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1. Does a physical pendulum ever act like a simple pendulum?

   https://doi.org/10.1088/1361-6404/ad1f95  

   Fulton, E. L.; Gay, T. J. (February 2024, European Journal of Physics)

   Abstract We show that for a physical pendulum comprising a massive sphere swinging from a massive string, there is, in general, a length of string for which its oscillatory period equals the period calculated by the simple pendulum model with a point-like mass swinging from a massless string whose model length equals the summed length of the real string and the sphere’s radius. 

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2. Strong nonreciprocity in a bistable pendulum with contactless coupling to a monostable pendulum

   https://doi.org/10.1007/s11071-025-10992-w  

   Rouleau, Michael; Booker, Zachary; Shi, Chengzhi; Meaud, Julien (February 2025, Nonlinear Dynamics)

   Abstract This article studies the nonreciprocity of a system that consists of a bistable element coupled to a monostable element through a contactless magnetic interaction. To illustrate the concept, the bistable element is physically realized using a pendulum that interacts with a stationary magnet and the monostable element is a classical pendulum. A numerical model is implemented to simulate the nonlinear dynamics of the system. Both simulations and experiments show that the system exhibits a strong amplitude-dependent nonreciprocity in response to initial excitations. At small input amplitudes, the system has an intrawell response with minimal transmission of energy whether the excitation is exerted on the side of the bistable pendulum or on the other side. However, at high input amplitude, a strong nonreciprocal behavior is observed: excitation of the bistable pendulum causes an interwell response which considerably reduces the distance between the two pendulums and allows energy to be efficiently transmitted through the contactless magnetic interaction; excitation of the monostable pendulum does not cause any interwell response and results in limited energy transmission. The combination of bistability and contactless nonlinear interactions allows the system to exhibit very strong amplitude-dependent nonreciprocity, which may be useful in a wide range of applications. 

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3. (2D+1) pendulum beams: non-diffracting optical spatial wavepackets that simulate quantum pendulum dynamics

   https://doi.org/10.1117/12.2627732  

   Nguyen, Thao; Rodríguez-Fajardo, Valeria; Galvez, Enrique J. (March 2022, Complex Light and Optical Forces XVI, 1201704)

   Andrews, David L.; Galvez, Enrique J.; Rubinsztein-Dunlop, Halina (Ed.)

   The similarity between the 2D Helmholtz equation in elliptical coordinates and the Schr¨odinger equation for the simple mechanical pendulum inspires us to use light to mimic this quantum system. When optical beams are prepared in Mathieu modes, their intensity in the Fourier plane is proportional to the quantum mechanical probability for the pendulum. Previous works have produced a two-dimensional pendulum beam that oscillates as a function of time through the superpositions of Mathieu modes with phases proportional to pendulum energies. Here we create a three-dimensional pendulum wavepacket made of a superposition of Helical Mathieu-Gaussian modes, prepared in such a way that the components of the wave-vectors along the propagation direction are proportional to the pendulum energies. The resulting pattern oscillates or rotates as it propagates, in 3D, with the propagation coordinate playing the role of time. We obtained several different propagating beam patterns for the unbound-rotor and the bound-swinging pendulum cases. We measured the beam intensity as a function of the propagation distance. The integrated beam intensity along elliptical angles plays the role of quantum pendulum probabilities. Our measurements are in excellent agreement with numerical simulations. 

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4. Quantum inspired 3D Pendulum beams

   https://doi.org/10.1088/2040-8986/adc141  

   Rodriguez-Fajardo, Valeria; Nguyen, Thao_P; Galvez, Enrique_J (March 2025, Journal of Optics)

   Abstract The technologies used in the manipulation of light can be used to do analogue simulations of physical systems with wave-like equations of motion. This analogy is maximized by the use of all the degrees of freedom of light. The Helmholtz equation in physical optics and the Schodinger equation in quantum mechanics share the same mathematical form. We use this connection to prepare non-diffracting optical beams representing the spatial and temporal dynamics of a nonlinear physical system: the quantum pendulum. By using the propagation coordinate to represent time in the quantum problem, we are able to analogue-simulate quantum wavepacket dynamics. These manifest themselves in novel optical beams with rich three-dimensional structures, such as rotation and sloshing of the light's intensity as it propagates. Our experimental results agree very well with the predictions from quantum theory, thus demonstrating that our system can be used as a platform to simulate the quantum pendulum dynamics. This three-dimensional light-sculpting capability has the potential to impact fields such as manipulation with light and imaging. 

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5. Physical Pendulum without Small Angle Approximation

   https://doi.org/10.1119/PICUP.Exercise.PWSAA  

   Vemuri, Gautam; Gavrin, Andy (November 2020, American Association of Physics Teachers)

   The simple pendulum usually studied by analytic methods invokes the small angle approximation (SAA) so that one can easily reduce the equation of motion to Hooke’s law and thereby obtain the period of the pendulum and other associated quantities. If the approximation is relaxed, the problem becomes analytically intractable and one must resort to computational methods. In this exercise, the pendulum with and without the SAA are compared to allow students to discover what happens to the temporal behavior for larger angles of displacement. The students will also be able to obtain quantitative estimates of what a “small angle” means, and the limits of validity of the SAA. In the computations, students will learn that the second order differential equation that describes the motion of the pendulum can be reduced to two coupled first order differential equations, which can then be solved by the Euler-Cromer algorithm. In this specialized exercise set, students will also learn to use the ODE45 package in MATLAB to solve differential equations and its advantages over Euler-Cromer method, as well as ‘findpeaks’ command in MATLAB. 

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