The Formulations of Classical Mechanics with Foucault’s Pendulum

Since the pioneering works of Newton (1643–1727), mechanics has been constantly reinventing itself: reformulated in particular by Lagrange (1736–1813) then Hamilton (1805–1865), it now offers powerful conceptual and mathematical tools for the exploration of dynamical systems, essentially via the action-angle variables formulation and more generally through the theory of canonical transformations. We propose to the (graduate) reader an overview of these different formulations through the well-known example of Foucault’s pendulum, a device created by Foucault (1819–1868) and first installed in the Panthéon (Paris, France) in 1851 to display the Earth’s rotation. The apparent simplicity of Foucault’s pendulum is indeed an open door to the most contemporary ramifications of classical mechanics. We stress that adopting the formalism of action-angle variables is not necessary to understand the dynamics of Foucault’s pendulum. The latter is simply taken as well-known and simple dynamical system used to exemplify and illustrate modern concepts that are crucial in order to understand more complicated dynamical systems. The Foucault’s pendulum first installed in 2005 in the collegiate church of Sainte-Waudru (Mons, Belgium) will allow us to numerically estimate the different quantities introduced.

Predictive monitoring of secondary epidemic waves of COVID-19 in Iran, Russia and other countries

In the last decade of April 2020, the second coronavirus epidemic wave in Iran has bloomed. The new wave has started in the vicinity of the critical point, marked by approximately 44,000 infections, where the rate of increase of the primary epidemic that appeared in Iran in mid-February 2020 was the highest. Today, this secondary wave almost has doubled the peak of the primary, and, passing the epidemic threshold of about 70,000 total cases in early June, generated the new third epidemic wave developing unpredictably and dynamically. The purpose of this work was to call into use a simple dynamical system represented by the discrete logistic equation with unknown parameters to predict secondary waves using the official statistical data. The mathematical modelling reveals the secondary epidemic waves in Sweden, the United States, Ukraine, Serbia, Romania, Czech Republic, Portugal, Luxembourg, Poland, and Ecuador. Also, the second waves appear in Russia and other countries. Despite many individual differences in the epidemic spread in different countries, we have traced regularity in the rise of secondary waves. The beginning of each new wave, if focusing on the number of total cases, practically coincides with the time of the maximum growth rate of the previous early epidemic. Thus, the passing through the threshold of the current wave should be the most responsible for strict observance of the rules of self-isolation and other sanitary standards.

Periodic Motions in a First-Order, Time-Delayed, Nonlinear System

In this paper, periodic motions in a first-order, time-delayed, nonlinear system are investigated. For time-delay terms of non-polynomial functions, the traditional analytical methods have difficulty in determining periodic motions. The semi-analytical method is used for prediction of periodic motion. This method is based on implicit mappings obtained from discretization of the original differential equation. From the periodic nodes, the corresponding approximate analytical expression can be obtained through discrete finite Fourier series. The stability and the bifurcations of such periodic motions are determined by eigenvalue analysis. The bifurcation tree of period-1 to period-4 motions are obtained and the numerical results and analytical predictions are compared. The complexity of periodic motions in such a simple dynamical system is discussed.

Mechanical Representation and Stability of Dynamical Systems Containing Fractional Springpot Elements

In this article we consider the Lyapunov stability of mechanical systems containing fractional springpot elements. We obtain the potential energy of a springpot by an infinite dimensional mechanical analogue model. Furthermore, we consider a simple dynamical system containing a springpot as a functional differential equation and use the potential energy of the springpot in a Lyapunov functional to prove uniform stability and discuss asymptotic stability of the equilibrium with the help of an invariance theorem.

Dynamic Evolution Analysis of Stock Price Fluctuation and Its Control

This paper studies a simple dynamical system of stock price fluctuation time series based on the rule of stock market. When the stock price fluctuation system is disturbed by external excitations, the system exhibits obviously chaotic phenomena, and its basic dynamic properties are analyzed. At the same time, a new fixed-time convergence theorem is proposed for achieving fixed-time control of stock price fluctuation system. Finally, the effectiveness of the method is verified by numerical simulation.