Solutions of the Mathieu–Hill Equation for a Trapped-Ion Harmonic Oscillator—A Qualitative Discussion

Bogdan M. Mihalcea ()
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Bogdan M. Mihalcea: National Institute for Laser, Plasma and Radiation Physics (INFLPR), Atomiştilor Str. Nr. 409, 077125 Măgurele, Romania

Mathematics, 2024, vol. 12, issue 19, 1-20

Abstract: We investigate solutions of the classical Mathieu–Hill (MH) equation that characterizes the dynamics of trapped ions. The analytical model we introduce demonstrates the equations of motion are equivalent to those of a harmonic oscillator (HO). Two independent approaches are used, based on two classes of complex solutions of the MH equation. This paper addresses both a damped HO and parametric oscillator (PO) for an ion confined in an electrodynamic (Paul) trap, along with stability and instability regions for the associated periodic orbits.

Keywords: Mathieu–Hill equation; Floquet theory; Sturm–Liouville theorem; electrodynamic trap; stability diagram (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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