Continuous piecewise linearization method for approximate periodic solution of the relativistic oscillator

This paper presents an approximate periodic solution to the vibration of the relativistic oscillator using a novel analytical method called continuous piecewise linearization method. First, an equivalent conservative equation for the vibration of the relativistic oscillator was derived in a simple straightforward manner that elucidates the physical meaning of the conservative equation. The continuous piecewise linearization method was then applied to derive periodic solutions for the displacement and velocity of the relativistic oscillator based on the conservative equation. The results of the present method were compared with results of published methods and exact numerical solution and the maximum error of the present method was less than 0.002%. The model derivations and the solutions presented in this paper are considerably simple and very accurate and can be used to introduce the relativistic oscillator in relevant undergraduate courses on dynamics. Essentially, knowledge of freshman calculus is sufficient to comprehend and implement the continuous piecewise linearization method for the relativistic oscillator.

ON PERIODIC SOLUTIONS OF LIÉNARD TYPE EQUATIONS

The Liénard type equation x'' + f(x, x')x' + g(x) = 0 (i) is considered. We claim that if the associated conservative equation x'' + g(x) = 0 has period annuli then a dissipation f(x, x') exists such that a limit cycle of equation (i) exists in a selected period annulus. Moreover, it is possible to define f(x, x') so that limit cycles appear in all period annuli. Examples are given. A particular example presents two limit cycles of non-convex shape in two disjoint period annuli.

HYDRODYNAMICS OF SELF-ALIGNMENT INTERACTIONS WITH PRECESSION AND DERIVATION OF THE LANDAU–LIFSCHITZ–GILBERT EQUATION

We consider a kinetic model of self-propelled particles with alignment interaction and with precession about the alignment direction. We derive a hydrodynamic system for the local density and velocity orientation of the particles. The system consists of the conservative equation for the local density and a non-conservative equation for the orientation. First, we assume that the alignment interaction is purely local and derive a first-order system. However, we show that this system may lose its hyperbolicity. Under the assumption of weakly nonlocal interaction, we derive diffusive corrections to the first-order system which lead to the combination of a heat flow of the harmonic map and Landau–Lifschitz–Gilbert dynamics. In the particular case of zero self-propelling speed, the resulting model reduces to the phenomenological Landau–Lifschitz–Gilbert equations. Therefore the present theory provides a kinetic formulation of classical micromagnetization models and spin dynamics.