Analytical Approximate Solution of a Coupled Two Frequency Hill\'s Equation

Abstract

A coupled two frequency Hill\'s equation is solved. Analytically approximate solution correct up-to first order is derived using modified Lindstedt-Poincare perturbation method. For a wide range of controlling parameters, we compare the numerical and analytical solutions. The solution is the first step towards developing a comprehensive understanding of the electrodynamics of charged particles in a combinational ion trap utilizing both electrostatic DC and RF fields along with a constant static magnetic field with prospects of confining antimatter such as anti-hydrogen for a reasonably long durations of time.

Introduction

Conclusion

For particle trajectory in x-y plane, the analytical approximate solution correct up-to first order, is derived for the coupled two frequency Hill\'s equation using modified Lindstedt-Poincare method. The analytical solution matches well with the numerical solution obtained by numerical simulating the system of coupled differential equations given in Eq. (3). The analytical solution has a limited number of harmonic terms, i.e., (?±?_1 ) and (?±??_1 ) terms, whereas the numerical solution encompasses the effect of all the harmonic terms which make up the complete solution. Therefore, the matching is observed for some range of controlling parameters only. If the order of the analytical solution is increased, the range of operating parameters for which the two solutions coincide will widen. However, the derivation of such higher order terms will be mathematically challenging. It is important to see that the solution described by Eq. 16 and Eq. 17 will blow up when ??1. To obtain single frequency solutions one can simply substitute q_r=0 and keep ? away from unity. In most of the practical settings [7,15], the value of ? is substantially higher than unity, a regime wherein the analytical solutions are a good match to the numerical solutions. Experience guides us that analytical solution correct up-to first and second order are usually sufficient to provide deeper insights to both individual particle as well as collective dynamics inside the trap [17-19]. The relevance of an analytical solution cannot be understated when one must study the collective dynamics inside such combinational traps. Since the fields are spatially linear in this set up, one must see if a distribution function can be constructed for the particles by the method of inversion [17]. It is well known that RF heating on account of applied RF fields will increase the temperature of the charged particles. The analytical tracking of temperature variation for each species inside such a trap is therefore important [18-19]. Temperature can be evaluated as the second order moment of the distribution function. To the best of my knowledge, such analytical work on collective dynamics for combinational traps has not been undertaken. Going ahead in this direction will require us to choose some operating parameters for stable configuration. The analytical expressions for particle dynamics derived in this work assumes importance as a vital starting point. Imperfections in electrode geometry of the trap introduce deviations from the quadrupole potential. It would be interesting to see if analytical solutions can be derived for particle dynamics in such a scenario. Study of nonlinear resonances, deviation in the values of secular frequencies, changes in the stability regimes of the dynamics are all very interesting problems that could be taken up as future work.

References

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Copyright

Copyright © 2024 Namratha Dongari, Anuranjan Kansal, Varun Saxena. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.