Table of contents

We provide a short introduction to the field of topological data analysis (TDA) and discuss its possible relevance for the study of complex systems. TDA provides a set of tools to characterise the shape of data, in terms of the presence of holes or cavities between the points. The methods, based on the notion of simplicial complexes, generalise standard network tools by naturally allowing for many-body interactions and providing results robust under continuous deformations of the data. We present strengths and weaknesses of current methods, as well as a range of empirical studies relevant to the field of complex systems, before identifying future methodological challenges to help understand the emergence of collective phenomena.

The idea that democracy is under threat, after being largely dormant for at least 40 years, is looming increasingly large in public discourse. Complex systems theory offers a range of powerful new tools to analyse the stability of social institutions in general, and democracy in particular. What makes a democracy stable? And which processes potentially lead to instability of a democratic system? This paper offers a complex systems perspective on this question, informed by areas of the mathematical, natural, and social sciences. We explain the meaning of the term 'stability' in different disciplines and discuss how laws, rules, and regulations, but also norms, conventions, and expectations are decisive for the stability of a social institution such as democracy.

Active matter, as other types of self-organizing systems, relies on the take-up of energy that can be used for different activities, such as active motion or structure formation. Here we provide an agent-based framework to model these processes at different levels of organization, physical, biological and social, using the same dynamic approach. Driving variables describe the take-up, storage and conversion of energy, whereas driven variables describe the energy consuming activities. The stochastic dynamics of both types of variables follow a modified Langevin equation. Additional nonlinear functions allow one to encode system-specific hypotheses about the relation between driving and driven variables. To demonstrate the applicability of this framework, we recast a number of existing models of Brownian agents and active Brownian particles. Specifically, active motion, clustering and self-wiring of networks based on chemotactic interactions, online communication and polarization of opinions based on emotional influence are discussed. The framework allows one to obtain critical parameters for active motion and the emergence of collective phenomena. This highlights the role of energy take-up and dissipation in obtaining different dynamic regimes.

We discuss the failure dynamics of the fiber bundle model, especially in the equal-load-sharing scheme. We also highlight the 'critical' aspects of their dynamics in comparison with those in standard thermodynamic systems undergoing phase transitions.

Understanding systems level behaviour of many interacting agents is challenging in various ways. In this review we will focus on how the interaction between components can lead to hierarchical structures with different types of dynamics, or causations, at different levels. We use the Tangled Nature model to discuss the co-evolutionary aspects of the connection between the microscopic level of the individual and the macroscopic systems level. At the microscopic level the individual agent may undergo evolutionary changes due to 'mutations of strategies'. The micro-dynamics always run at a constant rate. Nevertheless, the systems level dynamics exhibit a completely different type of intermittent abrupt dynamics where major upheavals keep throwing the system between metastable configurations. These dramatic transitions are described by a log-Poisson time statistics. The long time effect is a collective adaptation of the ecological networks. We discuss the ecological and macro-evolutionary consequences of the adaptive dynamics and briefly describe work using the Tangled Nature framework to analyse problems in economics, sociology, innovation and sustainability.

We study the planar motion of a self-propelled object against a withstanding flow, such as wind or sea current, whose speed is capable of perturbing the trajectory of the object, heading it off its path. Seven cases are dealt with, corresponding to fairly general wind types and all leading to nonlinear ordinary differential equations for the object motion. The first two cases concern motions described in Cartesian coordinates. They are followed by motions under a cyclonic wind, analysed in a polar reference frame; the wind is modelled as a logarithmic or hyperbolic spiral and closed-form solutions are obtained by means of the Gauss hypergeometric function and the Lerch transcendent. Further wind instances, treated in a Cartesian or in a polar reference system, require some special functions, such as the Lambert function, while finding the motion time law leads to a trinomial equation, in the most interesting situation of a mobile object that is faster than the wind. In the last case of a horizontal hyperbolic wind, time law and trajectory are computed in a polar reference frame by means of the Lambert function.

We present the solutions of the well-known riverboat problem in Poiseuille flow approximation. We find that the boat trajectory should be a cubic parabola for crossing the river in the shortest time (the first variant of the riverboat problem). We also derive and analyze the expressions for the heading angle, the total drift distance and the time to cross the river in both cases for crossing the river in the shortest time and by the shortest path (the second variant of the riverboat problem). We conclude that for the first variant it is possible to introduce an effective constant flow velocity which describes a boat crossing a river with a variable flow velocity and depends only on the flow velocity at the centre of the river. For the second variant this value additionally depends on the relative velocity of the boat.

An amusement ride involving rotations around three axes offers a large-scale illustration for the study of dynamics in three dimensions. The forces on the rider can be measured with co-moving accelerometers. As a first approximation, the different motions can be treated separately, but the combination of rotations is found to lead to a relatively large Coriolis effect. A mathematical description of the motion automatically combines all the different effects and offers a useful programming exercise.

We discuss the coupling of a rotating flywheel with a nonrotating flywheel. We find an expression for the tangential force between the wheels and an expression for the change in angular momentum of the system. Then, we calculate the fraction of rotational kinetic energy which is transferred from the rotating flywheel to the nonrotating flywheel. The theoretical results are compared with the experimental results in the article Mário S M N F Gomes et al, The 'Spinning disk touches stationary disk' problem revisited: an experimental approach (2018 Eur. J. Phys.39 045709). We then introduce a series of initially nonrotating flywheels between the two original flywheels which are coupled and decoupled, beginning with coupling between the original rotating flywheel and the first wheel in the series and finally coupling between the last flywheel in the series with the original nonrotating flywheel. The condition for maximum energy transfer from the original rotating flywheel to the original nonrotating flywheel for a given number of flywheels will be shown to be achieved when the masses of the flywheels constitute a geometric series if all the flywheels are of the same type. If the flywheels are not of the same type the maximum is achieved when the product of the K-factors of the inertial moments and the masses constitute a geometric series.

Experimental results are presented on the rolling of a steel ball over a small step. The experiment was designed to simulate the effects of surface roughness on rolling motion, where collisions between asperities on the ball and the rolling surface act to enhance the coefficient of rolling friction. In the present experiment, the ball was launched as a projectile when colliding with the step, with a consequent increase in vertical speed and a decrease in horizontal speed. Results are presented for four different diameter balls and four different steps, at incident rolling speeds between 0.3 and 1.2 m s⁻¹.

We have constructed and tested a tuned vibration absorber in an HO-scale model building structure to demonstrate dynamic absorption of structural oscillations due to simulated harmonic earthquake tremors. The apparatus is simple and inexpensive. A mathematical model including coupling of the absorber and the building structure is presented. Experimental results show strong agreement with the mathematical model. The experiment, suitable for an undergraduate laboratory, demonstrates principles of physics as well as mechanical engineering.

The Saha–Basu equation of state, which has attracted little attention from the scientific community, is here revisited. This equation has been derived using an approach that is different from the original approach. Expressions for some physical parameters, such as critical constants, Boyle's temperature, critical coefficients etc, have been derived and the equation has been expressed in terms of reduced parameters. Amagat's curves have been explained using the virial form of the equation. Some thermodynamic quantities, such as the Joule–Thomson coefficient, inversion temperature and heat capacity relations, have been arrived at from this equation of state. The comparison has been made with other related equations of state.

A numerical calculation for the stationary magnetic field produced by arrangements of non-concentric and non-coplanar loop current circuits is presented. The calculation is done by superposing the solution of the magnetic field produced by a set of loops with constant currents that mimic two and three-dimensional systems. In the three-dimensional cases, this is achieved by rotating the magnetic field produced by the non-coplanar loops and adding all the contributions at any arbitrary point in the space. We report the case of two coplanar non-concentric loops that do not overlap and two concentric coplanar rings with different radii carrying currents in the same and opposite directions. Then we consider two non-coplanar rings that are tilted by an angle. More complicated systems consist of a set of loops forming a semi-doughnut. As an extension, we add at the two ends of this system concentric loops to form a horseshoe magnet with a circular cross-section and analyze the results as a function of its geometric characteristics. We can calculate the solutions of the magnetic field in all the space and plot their field lines using a technique that makes use of the Runge–Kutta fourth-order method. In all the cases we plot with different colors the field lines to give information on their strength.

Following the initial idea presented in Amaral et al (2017 Eur. J. Phys38 025206), we address Kelvin's inversion of coordinates, which exchanges the radial coordinate for its inverse, in arbitrary dimensional electrostatics. For two- and three-dimensional configurations (planar and cylindrical symmetries) detailed analyses are provided. Using this symmetry, which is rather overlooked in usual electromagnetism textbooks, a wide range of problems is correlated showing that some complicated situations are much simpler in a dual description. We also show that the Lagrangian changes by a total derivative term under the studied inversion leading to action invariance.

One hundred and thirty years ago Oliver Heaviside (1850–1925) first published the exact expression of the electromagnetic field generated by a uniformly moving electric charge. A didactical route to establish such an important result is here proposed via a new treatment of the Feynman formula for the electric field intensity of a moving charge, when the charge moves uniformly. We believe the present approach could be a useful starting point for introducing as gently as possible a complex analysis of electrodynamics even to undergraduates not yet familiar with Maxwell's equations and Lorentz transformation.

The classical experiment and exercise in electrostatics of finding the relation between the charges and the separation of two suspended charged balls is developed to the more practical problem of calculating the separation as function of the voltage. The case of more than two suspended balls is also studied, and experimental results are shown, demonstrating that the approximations used are good enough for relatively precise measurements, and that with more than three balls the solution for the disposition of the balls is not unique, with interesting stability problems arising.

The Fourier transform method can be applied to obtain electromagnetic knots, which are solutions of Maxwell equations in a vacuum with non-trivial topology of the field lines and special properties. The program followed in this work allows us to present the main ideas and the explicit calculations at undergraduate level, so they are not obscured by a more involved formulation. We make use of the helicity basis for calculating the electromagnetic helicity and the photon content of the fields.

We compute the exterior Green function for a grounded equi-potential circular ring in two-dimensional electrostatics by treating the system geometrically as a 'squashed wormhole' with an image charge located in a novel but obvious position, thereby implementing a method first suggested in 1897 by Sommerfeld. We compare and contrast the strength and location of the image charge in the wormhole picture with that of the conventional point of view where an image charge is located inside the circular ring. While the two viewpoints give mathematically equivalent Green functions, we believe they provide strikingly different physics perspectives that encourage students to think more broadly about the subject. We present our discussion at a level suitable for use in advanced undergraduate and introductory graduate courses on electrostatics, or for use as a supplement to introduce the methods of Riemannian geometry in the context of general relativity.

Following the success of the pedagogic tool for monochromatic holography realized for popular science purposes a few years ago (Voslion and Escarguel 2012 Eur. J. Phys. 33 1803), an upgrade has been developed. The resulting device includes all the necessary equipment to produce far more spectacular 4'' × 5'' color holograms for science outreach purposes. Graduate schools are also offered several experiments such as holographic diffraction gratings. Optimized to be extremely robust to vibrations, particularly by using the weight of the object to avoid unwanted vibrations during hologram exposure, the robustness of this device is demonstrated through optical microscopy measurements.

While the Berry phase is often introduced in a variety of contexts, it is generally neglected in solid state courses due to a lack of simple models and the difficulty of the required mathematical techniques. In this paper, we calculate the Berry phase for a simple model, a one-dimensional Dirac comb with a secondary peak treated as a perturbation, using relatively straightforward techniques. Interestingly, we find that for this specific model the phase exhibits a band-dependent beat-like phenomenon as the secondary peak sweeps across the unit cell. Using perturbation theory we obtain an explicit expression for these oscillations, in excellent agreement with numerical results.

In this work we analyze the low-energy nonrelativistic limit of Dirac theory in the framework of effective field theory. By integrating out the high-energy modes of the Dirac field, given in terms of a combination of the two-component Weyl spinors, we obtain a low-energy effective action for the remaining components, whose equation of motion can then be compared to the Pauli–Schrödinger equation after demanding normalization of the wave function. We then discuss the relevance of the terms in the effective action in the context of an anisotropic dimensional analysis which is suitable for nonrelativistic theories.

An earlier-proposed atomic geometrical model is used to estimate the inter-electronic repulsion in multi-electronic atoms. The model allows calculation of the energies for each shell. The resulting total energies of all atoms in the periodic table differ by less than 1% from Hartree–Fock calculations, except for He, where the difference is 1.8%. However, compared to the experimental total energy values for the first 36 elements, the largest difference is 1.04% for Ne. In this simplified model, the first ionization energies of the alkali metals and alkaline earths proved to be somewhat poor. A modification of the model is proposed to improve these ionization energies, keeping a very good agreement with the experimental average shell energies and with the Hartree–Fock calculations for the total energies.

Entanglement can modify light–matter interaction effects and, conversely, these interactions can change the non-classical correlations present in the system. We present an example where these mutual connections can be discussed in a simple way at the graduate and advanced undergraduate levels. We consider the process of light absorption by multi-atom systems in non-product states, showing first that the absorption rates depend on entanglement. The reverse is also true, absorption processes can generate an hyperentangled atomic state involving in a non-product form both internal and spatial variables. This behavior differs from that of spontaneous emission, which disentangles atomic systems.

The three-dimensional isotropic quantum harmonic oscillator (QHO) plays important roles in physics. Despite its importance, much of its rich mathematical structure and symmetry are often not sufficiently stressed, particularly at the advanced undergraduate or beginning graduate levels. In this paper, the well-known yet less apparent SU(3) symmetry of the three-dimensional isotropic QHO is used to yield its energy eigenvalues. In particular, the relatively less explored cubic Casimir operator of the SU(3) Lie group is exploited to obtain these well-established energy eigenvalues. The relationship between the degeneracy of each energy level of the three-dimensional QHO and the dimensions of irreducible representations of the SU(3) group is revisited and elaborated.

Uniformly charged regular bodies have always been of great interest to physics as well as to several other scientific disciplines. In this work, we consider a uniformly charged straight wire with finite length and calculate the electrostatic potential created at an arbitrary point in space. The resulting expression obtained from direct integration techniques is transformed in such a way as to lead to a very insightful geometrical interpretation. It is shown that the final result depends only on the distances of the two end points of the straight wire from of the arbitrary point of interest in space. This observation leads to the immediate identification of the nature of the equipotential surfaces around a uniformly charged straight wire.

Glassy-state structural relaxation is a physical phenomenon that occurs in most amorphous and semicrystalline polymers after their melt-processing. As such, after these industrial conformation processes, polymers are usually found in a thermodynamic non-equilibrium state. Therefore, polymer chains undergo thorough conformational changes on their way towards a thermodynamically stable state. These molecular-level events involve macroscopic-level modifications, which are manifested as changes in physical properties. Here we propose a simple approach to monitor the physical aging in polymers by differential scanning calorimetry (DSC). Accordingly, the β_H enthalpy relaxation rate is directly extracted from the DSC curves taking into account the enthalpy loss at different aging times. This method allows a simple quantification of the bulk aging dynamics in polymers, and can be extrapolated to systems containing reinforcing particles; i.e. composites and nanocomposites. This safe, low cost and simple experiment designed for undergraduate students of physical, chemical and engineering specialities serves as a guide to easily determine the physical aging suffered by polymeric materials during their storage and to understand its implications at macroscopic level, which is a relevant field in condensed-matter physics. Given the paramount relevance of physical aging in glassy materials, this experiment may also serve to raise awareness of the importance of the thermal history of materials on their resulting properties, which is overlooked too often.

Sector models are tools that make it possible to teach the basic principles of the general theory of relativity without going beyond elementary mathematics. This contribution shows how sector models can be used to determine geodesics. We outline a workshop for high school and undergraduate students that addresses gravitational light deflection by means of the construction of geodesics on sector models. Geodesics close to a black hole are used by way of example. The contribution also describes a simplified calculation of sector models that students can carry out on their own. The accuracy of the geodesics constructed on sector models is discussed in comparison with numerically computed solutions. The teaching materials presented in this paper are available online for teaching purposes at https://www.spacetimetravel.org/.

Sector models permit a model-based approach to the general theory of relativity. The approach has its focus on the geometric concepts and uses no more than elementary mathematics. This contribution shows how to construct the paths of light and free particles on a spacetime sector model. Radial paths close to a black hole are used by way of example. We outline two workshops on gravitational redshift and on vertical free fall, respectively, that we teach for undergraduate students. The workshop on redshift does not require knowledge of special relativity; the workshop on particles in free fall presumes familiarity with the Lorentz transformation. The contribution also describes a simplified calculation of the spacetime sector model that students can carry out on their own if they are familiar with the Minkowski metric. The teaching materials presented in this paper are available online for teaching purposes at https://www.spacetimetravel.org.

Energy-momentum diagrams are useful visual tools to analyze collision-type phenomena such as elastic and inelastic collisions between macroscopic objects, Compton scattering, photon absorption, pair creation, or particle–antiparticle annihilation. In this paper a specific example is given for the pedagogical use of energy-momentum diagrams in analyzing a continuous process: the relativistic rocket motion.

The formula E = mc² is an icon in physics but, in basic education, it tends to be approached just as a prescription for simple numerical conversions of mass into energy. In order to go beyond this narrow scope, in this work one considers some applications of the mass–energy equivalence to microscopic problems, since there it plays an essential role, especially in the description of bound systems. In this realm one may grasp how far-reaching its consequences are. The cases discussed, which range from larger to smaller scales, are carbon monoxide, a hydrogen atom, a deuteron and a proton. One begins by discussing the energy balances of exothermic reactions involving these bound systems so as to produce evidence that their masses are smaller than those of their isolated constituents. One then considers a bound system as an isolated body and moves into a more microscopic description, showing that the entities which effectively contribute to its mass are constituent masses, supplemented by both kinetic and potential energies. The action of the gravitational field of the Earth over these entities gives rise to the observed weight of the bound system. In the case of protons and neutrons, 99% of the weight is given by just internal kinetic and potential energies. One also indicates that this picture holds for the weight of ordinary objects.

Explaining physics content in an understandable way has been described as a vital skill for physics teachers in schools and universities. Studies suggest that physics teacher trainees regard explanation as one of the hardest challenges of their profession, and many scholars argue that explanation should be a part of physics teacher education. This paper presents an online test instrument for physics explanatory skills. It has been developed for diagnostic purposes and can be used—among other things—for teacher training and self-assessment (available via a weblink). The test takes about 45 min and calculates a score for explanatory skills as well as an indicator for underlying reasoning strategies. The individual results are compared with average scores from a study with physicists involved in science communication and physics teacher trainees. This comparison allows an estimation of the participants' skill to explain physics content successfully and gives hints on how to improve. The test consists of video scenes which show a teacher in the attempt to explain a physics phenomenon to a student. When the student asks a question, the video stops and the participant has to decide how to continue. Afterwards, he or she is asked to reason the choice made. The dialogue continues with the next video scene. In this paper, we summarise findings on how a science teaching explanation should be provided and discuss how the test instrument covers these aspects. Findings from a study with N = 154 participants are briefly discussed to show evidence for the validity of the instrument. Finally, possible uses of the test are proposed, including (self-)assessment, physics teacher education, and evaluation of training programs for all those who are involved in explaining physics.

Encouraging positive inquiry-focused behaviours within the constraints of a physics teaching laboratory environment can be challenging. Here, we report on an implementation, the 'working grade' (w-grade), designed to directly assess aspects of students' laboratory practice with the aim of encouraging first-year undergraduate students to look beyond the concept of a 'correct outcome' to a physics experiment. The w-grade is composed of the five aspects of group work, querying, exploration, attitude and progress which are each marked on a 0, 1, 2, 3 scale. The initial implementation is presented in full as well as a second, simpler variant. The w-grade emphasises and directly rewards inquiry behaviours and students were much more willing to explore the experiments than in previous years.

The classical equations used to describe the incompressible flow of an ideal fluid may be obtained on an elegant and compact structure with a single equation using geometric algebra. This article proposes a simple way to explain and introduce this approach to a reader unfamiliar with the subject.

If one is not familiar with the physics of the violin, it is not easy to guess, even for an experimental physicist, that the so-called Helmholtz motion can be obtained as a solution to the one-dimensional wave equation for the motion of a bowed violin string. It is worth visualising this aspect from a graphical perspective without recourse to ordinary Fourier analysis, as has customarily been done. We show in this paper how to obtain the shape of the Helmholtz trajectory, that is, two mirror-symmetric parabolas, in the ideal case of no losses from internal dissipation and no viscous drag from the air and the non-rigid end supports. We also show that the velocity profile of the Helmholtz motion is also a solution of the one-dimensional wave equation. Finally, we again derive the parabolic shape of the Helmholtz trajectory by applying the principle of energy conservation to a violin string.

Magnetic pickup loops or B-dot probes are one of the oldest known sensors of time-varying magnetic fields. The operating principle is based on Faraday's law of electromagnetic induction. However, obtaining accurate measurements of time-varying magnetic fields using these kinds of probes is a challenging task. A B-dot probe and its associated circuit are prone to electrical oscillations. As a result, the measured signal may not faithfully represent the magnetic field sampled by the B-dot probe. In this paper, we have studied the transient response of a B-dot probe and its associated circuit to a time-varying magnetic field. Methods of removing the oscillations pertaining to the detector structure are described. After removing the source of the oscillatory signal, we have shown that the time-integrated induced emf measured by the digitiser is linearly proportional to the magnetic field sampled by the B-dot probe, thus verifying the faithfulness of the measured signal.

We induced a variety of star-shaped oscillating droplets by using deionized water and different kinds of hot surfaces. We explained the generating mechanism of Leidenfrost stars with standing wave theory, and we discussed several factors that influence the oscillation such as evaporation, plate materials, plate temperature and plate roughness. Finally, we concluded that in order to easily induce more stable stars and more modes, one should choose a plate with a higher thermal conductivity, smooth surface and chemical stability. This topic is well adapted for organizing a teaching process including demonstration, observation and verification in classroom experiments because of this fascinating phenomenon and the interesting mechanism behind it.

A harmonically driven oscillatory fully developed laminar flow through a rectangular cross-sectioned duct is analytically and experimentally studied. Experimental data is gathered at low cost with a relatively easy to implement, non-commercial, do-it-yourself (DIY) particle image velocimetry (PIV) system, which is proposed here as a flexible pedagogic resource for use in physics/engineering fluid mechanics undergraduate or graduate level education. As a DIY system, its components can be generic and/or open source based, making it useful in developing countries as well. An analytical model was developed, solved, and compared to the experimental data while the key features of the oscillatory flow were recognised and reciprocally represented by theory and experiments. This validated the proposed system as a low-cost alternative to the commercially available systems and it also confirmed it as a viable pedagogical resource. Consequently, the activities described could be straightforwardly implemented into undergraduate/graduate fluid mechanics pedagogical contexts, such as lecture aids, course projects, and laboratory practices. The proposed DIY-PIV system constitutes a teaching with technology pedagogical resource on fluid mechanics physics/engineering education that is also able to function as a limited/auxiliary or even state of the art research tool, according to the budget and the objectives.

The importance of learning symbolic computation in research in theoretical sciences cannot be overemphasised. While Fortran and/or C programming laboratories have become an essential part of MSc curricula now, symbolic computing is almost never taught. We demonstrate how a freeware, SAGE, can be employed for the variational solution of simple (or complex) Hamiltonians encountered in quantum mechanics in one dimension. One can easily change the trial wavefunction and the Hamiltonian and obtain estimates of ground state energy. This should lead to a qualitative understanding of the physics of the problem. A brief extension to the first excited state for potentials with parity is discussed. Finally, we give a brief overview of the range of problems which can be introduced in a physics laboratory by using SAGE.

This comment addresses the paper by Sarasua and Abal (2016 Eur. J. Phys.37 055103). Driven by an incorrect interpretation of the Clausius relation, those authors demonstrated the equivalence of the Kelvin–Planck statement of the second law and the principle of entropy increase. In addition to the necessary clarification of the Clausius relation, we propose an improved version of the demonstration made in the aforementioned paper, since the one carried out therein is restrictive. Our aim is to clarify issues that, being didactically and pedagogically relevant, are also subtle.