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Generalization of Abel's mechanical problem: The extended isochronicity condition and the superposition principle

### Abstract

This paper presents a simple but nontrivial generalization of Abel's mechanical problem, based on the extended isochronicity condition and the superposition principle. There are two primary aims. The first one is to reveal the linear relation between the transit-time T and the travel-length X hidden behind the isochronicity problem that is usually discussed in terms of the nonlinear equation of motion with U(X) being an unknown potential. Second, the isochronicity condition is extended for the possible Abel-transform approach to designing the isochronous trajectories of charged particles in spectrometers and/or accelerators for time-resolving experiments. Our approach is based on the integral formula for the oscillatory motion by Landau and Lifshitz [Mechanics (Pergamon, Oxford, 1976), pp. 27–29]. The same formula is used to treat the non-periodic motion that is driven by U(X). Specifically, this unknown potential is determined by the (linear) Abel transform X(U) ∝ A[T(E)], where X(U) is the inverse function of U(X), is the so-called Abel operator, and T(E) is the prescribed transit-time for a particle with energy E to spend in the region of interest. Based on this Abel-transform approach, we have introduced the extended isochronicity condition: typically, τ = T A (E) + T N (E) where τ is a constant period, T A (E) is the transit-time in the Abel type [A-type] region spanning X > 0 and T N (E) is that in the Non-Abel type [N-type] region covering X < 0. As for the A-type region in X > 0, the unknown inverse function X A (U) is determined from T A (E) via the Abel-transform relation X A (U) ∝ A[T A (E)]. In contrast, the N-type region in X < 0 does not ensure this linear relation: the region is covered with a predetermined potential U N (X) of some arbitrary choice, not necessarily obeying the Abel-transform relation. In discussing the isochronicity problem, there has been no attempt of N-type regions that are practically of full use for the charged-particle spectrometers and/or accelerators. In this Abel-transform approach, the superposition principle simplifies the derivation of X A (U) satisfying the extended isochronicity condition. Although the extended isochronicity condition inevitably discards the low-energy particles, there is no problem for handling accelerated particles because they do not involve the small-amplitude oscillations around the potential minimum. We present analytic examples of X A (U) that are instructive. In Appendix B , Urabe's criterion is interpreted in the time domain, using the Abel-transform approach.

© 2014 AIP Publishing LLC

Received 24 December 2012
Accepted 05 February 2014
Published online 27 February 2014

Acknowledgments:
The author is grateful to Dr. Osamu Furuhashi of Shimadzu Corp. for drawing our attention to the present theme and Professor Kunio Yasue of Notre Dame Seishin University for reading the manuscript and making the indispensable comments. Our gratitude is also extended to one of the referees who pointed out the interesting references concerning the isochronicity problem, including the works on Urabe's criterion.

Article outline:

I. INTRODUCTION
A. Abel's mechanical problem posed and solved in an A-type region
B. Abel-transform approach to periodic motion by Landau and Lifshitz
C. The aims and the contents of this paper
II. THE N-TYPE REGION AND THE EXTENDED ISOCHRONICITY CONDITION
III. THE SUPERPOSITION PRINCIPLE TO DERIVE THE INVERSE POTENTIAL *X* _{ A }(*U*) IN THE A-TYPE REGION
IV. ANALYTIC EXAMPLES OF THE A-TYPE SOLUTION *X* _{ A }(*U*)
A. *X* _{ A }(*U*) for the unbounded N-type region [Figure 3(a)]
B. *X* _{ A }(*U*) for the bounded N-type region [Figure 3(b)]
V. FINAL REMARKS

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