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In this work we highlight the importance of the simple harmonic oscillator as a fundamental mathematical model for the understanding of oscillatory phenomena in several areas of Physics and Engineering, emphasizing the need to know the methods for its solution, which serve as a starting point for studying systems more complex. Although this topic is relevant, many studies are limited to practical applications or general solutions of differential equations, with no significant connection between concepts. In this sense, an approach is proposed that explores the relationship between the harmonic oscillator, series and Fourier transform. Through the theory of the Sturm-Liouville linear operator, we introduce concepts such as the eigenfunctions of a Hermitian operator, orthogonality, and the formation of vector spaces. While the series appears as an expansion on an infinite basis of functions within a finite interval, the transform arises as a projection of mathematical objects such as functions and equations onto the space of the oscillator's eigenfunctions into an infinite interval. It is hoped that this approach will stimulate discussions and facilitate the understanding of more advanced topics in Physics and related areas, making them more accessible to undergraduate and graduate students. [ABSTRACT FROM AUTHOR]

Throughout the undergraduate course in Physics, the student frequently encounters problems involving oscillations. The primary model used to investigate oscillatory phenomena is the simple harmonic oscillator. As students advance in their academic education, they also learn various methods for solving differential equations. It is standard to employ such novel techniques in problems whose solution is previously known, to test the method's quality and for didactic purposes. Placing these two lines together, I present twelve different ways to solve the well-known simple harmonic oscillator problem. They fall into three categories: basic, intermediate, and advanced. This division is based on the level of knowledge necessary to solve the problem, progressively acquired throughout the student's academic education. Therefore, this material serves as a reference for students at distinct stages of their education. [ABSTRACT FROM AUTHOR]

The problem of underdamped classical harmonic motion through a force linearly proportional to velocity is revisited. Several properties of the system not usually studied are discussed, such as the role of constants of the displacement, the return time instants, the return positions, the maximum velocity time instants, the maximum velocity positions, the energies, the powers, and the total distance travelled. Expressions for these quantities are obtained showing the influence of the initial conditions of motion and the influence of damping. [ABSTRACT FROM AUTHOR]

In this work we present a formalism to describe the fractional Newtonian mechanics based in conformable fractional calculus. In this sense, a brief revision about integral and differential conformable calculus is implemented. In sequence, the conformable version to Lagrangian and Hamiltonian formalism are presented. As examples, the conformable harmonic oscillator and conformable wave equation are discussed. In all situations, usual results are obtained when we consider a = 1. An interesting result is that in the case 1 -- a 1 the conformable harmonic oscillator behaves like usual harmonic oscillator with time dependent and dissipation term. [ABSTRACT FROM AUTHOR]

In this work we will model the dynamics of the simple harmonic oscillator using discrete variables. The resulting recurrence relations were written as matrix transformations and, from a diagonalization process, we will show that the solution obtained converges to the well-known analytical solutions of the system. This approach avoids the continuity hypothesis and is relevant as it describes the dynamical of this particular system from a computational point of view. Furthermore, the algorithm we are proposing was implemented, and an error analysis was built from the principle of the conservation of mechanical energy. [ABSTRACT FROM AUTHOR]

Since the end of the 19th century, and later driven by aerospace development, the modeling of vibrations in metallic structures has demanded significant effort on the part of researchers. Add to this the importance of predicting earthquakes and minimizing the damage they generate, and there is one more significant purpose for the study of waves in structures. In the Classical Physics course, waves are approached from the perspective of the harmonic oscillator, waves in strings and sound tubes, analyzing longitudinal and transverse waves. However, surface waves and extensional waves are rarely mentioned, so the present work intends to contribute to the paving of this small gap, proposing a practical activity to calculate the modulus of the speed of extensional waves in steel and aluminum rods, using the frequencies of the harmonics of extensional standing waves. The theoretical development is fully understandable to the student who has basic knowledge of Physics and Differential and Integral Calculus. The experimental apparatus is low cost and fully available. The resonant frequencies are determined with a cell phone where the free Spectroid application has been installed. For extensional speeds, were obtained vext-Al = (5.05 ±0.02).10sm/s and vext-steel = (5.21 ± 0.08).103m/s, implying that the respective relative deviations from the literature were 1% and 0.6%, very good for a teaching laboratory. The activity also favors the deepening of the concepts related to oscillations and brings strong playful elements, such as the use of the cell phone and the pleasant timbre of the resonant rods, so that its application is fully feasible and recommendable. [ABSTRACT FROM AUTHOR]

This contribution sets out to present an advanced topic of major relevance in Physics: the spontaneous symmetry breaking, a crucial issue for the Higgs Mechanism, in a suitable approach for the high school level. The mathematical model of symmetry breaking is approached by exploiting the analogy between the simple harmonic and anharmonic oscillators. Next, we discuss how the Higgs boson's physical mass arises from the oscillation of the Higgs oscillatory coordinate, here playing the role of the Higgs field. We also propose the use of a computer simulation, with the Modellus software, to compute the energy of the oscillators. The purpose of carrying out simulation is to render the learning process more efficient and to increase the action of the student. [ABSTRACT FROM AUTHOR]

In this article, we investigate the dynamics of a charged particle when subjected to an electric field generated by two dielectric and homogeneous disks. We analyzed the construction of the motion equation for the accelerated charge, which in the linear approximation of the electric force results in a Simple Harmonic Oscillator (SHO). Subsequently, the Abraham-Lorentz radiation force is incorporated into the motion equation because the accelerated movement of the load results in the emission of energy radiation and, in this situation, behaves like a Damped Harmonic Oscillator (DHO). And we calculated Larmor's radiation power for this damped particle motion. [ABSTRACT FROM AUTHOR]

In this work we propose an analytical iterative approach in which the temporal evolution was replaced by the number of iterations. And for the case of a mass-spring system we obtained its velocity and its position as a function of time as partial sums of series that converged to the trigonometric functions that are the solutions of the simple harmonic oscillator. [ABSTRACT FROM AUTHOR]

The harmonic oscillator damped by frictional forces is revisited with a more robust and comprehensive physical and mathematical descriptions. Several system properties are presented and discussed, such as the solution to the general case of initial conditions, displacement envelopes, exact number of oscillations, instantaneous dissipation and average energy, quality factor, phase diagram and others. This text serves as a theoretical foundation for computer simulations and didactic experiments that explore oscillating systems damped by frictional forces. [ABSTRACT FROM AUTHOR]

In this paper we present an analysis of the asymptotic behavior of damped and forced harmonic oscillators considering the (x², x²) representation, we find that from the analysis of the asymptotic behavior of boundary conditions, taking as an example the case of damped and forced harmonic oscillators, which its possible infer the dynamical behavior in this region for more complex systems, such as the nonlinear oscillator described in section 5.Some implications of the asymptotic solution for this system are discussed at the end of this section. However, a more detailed analysis of the analytical behavior of this system would be beyond the scope of this paper. [ABSTRACT FROM AUTHOR]