $J(x)$.

Figure 1: Port-Hamiltonian two-port system.

###  III-A Topological Obstructions for Energy Conversion

Consider the subclass of port-Hamiltonian systems with two ports given as

$\displaystyle\dot{x}_{1}$
$\displaystyle=J_{1}(x_{1},x_{2})e_{1}-R_{1}(x_{1},x_{2})e_{1}+G_{1}u_{1},$
(13) $\displaystyle\dot{x}_{2}$
$\displaystyle=J_{2}(x_{1},x_{2})e_{2}-R_{2}(x_{1},x_{2})e_{2}+G_{2}(x_{1},x_{2})u_{2},$
$\displaystyle y_{1}$ $\displaystyle=G_{1}^{\top}e_{1},\qquad\quad
e_{1}=\frac{\partial H}{\partial x_{1}}(x_{1},x_{2}),$ $\displaystyle y_{2}$
$\displaystyle=G_{2}^{\top}(x_{1},x_{2})e_{2},\quad e_{2}=\frac{\partial
H}{\partial x_{2}}(x_{1},x_{2}),$

where $G_{1}$ is an invertible constant matrix, and the partial Hessian
$\frac{\partial^{2}H}{\partial x_{1}^{2}}(x_{1},x_{2})$ has full rank
everywhere. Because of the first assumption by proper choice of $u_{1}$ we can
ensure that $\dot{x}_{1}=0$, and thus $x_{1}=\bar{x}_{1}$ is constant. It
follows that for such $u_{1}$

$\frac{d}{dt}H(\bar{x}_{1},x_{2})=\frac{\partial H}{\partial
x_{2}}(x_{1},x_{2})\dot{x}_{2}\leq y_{2}^{\top}u_{2}$ (14)

Hence for such input $u_{1}$ the system is cyclo-passive at the second port
with storage function $H(\bar{x}_{1},x_{2})$. This directly implies that for
such input $u_{1}$ the system can not generate energy at port $2$ in a
recurrent manner, since by cyclo-passivity the net energy at port $2$ is
always flowing into the system.

Alternatively, by invoking both assumptions, it is clear that, instead of
keeping $x_{1}$ constant, by a proper choice of $u_{1}$ we can also ensure
that $e_{1}$, or equivalently $y_{1}$, is constant.

Recall now that the partial Legendre transformation [6] of $H$ with respect to
$x_{1}$ is given as

$H^{*}_{1}(e_{1},x_{2})=H(x_{1},x_{2})-e_{1}^{\top}x_{1},\
e_{1}=\frac{\partial H}{\partial x_{1}}(x_{1},x_{2}),$

where $x_{1}$ is expressed as a function of $e_{1},x_{2}$ by means of the
equation $e_{1}=\frac{\partial H}{\partial x_{1}}(x_{1},x_{2})$ (locally
guaranteed by the full rank assumption on the Hessian matrix). The following
properties of the partial Legendre transformation are well-known [6]

$\frac{\partial H^{*}_{1}}{\partial e_{1}}(e_{1},x_{2})=-x_{1},\
\frac{\partial H^{*}_{1}}{\partial x_{2}}(e_{1},x_{2})=\frac{\partial
H}{\partial x_{2}}(x_{1},x_{2}).$

Hence, with $u_{1}$ such that $y_{1}=\bar{y}_{1}$ is constant, and,
equivalently, $e_{1}=\bar{e}_{1}$ is constant, we have that

$\displaystyle\frac{d}{dt}H^{*}_{1}(\bar{e}_{1}$
$\displaystyle,x_{2})=-x_{1}^{\top}\dot{\bar{e}}_{1}+e_{2}^{\top}\dot{x}_{2}=e_{2}^{\top}J_{2}(x_{1},x_{2})e_{2}$
$\displaystyle-
e_{2}^{\top}R_{2}(x_{1},x_{2})e_{2}+e_{2}^{\top}G_{2}(x_{1},x_{2})u_{2}\leq
y^{\top}_{2}u_{2}.$

This implies that the system for any constant $y_{1}$ is cyclo-passive at the
second port $(u_{2},y_{2})$, with respect to the storage function
$H^{*}_{1}(e_{1},x_{2})$ with corresponding constant $e_{1}$. This property
was called one-port cyclo-passivity in [12], and implies that for port-
Hamiltonian systems of the form (13) we cannot convert energy from port $1$ to
port $2$ while keeping $y_{1}$ constant. Furthermore we note that, in view of
(13), for constant $y_{1}=\bar{y}_{1}=G_{1}^{\top}\bar{e}_{1}$ along all
cyclic motions

$\displaystyle\int\limits_{0}^{\tau}\bar{y}_{1}^{\top}u_{1}(t)dt$
$\displaystyle=\int\limits_{0}^{\tau}\bar{e}_{1}^{\top}G_{1}u_{1}(t)dt$
$\displaystyle=\int\limits_{0}^{\tau}\bar{e}_{1}^{\top}\big{[}\dot{x}_{1}$
$\displaystyle(t)-J_{1}(x_{1},x_{2})\bar{e}_{1}+R_{1}(x_{1},x_{2})\bar{e}_{1}\big{]}dt$
$\displaystyle\geq\bar{e}_{1}^{\top}\left(x_{1}(\tau)-x_{1}(0)\right)=0.$

Together this results in the following theorem, extending a main result of
[12].

###### Theorem 2

Consider a port-Hamiltonian system of the form (13). Then, for all cyclic
motions along which $y_{1}=\bar{y}_{1}$ is constant

$\int\limits_{0}^{\tau}\bar{y}_{1}^{\top}u_{1}(t)dt\geq 0\ \ \text{and}\
\int\limits_{0}^{\tau}y_{2}^{\top}(t)u_{2}(t)dt\geq 0.$ (15)

###  III-B Carnot-Cycles and Efficiency

Within a thermodynamics context, with $(u_{1},y_{1})$ corresponding to the
thermal port and $(u_{2},y_{2})$ corresponding to the mechanical port, the
trajectories for which $x_{1}$ is constant correspond to isentropic curves
(the entropy $S=x_{1}$ is constant), also called adiabatics, while the
trajectories for which $e_{1}=T$ (temperature) are constant are the
isothermals. We will sometimes use this nomenclature for port-Hamiltonian
systems as in (13) as well.

In order to convert energy from port $1$ to $2$ we need by Theorem 2 more than
one value of $e_{1}$. Within thermodynamics this amounts to the observation by
Carnot that at least two different temperatures $T$ are needed in order to
convert thermal energy into mechanical energy (‘work’) in a recurrent manner.
This gives rise to the well-known Carnot cycle, consisting of an isothermal at
high temperature, followed by an adiabatic leading to a lower temperature,
another isothermal, and finally an adiabatic which brings the system back to
its original state; see Fig. 2.

Figure 2: Carnot-cycle in the temperature-entropy plane.

The notion of the Carnot cycle can be extended to general port-Hamiltonian
systems of the form (13) as follows:

  1. 1.

On the time-interval $[0,\tau_{1}]$ consider an ‘isothermal’ with respect to
port $1$, corresponding to a constant $e_{1}=e_{1}^{h}$. Then

$\frac{d}{dt}H^{*}_{1}(e^{h}_{1},x_{2})\leq y_{2}^{\top}u_{2}.$

  2. 2.

On the time-interval $[\tau_{1},\tau_{2}]$ consider an ‘adiabatic’
corresponding to a constant $x_{1}$. Then

$\frac{d}{dt}H(x_{1},x_{2})\leq y_{2}^{\top}u_{2}.$

  3. 3.

On the time-interval $[\tau_{2},\tau_{3}]$ consider an ‘isothermal’
corresponding to a constant $e_{1}=e_{1}^{c}$. Then

$\frac{d}{dt}H^{*}_{1}(e^{c}_{1},x_{2})\leq y_{2}^{\top}u_{2}.$

  4. 4.

Finally, on the time-interval $[\tau_{3},\tau]$ consider an ‘adiabatic’
corresponding to a constant $x_{1}$. Then

$\frac{d}{dt}H(x_{1},x_{2})\leq y_{2}^{\top}u_{2}.$

Note that on both isothermals:

$\displaystyle H(x_{1},x_{2})$
$\displaystyle=H^{*}_{1}(e^{h}_{1},x_{2})+e^{h}_{1}x_{1},$ $\displaystyle
H(x_{1},x_{2})$ $\displaystyle=H^{*}_{1}(e^{c}_{1},x_{2})+e^{c}_{1}x_{1},$

respectively. Since the total process is a cycle, i.e., $x(\tau)=x(0)$,
addition of these inequalities yields

$\displaystyle 0$ $\displaystyle=H(x(\tau))-H(x(0))$
$\displaystyle\qquad\leq\int\limits_{0}^{\tau}y_{2}^{\top}(t)u_{2}(t)dt+e_{1}^{h}\Delta^{h}x_{1}+e_{1}^{c}\Delta^{c}x_{1},$

where $\Delta^{h}x_{1}$ and $\Delta^{c}x_{1}$ are the changes in $x_{1}$
during the isothermals on $[0,\tau_{1}]$ and $[\tau_{3},\tau]$, respectively.
Thus, the total energy delivered to the environment via port $2$ satisfies

$-\int\limits_{0}^{\tau}y_{2}^{\top}(t)u_{2}(t)dt\leq
e_{1}^{h}\Delta^{h}x_{1}+e_{1}^{c}\Delta^{c}x_{1},$ (16)

with equality in case there is no dissipation, i.e., when $R_{1}$ and $R_{2}$
are zero. Note that the right-hand side of (16) is the total energy provided
at port $1$ to the system during the two isothermals. Because of
$x(\tau)=x(0)$ necessarily $\Delta^{h}x_{1}+\Delta^{c}x_{1}=0$. Hence the
right-hand side of (16) can be also written as
$\left(e_{1}^{h}-e_{1}^{c}\right)\Delta^{h}x_{1}$. Assuming that both
$e_{1}^{h}$ and $e_{1}^{c}$ are positive, then this expression is positive if
$e_{1}^{h}>e_{1}^{c}$ and $\Delta^{h}x_{1}$. (This is the case considered in
thermodynamics where $e_{1}^{h}$ and $e_{1}^{c}$ are the temperatures of the
hot and cold reservoirs.) Furthermore, the efficiency of the Carnot cycle (in
case $R_{1}$ and $R_{2}$ are zero) is given as

$\frac{e_{1}^{h}\Delta^{h}x_{1}+e_{1}^{c}\Delta^{c}x_{1}}{e_{1}^{h}\Delta^{h}x_{1}}=1-\frac{e_{1}^{c}}{e_{1}^{h}},$
(17)

recovering the well-known expression from reversible thermodynamics [2, 5].

Figure 3: Gas-piston system with heat port.

###### Example 3

Consider a gas-piston system with incoming entropy flow (due to incoming heat)
as in Fig. 3. The system can be considered as a precursor to the Stirling
engine, cf. [13]. Its port-Hamiltonian description is given by

$\begin{bmatrix}\dot{S}\\\ \dot{V}\\\
\dot{\pi}\end{bmatrix}=\begin{bmatrix}0&0&0\\\ 0&0&A\\\
0&-A&0\end{bmatrix}\\!\\!\begin{bmatrix}\dfrac{\partial H}{\partial
S}\\\\[7.59995pt] \dfrac{\partial H}{\partial V}\\\\[7.59995pt]
\dfrac{\partial H}{\partial\pi}\\\ \end{bmatrix}+\begin{bmatrix}1&0\\\ 0&0\\\
0&1\end{bmatrix}\\!\begin{bmatrix}u_{1}\\\ u_{2}\end{bmatrix},$ (18)

with Hamiltonian $H(S,V,\pi)=U(S,V)+\frac{1}{2m}\pi^{2}$ and corresponding
natural outputs

$\displaystyle y_{1}$ $\displaystyle=\frac{\partial U}{\partial S}\ (=T),$
$\displaystyle y_{2}$ $\displaystyle=\frac{\pi}{m}.$

Here, $S$ represents entropy, $T$ temperature, $V$ the volume of the gas
enclosed by the piston, $A$ the area of the piston, and $\pi$ the momentum of
the piston with mass $m$. Finally, $U(S,V)$ is the internal energy of the gas,
$u_{1}$ the entropy flow, $y_{1}$ the temperature (so that $y_{1}u_{1}$ is the
heat flow entering the gas), $u_{2}$ is the external force on the piston, and
$y_{2}$ its velocity (so that $-y_{2}u_{2}$ is the mechanical work done by the
system). This is a port-Hamiltonian system of the form (13) with
$R_{1}=R_{2}=0$. Thus for $u_{1}$ such that either $S$ is constant (isentropic
curves), or $y_{1}=T$ is constant (isothermals), the system is cyclo-passive
at its mechanical port. In order to perform mechanical work in a cyclic manner
one needs at least two temperatures $T_{h}$ (hot) and $T_{c}$ (cold).

By using a Carnot cycle consisting of an isothermal at $T_{h}$, adiabatic,
isothermal at $T_{c}$, and finally an adiabatic to return to the original
state $(S,V,\pi)$ the mechanical work delivered to the surrounding is

$-\oint y_{2}(t)u_{2}(t)dt=\oint y_{1}(t)u_{1}(t)dt=Q_{h}+Q_{c},$ (19)

where $Q_{h}$ is the heat entering the system during the isothermal at
temperature $T_{h}$, and $-Q_{c}$ is the heat leaving the system during the
isothermal at temperature $T_{c}$.

###### Example 4

Consider an electromagnetic actuator as depicted in Fig. 4. Such device
converts electrical energy into mechanical motion (work).

Figure 4: Electromagnetic actuator. [Note that in practice a diode must be
placed across the inductor terminals.]

The port-Hamiltonian description (with $q$ and $p$ the displacement and the
momentum of the armature $m$, respectively, and $\varphi$ the magnetic flux of
the coil) is given as

$\displaystyle\begin{bmatrix}\dot{\varphi}\\\ \dot{q}\\\ \dot{p}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}0&0&0\\\ 0&0&1\\\
0&-1&0\end{bmatrix}\\!\\!\begin{bmatrix}\dfrac{\partial
H}{\partial\varphi}\\\\[7.59995pt] \dfrac{\partial H}{\partial
q}\\\\[7.59995pt] \dfrac{\partial H}{\partial
p}\end{bmatrix}+\begin{bmatrix}1&0\\\ 0&0\\\
0&1\end{bmatrix}\\!\begin{bmatrix}u_{1}\\\ u_{2}\end{bmatrix},$ (20)
$\displaystyle y_{1}$ $\displaystyle=\dfrac{\partial H}{\partial\varphi}\
(=I),$ $\displaystyle y_{2}$ $\displaystyle=\dfrac{\partial H}{\partial p},$

where $u_{1}$ is the supplied voltage and $y_{1}$ the associated current,
while $u_{2}$ is a mechanical force and $y_{2}$ the velocity of the armature.
This is a port-Hamiltonian system of the form (13), with the mechanical and
electromagnetic part of the system coupled via the Hamiltonian

$H(\varphi,q,p)=\frac{\varphi^{2}}{2L(q)}+\frac{p^{2}}{2m},$

where the electrical inductance ${L(q)>0}$ depends inversely on the horizontal
mechanical displacement.

Note that the system (20) has the same structure as the gas-piston system
(18). Thus, (20) is one-port cyclo-passive at the mechanical port, and a
similar Carnot cycle can be used in order to convert energy from the
electrical port to the mechanical port. Let $V_{a}>V_{b}$. The four processes
described in Subsection III-B are represented as follows:

  1. 1.

Switch $s_{a}$ is closed and $s_{b}$ is open. The coil absorbs an amount of
energy, say $E_{a}$, from the upper source $V_{a}$, while the current
${I=I_{a}}$ remains constant. This results in a decreasing $q$ and a certain
amount of external work done by the armature (and port 2).

  2. 2.

The coil is disconnected from the voltage source (both switches open) and
undergoes an ‘adiabatic’ transformation in which the magnetic flux remains
constant and the current $I$ decreases from $I_{a}$ to $I_{b}$ as the armature
recedes further to the coil.

  3. 3.

Switch $s_{b}$ is now closed and ${I=I_{b}}$. The armature is turning back to
its initial position with energy, say $E_{b}$, being transferred to the lower
voltage source $V_{b}$.

  4. 4.

Both switches open and ${I=I_{b}}$. The armature displaces to change the
inductance $L(q)$ until the current equals ${I=I_{a}}$. The magnetic flux
remains constant.

Now, since the energy

$E_{a}=\int\limits_{\varphi_{a}}^{\varphi_{b}}I_{a}d\varphi=I_{a}(\varphi_{b}-\varphi_{a})$

and, likewise, $E_{b}=-I_{b}(\varphi_{b}-\varphi_{a})$, we get that

$\frac{E_{b}}{E_{a}}=-\frac{I_{b}}{I_{a}}.$

Hence, the efficiency of the system in Fig. 4 equals

$\frac{E_{a}+E_{b}}{E_{a}}=1+\frac{E_{b}}{E_{a}}=1-\frac{I_{b}}{I_{a}},$

which is analogous to the efficiency of the Carnot-cycle (17).

##  IV ENERGY CONVERSION BY DIRECT INTERCONNECTION AND ENERGY-ROUTERS

The essential assumption in (13) is the absence of off-diagonal blocks in the
$J$-matrix. Since $J$ captures the ‘topology’ of the system, this means that
in port-Hamiltonian systems of the form (13) there is no direct topological
connection between the dynamics of the sub-vectors $x_{1},x_{2}$. Instead, the
interaction between the two subsystems is via the Hamiltonian $H(x_{1},x_{2})$
(and to a lesser extent through possible dependence of $J_{1}$ and $J_{2}$ on
$x_{1},x_{2}$, and similarly for $\mathcal{R}_{1},\mathcal{R}_{2}$). In this
section, we consider an example where we do have non-zero off-diagonal terms
(the heat exchanger), and two examples where off-diagonal elements are
introduced through the use of feedback.

###  IV-A Heat Exchanger

Consider a heat exchanger comprising two heat reservoirs connected via a
conducting wall; see Fig. 5.

Figure 5: Heat exchanger with temperatures $T_{1}$ and $T_{2}$.

The dynamics is given as

$\displaystyle\begin{bmatrix}\dot{S}_{1}\\\ \dot{S}_{2}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}0&\\!\\!\\!\\!\lambda\frac{E_{2}^{\prime}-E_{1}^{\prime}}{E_{1}^{\prime}E_{2}^{\prime}}\\\
\lambda\frac{E_{1}^{\prime}-E_{2}^{\prime}}{E_{1}^{\prime}E_{2}^{\prime}}&0\end{bmatrix}\\!\\!\begin{bmatrix}\dfrac{\partial
E}{\partial S_{1}}\\\\[6.60004pt] \dfrac{\partial E}{\partial
S_{2}}\end{bmatrix}+\begin{bmatrix}1&0\\\
0&1\end{bmatrix}\\!\begin{bmatrix}u_{1}\\\ u_{2}\end{bmatrix},$ (21)
$\displaystyle y_{1}$ $\displaystyle=E_{1}^{\prime}(S_{1}),\quad
y_{2}=E_{2}^{\prime}(S_{2}),$

where $\lambda$ is Fourier’s conduction coefficient, $E_{1}(S_{1})$ and
$E_{2}(S_{2})$ are the (thermal) energies of heat reservoirs $1$ and $2$ with
temperatures $E_{1}^{\prime}(S_{1})$ and $E_{2}^{\prime}(S_{2})$, and

$E(S_{1},S_{2})=E_{1}(S_{1})+E_{2}(S_{2})$

represents the total energy of the heat exchanger. Furthermore, $u_{1}$ and
$u_{2}$ are the external entropy flows entering the reservoirs $1$ and $2$, so
that $y_{1}u_{1}$ and $y_{2}u_{2}$ are the external heat flows entering the
two reservoirs. The equations (21) constitute a quasi port-Hamiltonian
system.222‘Quasi’ since the $J$-matrix does not directly depend on the state
variables $(S_{1},S_{2})$, but through the temperatures
$T_{1}=E_{1}^{\prime}(S_{1})$ and $T_{2}=E_{2}^{\prime}(S_{2})$.

Although (21) is a thermodynamic system, it is not in the form (13), and,
indeed, heat flow occurs from reservoir $1$ to reservoir $2$ for constant
$T_{1}$ as long as $T_{1}>T_{2}$. As an additional property we note that the
total entropy $S_{1}+S_{2}$ is always non-decreasing.

###  IV-B Energy Transfer by Energy-Routing

Consider two cyclo-lossless port-Hamiltonian systems

$\Sigma_{i}:\left\\{\begin{aligned} \dot{x}_{i}&=J_{i}(x_{i})\dfrac{\partial
H_{i}}{\partial x_{i}}(x_{i})+g_{i}(x_{i})u_{i},\\\
y_{i}&=g_{i}^{\top}(x_{i})\dfrac{\partial H_{i}}{\partial
x_{i}}(x_{i}),\end{aligned}\right.$

with $i\in\\{1,2\\}$. The two systems may be coupled to each other by the
output feedback (an elementary version of the Duindam-Stramigioli energy-
router [1, 8])

$\begin{bmatrix}u_{1}\\\
u_{2}\end{bmatrix}=\begin{bmatrix}0&-y_{1}y_{2}^{\top}\\\
y_{2}y_{1}^{\top}&0\end{bmatrix}\\!\begin{bmatrix}y_{1}\\\
y_{2}\end{bmatrix}+\begin{bmatrix}v_{1}\\\ v_{2}\end{bmatrix},$

with new inputs $v_{1}$ and $v_{2}$. Then, the closed-loop system
$\Sigma_{1}\circ\Sigma_{2}$ is a lossless port-Hamiltonian system with
Hamiltonian

$H(x_{1},x_{2})=H_{1}(x_{1})+H_{2}(x_{2}),$

but clearly not of the form (13). Due to the special form of the output
feedback, we have that

$\displaystyle\frac{d}{dt}H_{1}$
$\displaystyle=-||y_{1}||^{2}||y_{2}||^{2}+y_{1}^{\top}v_{1},$
$\displaystyle\frac{d}{dt}H_{2}$
$\displaystyle=||y_{1}||^{2}||y_{2}||^{2}+y_{2}^{\top}v_{2}.$

Hence, if energy is pumped into $\Sigma_{1}\circ\Sigma_{2}$ through port $1$,
then the stored energy in $\Sigma_{2}$ will increase—enabling its release
through port $2$.

###  IV-C Energy Transfer Using IDA-PBC

Consider again the electromagnetic actuator (20). In order to establish an
effective topological coupling between the electrical and mechanical
subsystems, we can try to enforce a coupling between the magnetic flux and the
momentum; thus realizing a desired interconnection matrix of the form

$J_{d}=\begin{bmatrix}0&0&\alpha\\\ 0&0&1\\\ -\alpha&-1&0\end{bmatrix},$ (22)

where $\alpha$ is a constant or function to be defined.

The easiest way to accomplish this is when the second input is available for
direct manipulation. Indeed, introducing the feedback

$\begin{bmatrix}u_{1}\\\ u_{2}\end{bmatrix}=\begin{bmatrix}0&\alpha\\\
-\alpha&0\end{bmatrix}\\!\begin{bmatrix}y_{1}\\\
y_{2}\end{bmatrix}+\begin{bmatrix}v_{1}\\\ v_{2}\end{bmatrix},$

with new inputs $v_{1}$ and $v_{2}$, readily replaces the $J$-matrix in (20)
by the interconnection matrix (22). However, usually only the first input is
available for control.

On the other hand, invoking the IDA-PBC methodology [7], and selecting a state
feedback $u_{1}=\alpha(\varphi,q)\frac{p}{m}+v_{1}$, with

$\alpha(\varphi,q)=\frac{1}{4}\frac{L^{\prime}(q)}{L(q)}\varphi,$

and $u_{2}=v_{2}$, yields (see the Appendix for details)

$\displaystyle\begin{bmatrix}\dot{\varphi}\\\ \dot{q}\\\
\dot{p}\end{bmatrix}=\begin{bmatrix}0&0&\alpha(\varphi,q)\\\ 0&0&1\\\
-\alpha(\varphi,q)&-1&0\end{bmatrix}\\!\\!\begin{bmatrix}\dfrac{\partial
H_{d}}{\partial\varphi}\\\\[7.59995pt] \dfrac{\partial H_{d}}{\partial
q}\\\\[7.59995pt] \dfrac{\partial H_{d}}{\partial
p}\end{bmatrix}+\begin{bmatrix}1&0\\\ 0&0\\\
0&1\end{bmatrix}\\!\begin{bmatrix}v_{1}\\\ v_{2}\end{bmatrix},$

in which the total stored energy is modified into

$H_{d}(\varphi,q,p)=\frac{\varphi^{2}}{L(q)}+\frac{p^{2}}{2m}.$

This (partially) controlled system is clearly not in the form (13) anymore,
and a direct interaction between the mechanical and magnetic subsystem is
achieved through the lossless modulation $\alpha(\varphi,q)$. Moreover, the
magnetic energy storage is doubled, enabling an increased energy conversion
rate.

##  V CONCLUSIONS AND OUTLOOK

Energy conversion in physical systems is related to the structure of the
$J$-matrix in its port-Hamiltonian formulation. In particular, if the off-
diagonal blocks of $J$ are zero we generally have one-port cyclo-passivity
[12], and need more than one value of the output at port $1$ in order to
convert energy from port $1$ to port $2$. This may be achieved by a direct
generalization of the Carnot cycle for heat engines. Another strategy may be
to use feedback in order to introduce non-zero off-diagonal blocks.

Among the many open problems we mention:

  * •

Consider a port-Hamiltonian system in the form (13). Can we generalize Carnot
cycles to more general cyclic processes, while retaining a notion of
efficiency?

  * •

How to formulate optimal energy conversion from one port to another?

  * •

The physical energy $H$ of any port-Hamiltonian system is somewhere ‘in
between’ $S_{ac}$ and $S_{rc}$. If we choose another storage function this
will generally lead to a different $J$-matrix (as well as different
dissipation structure $\mathcal{R}$). How to exploit this for energy
conversion?

## References

  * [1] V. Duindam, S. Stramigioli, “Port-Based Asymptotic Curve Tracking for Mechanical Systems,” Eur. J. Control, 10(5), 411–420, 2004.
  * [2] E. Fermi, Thermodynamics, Prentice-Hall, 1937 (Dover edition 1956).
  * [3] D.J. Hill, P.J. Moylan, “Cyclo-Dissipativeness, Dissipativeness, and Losslessness for Nonlinear Dynamical Systems,” Technical Report No. EE7526, Univ. of Newcastle, 1975.
  * [4] D.J. Hill, P.J. Moylan, “Dissipative Dynamical Systems: Basic Input-Output and State Properties”, Journal of the Franklin Institute, 309(5):327-357, 1980.
  * [5] D. Kondepudi and I. Prigogine, Modern Thermodynamics; From Heat Engines to Dissipative Structures, 2nd edition, Wiley, 2015.
  * [6] C. Lanczos, The Variational Principles of Mechanics, 4th Edition, Dover Publications Inc., 1986.
  * [7] R. Ortega, A.J. van der Schaft, I Mareels, and B. Maschke, “Putting Energy Back in Control,” IEEE Control Systems Magazine, Vol. 21, pp. 18–33, 2001.
  * [8] A. Sanchez-Squella, R. Ortega, R. Grino, S. Malo, “Dynamic Energy Router,” IEEE Control Systems Magazine, Dec. 2010, pp. 72–80.
  * [9] A.J. van der Schaft, $L_{2}$-Gain and Passivity Techniques in Nonlinear Control, 3rd Edition, Springer International, 2017.
  * [10] A.J. van der Schaft, “Cyclo-Dissipativity Revisited,” IEEE Trans. Autom. Contr. (Early Access; DOI: 10.1109/TAC.2020.3013941), 2020.
  * [11] A.J. van der Schaft and D. Jeltsema, Port-Hamiltonian Systems Theory: An Introductory Overview, Now Publishers, 2014.
  * [12] A.J. van der Schaft and D. Jeltsema, “Limits to Energy Conversion,” arXiv:2006.15953, submitted for publication, 2020.
  * [13] C. Mueller-Roemer and P. Caines, “An Isothermal Energy Function State Space Model of a Stirling Engine,” IFAC-PapersOnLine 48(11), 634–639, 2015.
  * [14] J.C. Willems, “Dissipative Dynamical Systems, Part I: General Theory,” Arch. Rat. Mech. and Analysis, 45(5):321–351, 1972.
  * [15] J.C. Willems, “Qualitative Behavior of Interconnected Systems,” Annals of Systems Research, 3:61–80, 1973.

## APPENDIX

The IDA-PBC methodology [7] entails to solving the so-called matching equation

$J_{d}(x)\frac{\partial H_{a}}{\partial x}=-J_{a}(x)\frac{\partial H}{\partial
x}+G(x)u,$

with $J_{a}(x)=J_{d}(x)-J(x)$. For the system (20), desired interconnection
structure (22), and $u_{1}=\beta(\cdot)+v_{1}$, this amounts to solving

$\displaystyle\alpha(\cdot)\frac{\partial H_{a}}{\partial p}$
$\displaystyle=-\alpha(\cdot)\frac{\partial H}{\partial p}+\beta(\cdot),$
$\displaystyle\frac{\partial H_{a}}{\partial p}$ $\displaystyle=0,$
$\displaystyle-\alpha(\cdot)\frac{\partial
H_{a}}{\partial\varphi}-\frac{\partial H_{a}}{\partial q}$
$\displaystyle=\alpha(\cdot)\frac{\partial H}{\partial\varphi}.$

Hence, selecting $H_{a}(\varphi,q)=\frac{1}{2L(q)}\varphi^{2}$, we have that

$\frac{1}{2}\frac{L^{\prime}(q)}{L^{2}(q)}\varphi^{2}=2\alpha(\cdot)\frac{\varphi}{L(q)}\Rightarrow\alpha(\varphi,q)=\frac{1}{4}\frac{L^{\prime}(q)}{L(q)}\varphi,$

together with the full state feedback control

$\beta(\varphi,q,p)=\alpha(\varphi,q)\frac{\partial H}{\partial p},$

and energy storage $H_{d}(\varphi,q,p)=H(\varphi,q,p)+H_{a}(\varphi,q)$.


# Using storytelling to foster the teaching and learning of gravitational
waves physics at high-school

M Tuveri13, A Steri23 and D Fadda4 1 Dipartimento di Fisica, Università di
Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy  2 Dipartimento di
Ingegneria Meccanica, Chimica e dei Materiali, Università di Cagliari, 09123
Cagliari, Italy  3 Istituto Nazionale di Fisica Nucleare, Sezione di Cagliari,
Cittadella Universitaria, 09042 Monserrato, Italy  4 Dipartimento di
Pedagogia, Psicologia, Filosofia, Università di Cagliari, 09123 Cagliari,
Italy<EMAIL_ADDRESS>
###### Abstract

Studies in Physics Education Research show that interdisciplinary approaches
in education foster students’ motivation, creativity, curiosity, and interest
in physics. We discuss their features and potential role in bringing
contemporary physics topics to high school, and how to use them to integrate
formal educational programs. We make an explicit example of the use of
storytelling and theatrical techniques to introduce secondary school students
to black holes and gravitational waves topics. The activity has been designed
by the Educational Division of the Physics Department at the University of
Cagliari. Participants were 200 high-school students (17 to 19 years old) from
five schools (scientific, humanities) in Sardinia. A measure of the efficacy
in the use of artistic tools to communicate and teach the proposed subjects
has been done utilizing a research questionnaire. We collected 76 answers.
Results show that our methodology is useful to introduce students to
contemporary physics themes, fostering their interest and learning of such
contents. Students from humanities significantly appreciated more the use of
poetry and artistic tools than their scientific peers. Finally, we discuss the
potentiality of our approach in orientating students towards a STEAM (STEM and
Arts) career.

††:

Keywords: informal learning; interdisciplinary; storytelling; STEAM; general
relativity; gravitational waves

##  1 Introduction

Offering an interdisciplinary vision of science to high school students and
teachers is becoming a common trend of learning (see [1, 2, 3] and refs
therein). Interdisciplinary, in its most simple form, constitutes an
appreciation of different discipline fields and the value of communicating
across disciplinary boundaries to find ways to work together. As emphasized in
[4], playing and learning contents and methods from different disciplines
bring educators and students to develop new forms of scientific reasoning and
toolkits. Learning experiences and skills acquired through interdisciplinary
apply physics fundamentals to real-world solutions making a connection between
the world of research and society. To show how science is evolving and to
provide new instruments to learn science and physics in an enlarged context,
mixing knowledge, techniques, and methods from different disciplines should be
part of science and education curricula, developing an integrated model of
learning and teaching [5]. Interdisciplinary approaches in education also
enhance students’ engagement in physics [6].

In general, this kind of educational experiments are carried out in informal
context, such as out-of-school time, and outdoor activities, science outreach
labs, visits to museums, summer camps, where they keep in contact with science
through different media and activities [7, 8, 9, 10]. Researches show that
teaching science in informal contexts increases experience excitement,
interest, and motivation to learn about phenomena in the natural and physical
world, improves scientific awareness and literacy [11, 12, 13, 14, 15],
bolstering motivation to learn science and physics [16, 17, 18]. It also helps
them in building a “culturo-scientific thinking”, that is learning how to
think scientifically and creatively, with an understanding of the evolution of
the nature of science [19].

Integrating the good practices of learning in informal contexts even in formal
curricula at school could bring some benefits in students’ learning of
science, engagement and motivation. Mixing arts and science seem to be very
promising in this direction [6, 7]. Research results show that arts,
narrative, and drama help students in reflecting on scientific concepts and
science evolution from epistemological and historical perspectives. It
promotes peer interaction at school and interaction among school, family, and
society [20], also preventing students from school abandon [21]. Moreover, it
helps teachers better understand what students are thinking [22]. This kind of
approach involves emotional involvement to foster students’s development of
scientific imagination and creativity, allowing personal learning styles, and
cognitively mediating in approaching physics (see [6] and refs therein).
Moreover, when followed by minds-on activities, such as a debate, students re-
elaborate what they experienced and/or re-propose contents in a personal way,
bettering their learning process [11, 12], developing their critical thinking
skills. These activities also orientate students towards a STEM career [23,
24].

To show how science is evolving and to provide new instruments to learn
science and physics in an enlarged context, mixing knowledge, techniques, and
methods from different disciplines should be part of science and education
curricula, developing an interdisciplinary and integrated model of learning
and teaching [1, 2, 3, 5, 25]. Nevertheless, sometimes secondary science
teachers face some difficulties in inspiring their classes. The use of stories
can help improve teaching and students’ learning and physics is full of
stories that can afford for this job. The use of narrative and stories to
teach physics has some well-recognized benefits in education, presenting
science in a more “coherent, memorable, and meaningful format in order to
interest and engage pupils, as well as to unify the curriculum and to preserve
a sense of the bigger picture” (see [26], p.121). Storytelling can be used as
a stimulus to engage, excite, and emotionally involve experience from which to
learn, also bettering students’ capability to memorize concepts [26, 27].
Millar and Osborne [28] enlightened the role of sustaining and developing the
curiosity of young people attending a science curriculum, fostering a sense of
wonder, enthusiasm, and interest in science. It seems that stories could be a
way of achieving this sense of wonderment, enthusiasm, and interest in
science, provided they are relevant, fun, and interesting [26].

Despite its role in education is widely recognized in many contexts and, in
particular, in science education [27, 29, 30], the use of storytelling in
school is not so common. Since storytelling has its genesis in arts, someone
speculates that is not recognized as a rigorous pedagogical methodology to
teach and learn [31]. This idea can be mediated by the fact that instructional
approaches that are perceived to be subjective and less rigorous will only be
adopted with cautious reservation, basically for science communication [32],
not for science lessons [33]. However, as noted in [34], teaching itself is,
in reality, nothing more than an evolved and extremely codified form of
storytelling, the more effective the more it is linked to the telling of real
stories that highlight from time to time crucial elements of the discipline
being learned.

Studies in Physics Education Research (PER) underly that storytelling fosters
pupils’ creativity and positive attitude towards science [35]. It also offers
a methodology guide to teach science in the laboratory. For example, it can be
used to introduce a scientific problem in the form of a story for students to
solve, possibly incorporating real-life scientific issues that researchers
face in their everyday life [26, 32]. Storytelling has always been a medium to
transfer traditional knowledge, beliefs, values, and practices over time and
its core relies on sound education pedagogy [36, 37, 38]. Formal programs in
school included aspects of storytelling that were used to raise students’
interest and to encourage students to actively participate in learning [39],
but recently new forms of digital storytelling seem to replace them [34, 40,
41]. Nevertheless, role-play, dramatization, and storytelling will continue to
hold promise for the creation of exciting, engaging, meaningful, and rewarding
classroom experiences even in a technological environment storytelling will
remain a versatile option available to teachers to introduce science in the
classroom.

The specific focus of this work is on the use of storytelling to bring
contemporary physics topics in high school and, in particular, black hole and
gravitational waves physics. Specifically, the aim of this work is to explore
the effect of storytelling on students’ levels of classroom participation,
motivation, and interest in the proposed topics. We also meant to measure
their engagement and, most interestingly, their views about the effectiveness
of storytelling as a teaching/learning strategy in the science classroom. The
research can give instructors a methodological tool to encourage them to bring
these topics to school, using storytelling to optimum advantage in science.
This introduces students to current trends in research, trying to bypass
content-related difficulties (both physical and mathematical), but still
making them explore our universe with inquiry and minds-on activities,
improving their motivation, curiosity, and interest in physics.

In this paper, we address our research goal by reporting from an educational
program developed by the PER group at the University of Cagliari which makes
use of storytelling and theatrical techniques (a monologue) to bring general
relativity and gravitational-related topics such as gravitational waves and
black hole physics at high-school. We briefly illustrate the physical
background and the design of the educational program. The activity involved
200 students (17 - 19 years old) from five schools in Sardinia. We measured
the efficacy of our activity according to our research goals by using a
questionnaire. We report and discuss our results, pointing the potentiality of
our approach out in orientating students towards a STEAM (STEM and arts)
career.

##  2 Physics Framework and the monologue

In the context of contemporary physics, General Relativity (GR) gives the
possibility to carry out some educational experimentation taking into account
the use of stories and storytelling. Indeed, in a broader sense, GR topics are
related to the world views held by students, such as our universe, stars, and
planets, thus providing conceptual, theoretical, and experimental means for
bridging the gap between the teaching of science and the teaching of
humanistic subjects. This is true, for example, for concepts pertaining to
cosmology like the Big Bang, which connects physics with philosophy [42, 43].
Students’ first approach to these contents is different: movies,
documentaries, media, and TV shows, all of which contribute to shaping their
conceptions about astrophysical phenomena. The work of experts should be that
of guiding them toward the understanding of our universe, stimulating the
debate, and offering a modern vision of phenomena around us [7, 12]. This also
offers the possibility to debate about what is science and what is not, what
can be measurable and what is not, thus re-enforcing their critical thinking
[43].

Let us make an explicit example for the sake of completeness. The concepts of
“Big Bang” and “black hole” are metaphors, analogies used by physicists to
explain something invisible whose physical meaning is still obscure, but which
is useful to imagine a given phenomenon. They are a linguistic expedient that
represents a way to move from a provisional ad hoc model (for example an
explosion or a dark, deep hole, respectively) to a permanent theory (GR) [6,
44, 25]. From the educational point of view, they have the potential to grasp
and hold students’ attention while easily encouraging them to participate,
share, and collaborate to learn. Telling the universe by means of such
metaphors is not only an artistic and linguistic tool, but it has also an
educational purpose, giving the possibility to teachers to explore
contemporary physics topics in class, living the contemporaneity of research
and its methodologies.

More in detail, the word “black hole”, invented by John A. Wheeler [45],
describes the final state of matter when a massive star (with more than 1.4
Solar masses) explodes. Certainly, there are not any “holes” in our universe,
but the idea of an obscure region of spacetime where everything falls down and
objects cannot escape from there is pictorially represented by that metaphor.
Nevertheless, even if we cannot observe them, we can study them by the motion
of stars and matter surrounding them or, by detecting the gravitational waves
produced by the merging of two black holes. Indeed, in 2016, after 100 years
of Einstein paper [46], the LIGO and Virgo collaborations first announced the
discovery of gravitational waves, ripples of space-time traveling all around
the universe at the speed of light produced by violent astrophysical events
[47]. An interesting feature of such waves is that their effect is to modify
space and time, measuring changes in length much smaller than the diameter of
an atomic nucleus (for a merging of black holes with masses of the order of
ten Solar masses, the change in length is of the order of $10^{-21}$ cm, see
[47]). Moreover, in 2019, researchers from the Event Horizon Telescope
collaboration published a “picture” of the first image of the shadow of the
supermassive black hole at the center of the M87 galaxy [48], replicated in
2022 with the first image of the black hole at the center of our galaxy, the
Milky Way [49].

The search for understanding our universe is still ongoing and researchers are
looking for new technologies and devices to study its features, mainly in its
initial stages. Indeed, when stars were not formed at all, no light signal
could be detected from satellites or telescopes on the ground, so standard
astronomical observation cannot give us any information about the first
million years of our universe. At that time, no matter or light signal was
there and, basically, only primordial quantum fluctuations populated the
universe, probably growing, collapsing, and forming primordial black holes.
Thus some primordial gravitational signal could, in principle, be detectable
[50, 51]. This is, for example, one of the goals of the Einstein Telescope
(ET) experiment [52], a ground-based third-generation gravitational wave
interferometer aimed to explore the universe with gravitational waves up to
cosmological distances. ET has a reference configuration based on a triangular
shape consisting of three nested detectors with 10 km arms, where each
detector has a “xylophone ” configuration made of an interferometer tuned
toward high frequencies, and an interferometer tuned toward low frequencies
and working at cryogenic temperature [51]. Its scientific output is related to
compact binary coalescences, multi-messenger astronomy, and stochastic
backgrounds. The detector will be capable of observing the entire Universe
using gravitational waves aiming to increase a factor ten the sensitivity of
previous generation detectors [53]. It could be operating in the mid-2030s in
Europe, and one of the site candidates is in Sardinia (Italy). This site has
peculiar features (very low anthropomorphic, electromagnetic, and geophysical
noises in the range of frequencies at which ET will operate) which makes it
one of the best candidates to host the infrastructure [54].

###  2.1 The monologue

The possibility for Sardinia to host the experiment motivates us to design a
suitable activity to bring gravitational waves physics contents and ET physics
to high school. From a methodological point of view, we use storytelling and
theatrical techniques such as a monologue. The latter is called “Viaggio in
una storia lunga 14 miliardi di anni” (“A 14 billions years old journey”). The
story time is set in the middle of 2030 when ET should be operating. The
protagonist of the story is a physicist who is currently part of the ET
collaboration, working on gravitational wave physics. He is in the control
room of ET in Sardinia, checking for all the parameters coming from the
detector in the underground caves, from optics to lasers. While he is thinking
at his past, his career and his life as a researcher, his studies, and his
work in Sardinia, he is also checking for possible gravitational waves signals
appearing on a monitor in front of him. It was at that moment that he started
traveling all around the universe, from Sardinia to the origin of the
universe. In this 14 billion years journey back in time, he describes the
relativistic effects due to traveling back in time at the speed of light, the
structures in our universe and their formation, unless he encounters a
mysterious phenomenon he realizes to be a black hole merging. This offers him
the possibility to tell the physics of tidal disruption and the consequent
formation and propagation of gravitational waves. Finally, he travels back
home surfing such gravitational wave. Once went back to the control room, he
realizes having discovered a primordial gravitational wave signal.

The monologue (in Italian) can be found at the following link:
https://www.youtube.com/watch?v=SMaNFRKdy4c.

Figure 1: The panel shows students’ distribution as a function of school.

Figure 2: The panel shows students’ distribution as a function of the classes
(left) and the gender (right).

##  3 Methods

The activity was held in Sardinia, Italy, in 2023 along the lines of the
informal learning educational program at the Physics Department of the
University of Cagliari and the Cagliari Division of the National Institute of
Nuclear Physics (INFN). The activity involved 200 students (17-19 years old)
from five high schools in all the Region (4 scientific lyceums, 1 humanities).
In Italy high school lasts five years, students participating to the activity
were attending third to fifth year classes.

The activity was organized as follows: the first 20 minutes was dedicated to
the monologue, followed by a 20-minute session where the a researcher
explained gravitational-related contents and ET physics (gravity according to
general relativity, black holes, gravitation waves formation and propagation,
detection of gravitational waves with ET, cosmology). Finally, 20 minutes were
left for debate, discussions, and a question and answers section.

During the monologue, some poems both in Italian and Sardinian language 111The
Sardinian language, with its several variants all along the Island, descends
from Latin with some words remaining from the ancient Nuragic language, and it
is currently spoken in Sardinia. It is generally non longer taught in school,
a part from some specific project pertaining mostly primary schools, but
people learn it mostly at home. were used to tell some facts concerning
physics. This served as an artistic tool and also to reinforce the identity of
the character which is also related to the place where ET is located in the
story.

To address the aim of this work, which is to explore the effect of
storytelling on students’ levels of classroom participation, motivation, and
interest in the proposed topics, we built a specific research questionnaire.
In this way, we could quantitatively measure their engagement and their
perception about the efficacy of storytelling as a teaching/learning strategy
in the science classroom. We identified five dimensions corresponding to eight
aspects to investigate: general information (demography - 5 items; preliminary
knowledge on the proposed topic - 6 items); interest in STEM and engagement
(interest and passion in STEM - 2 items; engagement - 4 items); communication
and artistic tools (the monologue - 8 items; the poems - 5 items); motivation
(3 items); final remarks (8).

Students could answer using a 6-point Likert scale from 1 (completely
disagree) to 6 (completely agree). We conducted descriptive analysis of data.
After that, to determine whether there were statistically significant
differences between means of domains based on schools, classes, and gender we
carried out a multivariate analysis of variance (MANOVA). We also tested
correlations among the domains with the “r” of Pearson coefficient.
Reliability of the questionnaire were investigated in terms of internal
consistency by Cronbach’s alpha; mean value among the domains was 0.89 (higher
than the standard 0.8 threshold).

##  4 Results

We collected 76 answers, 23 from humanities and 52 from scientific high
schools students (1 missing answer), see Fig. 1. Considering the gender, 48
were females, 27 males, and 1 “other”. Concerning classes, 28 were attending
the third class, 39 the fourth, and 9 the fifth class, see Fig. 2. We first
show general results, then we will display school, class, and gender results.
After a general overview, we will focus on school, class, and gender results
by 75 students (removing the case with missing data; i.e., “other”).

Firstly, we asked students if before attending the activity they knew the ET
project: 45 answered “no”, 31 “yes”.

Figure 3: The panel shows students’ feedback on their level of preparation to
attend the activity.

Concerning the first macro domain, “general information”, we asked if
students’ preliminary knowledge was adequate to attend the activity: for 16
(21%) of them, their preparation was not adequate; 6 (8%) did not know; 29
(38%) answered that their preparation was slightly adequate; for 24 (32%)
students their preparation was adequate and 1 student reported to have a very
adequate preparation (see Fig 3).

Figure 4: The panel shows students’ interest in topics presented during the
activity. On the left, means and standard deviation; on the right, results on
the 6-point Likert scale for each topic. Figure 5: The panel shows means and
standard deviation (error bars) about students’ interest in the proposed
topics according to the type of school. Students rated using a 6-point Likert
scale, from 1 (completely disagree) to 6 (completely agree) Figure 6: The
panel shows means and standard deviation (error bars) about students’ interest
in the proposed topics according to the class. Students rated using a 6-point
Likert scale, from 1 (completely disagree) to 6 (completely agree) Figure 7:
The panel shows means and standard deviation (error bars) about students’
interest in the proposed topics according to their gender. Students rated
using a 6-point Likert scale, from 1 (completely disagree) to 6 (completely
agree)

We asked also for their interest in the topics we proposed. Results are shown
in Fig. 4. Concerning the type of school, results are shown in Fig. 5.
Accordingly, results concerning the class and the gender are shown in Figs 6
and 7.

Figure 8: The panel shows students’ interest in science communication.
Students rated using a 6-point Likert scale, from 1 (completely disagree) to 6
(completely agree).

Figure 9: The panel shows means and standard deviation (error bars) about
students’ interest in science communication according to the type of schools
(left) and classes (right). Students rated using a 6-point Likert scale, from
1 (completely disagree) to 6 (completely agree). Figure 10: The panel shows
means and standard deviation (error bars) about students’ interest in science
communication according to the gender. Students rated using a 6-point Likert
scale, from 1 (completely disagree) to 6 (completely agree).

Concerning STEM Engagement and interest (from now-on “STEM Engagement”), we
asked if students are interested in science communication, i.e. in outreach
and dissemination activities: 1 student (1.3%) is not interested at all, 12
(15.8%) are indifferent to such activities, 27 (35.5%) are slightly
interested, 23 (30.3%) are interested and 13 (17.1%) are very interested in
them, see Fig. 8. Concerning the schools and the classes means and standard
deviation are shown in Fig. 9. Finally, we also reported gender results in
Fig. 10.

Students also gave their feedback about the role of science communication and
outreach in society. In particular, we asked them if we need more science
communication and outreach activities at school: 65 (85.5%) students answered
“yes”, 1 (1.3%) said “no” and 10 (13.2%) “I do not know”. We also asked them
if we need more science communication and outreach activities locally, all
around the Region: 53 (69.7%) students rated “yes”, 3 (3.9%) answered “no” and
10 (26.3%) “I do not know”. Finally, we collected their feedback about the
need for more science communication and outreach activities in our society: 65
(85.5%) answered “yes”, 2 (2.6%) “no”, and 9 (11.8%) “I do not know”.

Figure 11: The panel shows means and standard deviation (error bars) about the
use of science communication as a tool to engage students in STEM. Figure 12:
The panel shows students’ feedback on the use of science communication as a
tool to engage students in STEM according to the 6-point Likert scale.

Figure 13: The panel shows means and standard deviation (error bars) about the
use of science communication as a tool to engage students in STEM according to
the type of school (above) and the classes (below).  Figure 14: The panel
shows means and standard deviation (error bars) about students’ interest in
science communication according to the gender.

We also investigated students’ feedback about the role of science
communication (outreach and dissemination) as a tool to learn science (a), to
orientate people in choosing (b) and pursuing (c) an academic career. Finally,
we also asked them if science communication drives their interest in science.
Students rated using a 6-point Likert scale, from 1 (completely disagree) to 6
(completely agree). Results are shown in Fig. 12. Results concerning means and
standard deviation for the four items are shown in Fig. 11. In Figs 13 and 14,
we show results divided per school, classes, and gender, respectively.

Figure 15: The panel shows means and standard deviation (error bars) about the
use of the monologue to engage students’ in physics. Students rated using a
6-point Likert scale, from 1 (completely disagree) to 6 (completely agree).

Figure 16: The panel shows means and standard deviation (error bars) about the
use of science communication as a tool to engage students in STEM according to
the type of school (above) and the classes (below).  Figure 17: The panel
shows means and standard deviation (error bars) about students’ interest in
science communication according to the gender. Students rated using a 6-point
Likert scale, from 1 (completely disagree) to 6 (completely agree).

Concerning the macro-domain related to the monologue, we investigated 8
different items to study the efficacy of the artistic tool on learning,
motivation, engagement, and the presence of research infrastructure in
Sardinia. The items were the following: a) I discovered new physics concepts
thanks to the monologue; b) The monologue fostered my interest in physics; c)
The monologue stimulated my creativity; d) The monologue is a useful tool to
communicate physics; e) The monologue is a useful tool to passionate people in
physics; f) The monologue brought me into researchers’ lives, making me live a
research-like experience; g) The monologue made me know about research
infrastructures in Sardinia: h) The monologue helped me in evaluating the
importance of local research (in Sardinia). Also in this case, students could
answer using a 6-point Likert scale, from 1 (completely disagree) to 6
(completely agree), see Table 1. Means and standard deviation, school, and
class results are shown in Figs. 15 and 16. Gender results are shown in Fig.
17.

The Monologue
---
|  Completely Disagree |  Disagree |  Indifferent |  Slightly Agree |  Agree |  Completely agree
I discovered new physics concepts thanks to the monologue |  1 (1.3%) |  8 (10.5%) |  11 (14.5%) |  32 (42.1%) |  14 (18.0%) |  10 (13.2%)
The monologue fostered my interest in physics |  1 (1.3%) |  12 (15.8%) |  12 (15.8%) |  27 (35.5%) |  16 (21.0%) |  8 (10.5%)
The monologue stimulated my creativity |  1 (1.0%) |  11 (14.5%) |  18 (23.7%) |  18 (23.7%) |  20 (26.7%) |  8 (10.5%)
The monologue is a usefull tool to communicate physics |  0 (0%) |  5 (6.6%) |  7 (9.2%) |  30 (39.5%) |  21 (27.6%) |  13 (17.1%)
The monologue is a usefull tool to passionate people in physics |  0 (0%) |  2 (2.6%) |  8 (10.5%) |  31 (40.8%) |  16 (25.0%) |  19 (25.3%)
The monologue brought me into researchers’ lives, making me live a research-like experience |  4 (5.3%) |  10 (13.2%) |  11 (14.5%) |  21 (27.6%) |  19 (25.0%) |  11 (14.5%)
The monologue made me know about research infrastructures in Sardinia |  2 (2.6%) |  2 (2.6%) |  7 (9.2%) |  29 (38.2%) |  25 (32.9%) |  11 (14.5%)
The monologue helped me in evaluating the importance of local research (in Sardinia) |  2 (2.6%) |  2 (2.6%) |  7 (9.2%) |  20 (26.3%) |  28 (36.8%) |  17 (22.3%)
Table 1: The table shows the number of answers (and corresponding percentage)
for each point of the Likert scale concerning students’ feedback about the
efficacy of the monologue on the selected domains. Figure 18: The panel shows
means and standard deviation (error bars) about the use of poems within the
monologue as an educational tool.

Figure 19: The panel shows means and standard deviation (error bars) about the
use of poems within the monologue as an educational tool according to the type
of school (above) and the classes (below).  Figure 20: The panel shows means
and standard deviation (error bars) about the use of poems within the
monologue as an educational tool according to the gender.

Concerning the use of poems as an educational tool, we asked students if a)
they liked the use of poetry within the monologue to talk about physics; b)
poems motivated them to carefully follow the performance, taking care of the
scientific contents of the monologue; c) poems and the use of poetry captured
their attention, fascinating them; d) poems are a useful instrument to
communicate science (physics); e) poems and the use of poetry stimulated their
creativity. Also in this case students rated items using a 6-point Likert
scale, from 1 (completely disagree) to 6 (completely agree). The means and
standard deviation for each item are shown in Fig. 18. We report results on
single items according to the Likert scale in Table 2. School, classes, and
gender results are shown in Figs. 19 and 20.

Poems
---
|  Completely Disagree |  Disagree |  Indifferent |  Slightly Agree |  Agree |  Completely agree
I liked the use of poetry within the monologue to talk about physics |  4 (5.2%) |  10 (13.1%) |  7 (9.2%) |  26 (34.2%) |  18 (23.7%) |  11 (14.5%)
Poems motivated me in carefully following the performance, taking care of the scientific contents of the monologue |  3 (3.9%) |  6 (7.9%) |  16 (21.1%) |  26 (34.2%) |  19 (25.0%) |  6 (7.9%)
Poems and the use of poetry captured my attention, fascinating me |  4 (5.2%) |  3 (3.9%) |  13 (17.1%) |  24 (31.6%) |  19 (25.0%) |  13 (17.1%)
Poems are a useful instrument to communicate science (physics) |  4 (5.2%) |  11 (14.5%) |  22 (28.9%) |  16 (21.0%) |  16 (21.0%) |  7 (9.2%)
Poems and the use of poetry stimulated my creativity |  6 (7.9%) |  6 (7.9%) |  19 (25.0%) |  20 (26.3%) |  20 (26.3%) |  5 (6.6%)
Table 2: The table shows the number of answers (and corresponding percentage)
for each point of the Likert scale concerning students’ feedback about the use
of poems as an educational tool. Figure 21: The panel shows means and standard
deviation (error bars) about students’ motivation in learning physics and the
proposed subjects after attending the activity.

Figure 22: The panel shows means and standard deviation (error bars) about
students’ motivation in learning physics and the proposed subjects after
attending the activity according to the type of school (above) and the classes
(below).  Figure 23: The panel shows means and standard deviation (error bars)
about students’ motivation in learning physics and the proposed subjects after
attending the activity according to the gender.

Concerning motivation, we investigated a) how the activity motivated students
to study physics; b) if after attending the activity, their interest in
physics grew; c) if the activity motivated them to study gravitational physics
related-contents in their future. Students rated items using a 6-point Likert
scale, from 1 (completely disagree) to 6 (completely agree). The means and
standard deviation for each item are shown in Fig. 21. We report results on
single items according to the Likert scale in Table 3. School, classes, and
gender results are shown in Figs. 22 and 23.

Motivation in learning physics
---
|  Completely Disagree |  Disagree |  Indifferent |  Slightly Agree |  Agree |  Completely agree
The activity motivated me in studying physics |  9 (11.8%) |  9 (11.8%) |  20 (26.3%) |  22 (28.9%) |  10 (13.2%) |  6 (7.9%)
The activity motivated me in studying physics |  3 (3.9%) |  10 (13.2%) |  20 (26.3%) |  14 (18.4%) |  20 (26.3%) |  9 (11.8%)
The activity motivated me in studying gravitational physics related-contents in the future |  4 (5.2%) |  5 (7.9%) |  18 (23.7%) |  19 (25.0%) |  16 (21.1%) |  13 (17.1%)
Table 3: The table shows the number of answers (and corresponding percentage)
for each point of the Likert scale concerning students’ feedback about the use
of poems as an educational tool.

Finally, when asked if they would participate again in similar activities, 66
students (86.8%) answered “yes”, 3 (3.9%) “no”, and 7 (9.2%) “I do not know”.
The overall feedback on the activity was as follows: 1 (1.3%) rated as bad, 4
(5.3%) were indifferent, 15 (19.7%) slightly good, 30 (39.5%) rated as good,
and 26 (34.2%) as very good.

###  4.1 Correlations and MANOVA

Concerning correlations, results are reported in Table 4. All of them are
significative for p $<$ 0.1 (p stands for p-value).

The results of the MANOVA concerning differences between means of dimensions
related to “Preliminary knowledges”, “STEM Engagement”, “The Monologue”,
“Poems”, and “Motivation” according to gender, school and classes showed a
significant multivariate difference only for the school (Wilkis lambda .704, F
$=$ 5.05, p $<$ 0.1).In particular, the differences are associate to “STEM
Engagement”, “The Monologue”, “Poems”. Concerning “STEM Engagement”, the means
are higher for scientific schools, whereas, means are higher for humanities in
the “Monologue“, and “Poems”domains, see Tables 5, 6, 7.

Correlations
---
|  |  Preliminary Knowledges |  STEM engagement |  Monologue |  Poems |  Motivation
Preliminary Knowledges |  Pearson correlation |  1 |  |  |  |
STEM engagement |  Pearson correlation |  .656 |  1 |  |  |
Monologue |  Pearson correlation |  .523 |  .483 |  1 |  |
Poems |  Pearson correlation |  .356 |  .320 |  .734 |  1 |
Motivation |  Pearson correlation |  .604 |  .600 |  .637 |  .564 |  1
Table 4: The table shows correlations between macro-domains investigated. All
of them are significative for p $<$ 0.1 .  STEM Engagement
---
| N | Mean | Std. Dev. | Std. Err.
Humanities | 23 | 4.26 | 0.92 | 0.19
Scientific | 52 | 4.54 | 0.79 | 0.11
Total | 75 | 4.46 | 0.84 | 0.09
Table 5: Means, standard deviation and standard error for the “STEM
Engagement”domain Monologue
---
| N | Mean | Std. Dev. | Std. Err.
Humanities | 23 | 4.48 | 0.96 | 0.20
Scientific | 52 | 4.11 | 0.97 | 0.13
Total | 75 | 4.23 | 0.97 | 0.11
Table 6: Means, standard deviation and standard error for the “Monologue
”domain Poems
---
| N | Mean | Std. Dev. | Std. Err.
Humanities | 23 | 4.41 | 0.99 | 0.21
Scientific | 52 | 3.68 | 1.21 | 0.17
Total | 75 | 3.91 | 1.19 | 0.14
Table 7: Means, standard deviation and standard error for the “Poems ”domain

##  5 Discussion

We have studied the use of storytelling and artistic tools to bring
contemporary physics topics and, in particular, black hole and gravitational
waves physics at high school. The aim of this work was to explore the effect
of storytelling on students’ levels of classroom participation, motivation,
and interest in the proposed topics. We also meant to measure their engagement
and, most interestingly, their views about the effectiveness of storytelling
as a teaching/learning strategy in the science classroom. To do so, we
designed a specific activity we proposed to 5 schools in Sardinia, Italy (200
students involved). The activity started with a monologue about the physics of
ET and its possible implementation in Sardinia. The monologue made use of
storytelling and poems, the latter being both in Italian and Sardinian
language and related to the physical content of the story. After that, a
20-minute session to explain the physics of ET and of gravitational waves was
done. In particular, we focused on gravity according to GR, black hole and
gravitational wave formation, cosmology, and the early universe as possibly
studied by ET. The activity ended with a 20-minute session of debate, where
students asked questions and raised their curiosity about the proposed topics.

To measure the efficacy of our activity we wrote a research questionnaire to
investigate specific domains. What emerges from our results is quite
encouraging in many aspects. Even if the majority of students did not know the
ET project, infrastructure and which physics will be studied, regardless of
the type of school or the class, they rated their average scientific
preparation as adequate to attend the activity. Means are slightly higher for
scientific and for the fifth class with respect to humanities and the third
and fourth classes, respectively, but there are not any significative
differences. This result is encouraging since it means that such topics can be
discussed at school even without specific preparation for them. If we take a
look at classes, students attending the third and fourth classes have a lower
mean than their colleagues in the fifth classes. This is probably due to the
fact that these contents are far away from the scholarly curriculum and that
their preparation in mathematics and physics was at a starting level. Students
did not know many concepts used during the monologue as the one of “field”
(e.g. electromagnetic field), or “wave”. Let us notice that during the
monologue no use of mathematics was done.

In particular, students’ higher interest was in black hole physics and
cosmology. These findings are confirmed also in the case of the type of
school, with means generally higher for humanities than the scientific. In the
case of classes, the general trend is respected (black holes and cosmology are
the preferred topics), and the means are higher for students attending the
fifth class than the others. However, students in the third class showed
higher means than their colleagues in the fourth class, even if there is not
any significant difference between the two. Concerning gender, males showed
slightly higher means than their female colleagues in all the items
investigated within this domain (no significant differences).

Students reported having a high interest in science communication, where we
intended outreach and dissemination activity, see Fig. 8. Also, in this case,
there are not any significant differences between schools, classes, and gender
(even if males exhibit a slightly higher mean on this item than females).
Science communication as a tool to learn science was rated with high votes by
most of the students, thus suggesting that activities like the one we designed
can be considered as a supplementary tool in teaching and learning. One half
of the sample rated with high votes the role of science communication in
driving their interest in science and in orientating them to university (see
Fig. 12). When asked if science communication activities motivated them to
start an academic career, the mean raises down with respect to the other
items. A possible interpretation of this result is that even if this kind of
activity can, in some sense, help in orientating students toward a particular
university course, raising their interest in science, it is not evident that
they will pursue such a career. Nevertheless, students affirmed that through
this kind of activity, they had entered into contact with science, learning
new concepts, see Fig. 15.

The use of an artistic tool such as the monologue which, in turn, meant using
storytelling to tell physics, passionated students towards the subject
proposed. They also evaluated it as a useful instrument to communicate
science. This suggests that implementing these strategies in the classroom
during formal curriculum can engage students in science. Moreover, the
majority of students affirmed to have learned new concepts of physics thanks
to the monologue, raising their interest in physics, see Table 1. Concerning
creativity, we did not collect very high votes, but if we take a look at the
type of schools, humanities showed higher means than scientific (even if there
are not any significative differences between the two). If we want to
understand better why the global mean is not so high, we can also take a look
at the class distribution (see Fig. 16). Students attending the third and
fourth classes exhibited lower means than their colleagues attending the fifth
class (and students attending the third and fourth classes represent the
majority of the sample). This can be related to their lower level of
preparation in physics, thus suggesting that the contents of the monologue
could be more adequate for students attending the last two years of high
school. However, if we take a look at the capacity of storytelling to
passionate people, no differences between classes appeared in the data.
Concerning gender, females reported slightly higher means than males (see Fig.
17).

A similar situation appears in the use of poems within the monologue. What is
really interesting is that when one takes a look at the school distribution of
data, humanities reported higher means for every item we investigated. Results
reveal that the difference is significant, thus suggesting that the use of
artistic tools to communicate physics, as well as an educational tool, is
welcome in humanities. This is an important result, also corroborated by the
results in the other domains we investigated, since humanities suffer from the
lack of science in their curriculum. Moreover, when we performed single
informal interviews with a small random sample of students, the ones attending
humanities reported that they are not good for pursuing a scientific career.
If interdisciplinary approaches based on storytelling can serve in bringing
science and, in particular, physics to school, teachers could think about the
implementation of specific programs based on this methodology during their in-
class activities. Students reported that the use of poems captured their
attention, motivated them to attend the activity and, in the case of
humanities, stimulated their creativity, too. The use of poems can offer
students the possibility to put humanistic and scientific knowledge together,
stimulating their critical thinking skills, as also suggested by [42] in the
case of cosmology. This is also confirmed by the MANOVA (see Tables 5, 6, 7),
where significative differences according to the school appeared between the
following domains: “STEM Engagement”, the “Monologue”, and “Poems”. Students
from humanities rated with higher votes the items appertaining to these
domains, thus suggesting the positive influence of our methodology in
engaging, interesting, and passionate them towards gravitational physics.

Finally, concerning motivation, even if students seemed to be not very
motivated in pursuing a career in physics (as also noted before in the case of
science communication items), they want to learn more about gravitational
waves physics. This, in some sense, seems to be counterintuitive. However, let
us notice that for high school students, studying physics is different from
studying astronomy or astrophysics, thus they can be interested in learning
more gravitational physics but not in studying physics as a whole subject.
This fact emerged during the debate phase of the activity, where students made
questions to satisfy their need for knowledge about physics. Students made
questions on black holes, white holes, dark matter and dark energy, the
expansion of the universe, and quantum gravity. The expert stimulated the
discussion, linking the answers to ET physics, whose fields of investigation
will be very promising to address many of the students’ questions [50, 51].
The role of debate after science theatre has been emphasized in [11, 12] as
important to stimulate critical thinking and engage students in learning
science, with particular attention to acquiring scientific literacy. Our
results go in the same direction, which is to be taken into account by
teachers during the design of a lesson.

If we look at correlations between the domains we investigated, we see that
“Motivation”correlates with the ‘Monologue”and “Poem”domains. The correlation
appears also with items related to “STEM Engagement”. This suggests that the
efficacy of an activity like the one we proposed relies on students’
sensibility towards science communication and its role in our society to learn
physics. Nevertheless, since motivation correlates also with the methodology
we used to bring physics into the classroom, a positive effect of storytelling
and interdisciplinary in this domain appeared. This corroborates what is
already known from literature on the positive effects of storytelling and
interdisciplinary approaches to bring STEM in high school, fostering
engagement and learning [27, 29, 30, 35, 34]. Studying correlations could
offer some insights into the mechanism of generating enthusiasm and fostering
motivation in students, possibly individuating the independent variables,
which in our case are related to students’ passion for science communication
(outreach and dissemination).

Finally, the overall feedback about the activity is very positive, showing
students’ appreciation of our methodology and its possible implementation at
school. In particular, we also suggest that this kind of approach could be
useful in orienting students toward a STEM or STEAM (Science Technology
Engineering Arts, and Mathematics) career, as also discussed in [23, 24]. In
fact, what emerges from our results is that the activity raised students’
motivation to learn gravitational-related topics. They also found poems and
the monologue a useful tool to communicate physics, thus their interest can
also be driven towards the study of arts and science communication (see also
[55] and refs therein). Teachers in collaboration with researchers can design
specific programs to integrate formal scholar curricula to engage students in
these fields.

##  6 Conclusion

In this work, we proposed storytelling and artistic tools (a theatrical
monologue) to bring gravitational waves physics to high school. The PER group
of the University of Cagliari and INFN Cagliari designed a specific program
devoted to 200 high school students (17 to 19 years old, third to fifth
classes) in Sardinia.

The aim of our work was to explore the effect of storytelling on students’
levels of classroom participation, motivation, and interest in the proposed
topics. We also meant to measure their engagement and, most interestingly,
their views about the effectiveness of storytelling as a teaching/learning
strategy in the science classroom. We investigated these domains by self-
report questionnaires administered to 76 students and data were analyzed both
qualitatively and quantitatively.

The results of our research are encouraging. We hope that they can also
motivate high school teachers to explore this kind of approach in their
classes, at least to introduce new arguments or to engage more students in
STEM. Moreover, if teachers do not feel adequate to explore by themselves this
methodology, they can promote students’ creation of stories to motivate them
to learn physics (see [27, 40, 41]). In particular, statistical analysis
showed that results are more positive for humanities than scientific high
school students. This is an important result since humanities suffer from the
lack of a proper scientific curriculum. If interdisciplinary approaches based
on storytelling can serve in bringing science and, in particular, physics to
school, teachers could think about the implementation of specific programs
based on a methodology like the one we proposed.

Offering an interdisciplinary vision of science to high school students and
teachers are becoming a common trend of informal learning (see [1, 2, 3] and
refs therein). To show how science is evolving and to provide new instruments
to learn science and physics in an enlarged context, mixing knowledge,
techniques, and methods from different disciplines should be part of science
and education curricula, developing an integrated model of learning and
teaching [5]. This research can give instructors a methodological tool to
encourage them to bring these topics to school, using storytelling to optimum
advantage in science. It can be used to introduce students to current trends
in research, trying to bypass content-related difficulties (both physical and
mathematical), but still making them explore our universe with inquiry and
minds-on activities, improving their motivation, curiosity, and interest in
physics.

The study has some limitations that future investigations should take into
account. The size of sample not allow to identify a model able to discriminate
the cognitive factors important to improve efficacy in our methodology. Carry
out interviews, also with a small sample, could help in this direction. A
measure of the level of learning pre and post-activity (formative evaluation)
could also give important feedback to quantitatively measure the efficacy of
the methodology on learning.

##  7 Acknowledgments

The authors acknowledge faculty members, teachers, and students who
participated in our studies.

##  8 Ethical statement

Informed consent to participate in the study has been obtained from
participants. Any identifiable individuals participating at the study have
been also aware of intended publication. Informed consent to publish has be
obtained from participants of the study. This work was carried out in
accordance with the principles outlined in the journal’s ethical policy and
with the “Codice etico e di comportamento” of the University of Cagliari.

##  9 Funding declarations and conflicts of interests

There are no known conflicts of interest associated with this publication and
there has been no significant financial support for this work that could have
influenced its outcome.

##  10 References

## References

  * [1] Spelt E. J. H. et al., (2009) Ed. Psyc. Rev., 21, 365;
  * [2] Gao X. et al., (2020) J. STEM Ed., 7, 24;
  * [3] Davies M. et al., (2007) in Melbourne: Cen. for the Stud. of High. Ed.;
  * [4] Thong C. et al., (2023) in Streit-Bianchi, M., Michelini, M., Bonivento, W., Tuveri, M. (eds), New Chall. and Opp. in Phys. Educ., Chall. in Phys. Ed., Springer, 343-359;
  * [5] Abd-El-Khalick F. et al., (2010) J. Res. Sci. Teach., 37(10), 1057-1095;
  * [6] Giliberti M., (2023) in Streit-Bianchi, M., Michelini, M., Bonivento, W., Tuveri, M. (eds), New Chall. and Opp. in Phys. Educ., Chall. in Phys. Ed., Springer, 175-189;
  * [7] Fazio C. et al., (2021) in Jarosievitz B., Sökösd C. (eds) Teaching-Learning Contemporary Physics. Challenges in Physics Education. Springer;
  * [8] National Research Council, (2009) National Academies Press;
  * [9] Ucko D. A., (2010) The Mus J. 53, 129-136;
  * [10] Tuveri M. et al., (2023) Nuo. Cim. C, 46(6);
  * [11] Giliberti M. et al., (2019) J. Phys: Conf. Series, 1286, 012065;
  * [12] Giliberti M. et al., (2022), J. Phys.: Conf. Ser, 2297 012020;
  * [13] Michelini M. et al., (2021) App. e Prop. per l’Ins. Appr della Fis. a Liv. Preuniv., Prin ’21;
  * [14] Corni F. et al., (2021) App. e Prop. per l’Ins. Appr della Fis. a Liv. Preuniv., Prin ’21;
  * [15] Michelini M., (2010)New Trends in Sc. and Tech. Ed., 1, 257-274;
  * [16] Michelini M., (2005) Proc. of the Third In. GIREP Sem. Inf. Learn. and Pub. Und. of Phys., 18-38;
  * [17] Affeldt F. et al., (2017) Chem. Ed. Res. Pract., 18(1), 13-25;
  * [18] Sokołovska D., (2018) Springer Nat. Switz. (D. Sokołowska, M. Michelini Eds), 243-255;
  * [19] Goorney S. et al, (2022), Ed. Sci., 12, 474;
  * [20] Ødegaard M., (2003) Stud. Sci. Ed., 39(1), 75-101;
  * [21] Scierri I. D. M. et al., (2019) J. of Th. and Res. in Ed., 14(1), 193-220;
  * [22] McGregor D., (2014), Pedag., 9(3), 216-232;
  * [23] Linh N. Q. et al., (2019) J. of Phys: Conf. Ser., 1340, 012025;
  * [24] Devins D. et al., (2015) High Ed., Skills and Work-Bas. Learn., 5(4), 352-368;
  * [25] Tuveri M. et al., (2023) in Streit-Bianchi, M., Michelini, M., Bonivento, W., Tuveri, M. (eds), New Chall. and Opp. in Phys. Educ., Chall. in Phys. Ed., Springer, 145-157;
  * [26] Rowcliffe S., (2004) Sch. Sci. Rev., 86(314), 121-126;
  * [27] Maharaj-Sharma R., (2022) Ed. Inq., 1-20;
  * [28] Millar R. et al., (1998) School of Ed., King’s College London;
  * [29] Engel A. et al., (2018) Child. Ed., 94(6), 4-12;
  * [30] Kaur G. J., (2020), A portfolio submitted in partial fulfillment of the requirements for the degree of Master of Education, 1-55;
  * [31] Denning S., (2005) The leader’s guide to storytelling. San Francisco, CA: Jossey-Bass;
  * [32] Richter A. et al., (2019) JCOM, 18(06), A02;
  * [33] Olson R., (2015) Houston, we have a narrative: Why science needs story. Chicago: The University of Chicago Press;
  * [34] Petrucco C., (2009) It. J. Soc. Ed., 17(1);
  * [35] Simon S., (2000) Students’ attitudes towards science. In Good pract. in sc. teach.: What research has to say, ed. Monk, M. and Osborne, J. Maidenhead: Open University Press;
  * [36] Abrahamson C. E., (2000) Educ., 118(3), 440-451;
  * [37] Farrel K et al., (1982) Effects of storytelling: An ancient art for modern classrooms.;
  * [38] McDonald J. K., (2009) Educ. Med. Int., 46(2), 111-122;
  * [39] Green M. C. et al., (2002) Narrative impact: Soc. and cogn. found., Mahwah, NJ: Erlbaum;
  * [40] Kotluk N. et al., (2016) Turk. Onl. J. of Dist. Ed., 17(1), 87-99;
  * [41] Marsico G. et al., (2019) Int. J. of Inn. in Sc. and Math. Ed., 27(6), 36-44;
  * [42] Kragh H., (2013) Astron. and Geophys., 54(2), 28-30;
  * [43] Kragh H., (2011) Sc. and Ed., 20, 343-357;
  * [44] Ogborn J., (2011) Rev. Bras. Pesqui. Educ. Cienc., 6(1);
  * [45] Thorne K. S., (2009) APS April Meeting Abstracts, B5.002;
  * [46] Einstein A., (1916) Koeniglich Preuss. Akad. der Wissens. (Berlin). Sitzungsb., 154-167;
  * [47] Abbott B. P. et al., (2016) Phys. Rev Lett., 116 061102;
  * [48] The Event Horizon Telescope Coll., (2016), The Astr. Phys. J. Lett., 875(1), L1;
  * [49] The Event Horizon Telescope Coll., (2022), The Astr. Phys. J. Lett., 930, L12;
  * [50] Maggiore M. et al., (2020) JCAP, 03, 050;
  * [51] Branchesi M. et al., (2023) JCAP, 07, 068;
  * [52] Punturo M. et al., (2010) Class. Quant. Grav, 27, 194002;
  * [53] Grado A., (2023), J. Phys.: Conf. Ser., 2429 012041;
  * [54] Saccorotti G. et al., (2023), Eur. Phys. J. Plus, 138(9), 793;
  * [55] Tuveri M. et al., (2022), Proc. of Sc., 40231 (043);