Abstract
The linear oscillator is a simple model that lies at the foundation of many physical phenomena and plays a crucial role in accelerator dynamics. Many systems can be viewed as an approximation to a set of independent linear oscillators. In this chapter, we will review the main properties of the linear oscillator including its response to resonant excitations, slowly varying forces, random kicks, and parametric variation of the frequency. We will discuss the impact of damping terms as well as how small, nonlinear terms in the oscillator equation modify the oscillator frequency and lead to nonlinear resonance.
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Notes
- 1.
Here we use the definition of the complete elliptic integral following the convention of the software package Mathematica [1], \(K(m)=\int _0^{\pi /2}\left( 1-m\sin ^2\theta \right) ^{-1/2}d\theta \).
References
Wolfram Research, Inc. Mathematica, Version 11.2, 2017
L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics, 3rd edn., Mechanics (Elsevier Butterworth-Heinemann, Burlington MA, 1976). (translated from Russian)
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Stupakov, G., Penn, G. (2018). Linear and Nonlinear Oscillators. In: Classical Mechanics and Electromagnetism in Accelerator Physics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-90188-6_4
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DOI: https://doi.org/10.1007/978-3-319-90188-6_4
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