arXiv subjects

Karlo Lelas

Publications and source records attributed to Karlo Lelas.

11 recordsLinked to original sources

arXiv2026

Damped harmonic oscillator revisited: a new approach to energy decay in the case of Coulomb, Stokes, and Newton damping

Approximate formulas are derived to describe energy loss in a harmonic oscillator that experiences three distinct damping mechanisms: constant-magnitude (Coulomb), velocity-proportional (Stokes), and velocity-squared (Newton), using fundamental mathematical methods and physical insight. Our methodology leverages an understanding of the free harmonic oscillator and the inherent link between energy dissipation rates and the power exerted by damping forces. We establish a direct analytical framework for assessing the energy of a damped harmonic oscillator, obviating the need for amplitude-based equations. The simplicity of our findings is accompanied by their remarkable accuracy when validated against exact or computational simulations. In addition to an excellent approximate description of the energy decay, we also show how to derive an exact solution in the case of Stokes damping without relying on the standard procedure for solving second-order differential equations. The theoretical underpinnings and mathematical strategies employed are well-suited for undergraduate-level or advanced high school physics instruction.

physics.class-ph↗

arXiv2026

An alternative approach to several important systems in classical mechanics: energy factorization

We show how several important classical problems, with positive definite potential energy, can be solved by starting from the factorization of the total mechanical energy using complex numbers. In particular, we derive in a new way exact analytical solutions for: simple harmonic oscillator, vertical projectile motion, motion under a repulsive inverse cube force, and damped harmonic oscillator (with linear damping). We also show how this approach easily yields an excellent approximation of the energy decay and a new approximate analytical solution in the case of a weakly damped harmonic oscillator. Our derivations are suitable for undergraduate physics teaching as an alternative to solving Newton's equations of motion. In addition, we comment on the limitations of our approach, but also on the insights it provides and opportunities for further research.

physics.class-ph↗

arXiv2025

New approach to approximate analytical solutions of a harmonic oscillator with weak to moderate nonlinear damping: Part I

We introduce a new approach to deriving approximate analytical solutions of a harmonic oscillator damped by purely nonlinear, or combinations of linear and nonlinear damping forces. Our approach is based on choosing a suitable trial solution, i.e. an ansatz, which is the product of the time-dependent amplitude and the oscillatory (trigonometric) function that has the same frequency but different initial phase, compared to the undamped case. We derive the equation for the amplitude decay using the connection of the energy dissipation rate with the power of the total damping force and the approximation that the amplitude changes slowly over time compared to the oscillating part of the ansatz. By matching our ansatz to the initial conditions, we obtain the equations for the corresponding initial amplitude and initial phase. Here we demonstrate the validity of our approach in the case of damping quadratic in velocity, Coulomb damping, and a combination of the two, i.e. in this paper we consider purely nonlinear damping, while the dynamics with combinations of damping linear in velocity and nonlinear damping will be analyzed in a follow-up paper. In the case of damping quadratic in velocity, by comparing our approximate analytical solutions with the corresponding numerical solutions, we find that our solutions excellently describe the dynamics of the oscillator in the regime of weak to moderately strong quadratic damping. In the case of Coulomb damping, as well as in the case of a combined Coulomb and quadratic damping, our approximate analytical solutions agree well with the corresponding numerical solutions until the last few half-periods of the motion. Therefore, for these two cases, we introduce improved variants of our approximate solutions which describe the dynamics well until the very end.

physics.class-ph↗

arXiv2025

Addendum: Modeling the amplitude and energy decay of a weakly damped harmonic oscillator using the energy dissipation rate and a simple trick (2025 Eur. J. Phys. 46(1) 015004)

We show how to adapt the approach introduced for viscous damping in [1] to derive the approximate amplitude decay in the case of damping by a force of constant magnitude (sliding friction) and in the case of damping by a force proportional to the square of velocity (air resistance). We obtain two first-order differential equations from which we obtain the approximate time-dependent amplitudes corresponding to the considered damping forces. Our approach is suitable for first-year undergraduates, as it relies on the physical concepts and mathematical techniques they are familiar with.

physics.class-ph↗

arXiv2025

The amplitude decay of a harmonic oscillator damped simultaneously by weak linear and nonlinear damping forces

We derive approximate expressions for the amplitude decay of harmonic oscillations weakly damped by the simultaneous action of three different damping forces: force of constant magnitude, force linear in velocity, and force quadratic in velocity. Our derivation is based on a basic understanding of the undamped harmonic oscillator and the connection between the energy dissipation rate and the power of the total damping force. By comparing our approximate analytical solutions with the corresponding numerical solutions, we find that our solutions excellently describe the dynamics of the oscillator in the regime of weak damping by combinations of these three forces for an experimentally relevant range of corresponding damping constants. The physical concepts and mathematical techniques we employ are suitable for undergraduate physics teaching.

physics.class-ph↗

arXiv2025

Optimal damping adapted to a set of initial conditions

Vibrating systems can respond to an infinite number of initial conditions and the overall dynamics of the system can be strongly affected by them. Therefore, it is of practical importance to have methods by which we can determine the damping that is in some sense optimal for all initial conditions, or for a given set of initial conditions. For a single and multi degree of freedom systems, we determine the optimal damping coefficients adapted to different sets of initial conditions using the known method of minimizing the (zero to infinity) time integral of the energy of the system, averaged over a set of initial conditions, and using two new methods that we introduce. One method is based on determining the damping for which the energy of the system, averaged over a set of initial conditions, drops the fastest to a given threshold value. The other method is based on determining the damping that gives minimal average settling time of the system, where we take that the system settled when its energy dropped to a given threshold value. We show that the two new methods give results for optimal damping that are in excellent agreement with each other, but are significantly different from the results given by the minimization of the average energy integral. More precisely, for considered multi degree of freedom systems and sets of initial conditions, the two new methods give optimal damping coefficients that converge to the critical damping of the first mode as the target energy threshold decreases. On the other hand, for these same systems and sets of initial conditions, the method of minimizing the average energy integral gives optimal damping coefficients which are deep in the overdamped regime with respect to the first mode.

physics.class-ph↗

arXiv2025

Driven damped harmonic oscillator revisited: energy resonance

We derive the exact expression for the resonant frequency of the time-averaged steady-state energy and show that this frequency is excellently approximated by the arithmetic mean of the amplitude and velocity resonant frequencies. In addition, we argue that the frequency of the amplitude resonance can be regarded as the energy resonance frequency, since it provides the maximal peak values of instantaneous energy.

physics.class-ph↗

arXiv2024

Modeling the amplitude and energy decay of a weakly damped harmonic oscillator using the energy dissipation rate and a simple trick

We demonstrate how to derive the exponential decrease of amplitude and an excellent approximation of the energy decay of a weakly damped harmonic oscillator without solving the associated equation of motion and without insight into the analytical form of its solution. This is achieved using a basic understanding of the undamped harmonic oscillator and the connection between the damping force's power and the energy dissipation rate. The trick is adding the energy dissipation rates corresponding to two specific pairs of initial conditions with the same energy. In this way, we obtain a first-order differential equation from which we get the time-dependent amplitude and the energies corresponding to each pair of considered initial conditions. Comparing the results of our model to the exact solutions and energies yielded an excellent agreement. The physical concepts and mathematical techniques we employ are well-known to first-year undergraduates.

physics.class-ph↗

arXiv2023

Damped harmonic oscillator revisited: the fastest route to equilibrium

Theoretically, solutions of the damped harmonic oscillator asymptotically approach equilibrium, i.e., the zero energy state, without ever reaching it exactly, and the critically damped solution approaches equilibrium faster than the underdamped or the overdamped solution. Experimentally, the systems described with this model reach equilibrium when the system's energy has dropped below some threshold corresponding to the energy resolution of the measuring apparatus. We show that one can (almost) always find an optimal underdamped solution that will reach this energy threshold sooner than all other underdamped solutions, as well as the critically damped solution, no matter how small this threshold is. We also comment on one exception to this for a particular type of initial conditions, when a specific overdamped solution reaches the equilibrium state sooner than all other solutions. We confirm some of our findings experimentally.

physics.class-ph↗

arXiv2020

Modulation instability in the nonlinear Schrödinger equation with a synthetic magnetic field: gauge matters

We theoretically investigate the phenomenon of modulation instability for systems obeying nonlinear Schrödinger equation, which are under the influence of an external homogeneous synthetic magnetic field. For an initial condition, the instability is detected numerically by comparing dynamics with and without a small initial perturbation; the perturbations are characterized in a standard fashion by wavevectors in momentum space. We demonstrate that the region of (in)stability in momentum space, as well as time-evolution in real space, for identical initial conditions, depend on the choice of the gauge (i.e., vector potential) used to describe the homogeneous synthetic magnetic field. This superficially appears as if the gauge invariance is broken, but this is not true. When the system is evolved from an identical initial condition in two different gauges, it is equivalent to suddenly turning on the synthetic magnetic field at $t=0$. This gives rise, via Faraday's law, to an initial instantaneous kick of a synthetic electric field to the wavepacket, which can differ for gauges yielding an identical uniform magnetic field at $t>0$.

cond-mat.quant-gas↗

arXiv2015

The Harper-Hofstadter Hamiltonian and conical diffraction in photonic lattices with grating assisted tunneling

We introduce a grating assisted tunneling scheme for tunable synthetic magnetic fields in photonic lattices, which can be implemented at optical frequencies in optically induced one- and two-dimensional dielectric photonic lattices. We demonstrate a conical diffraction pattern in particular realization of these lattices which possess Dirac points in $k$-space, as a signature of the synthetic magnetic fields. The two-dimensional photonic lattice with grating assisted tunneling constitutes the realization of the Harper-Hofstadter Hamiltonian.

physics.optics↗