Generalization of Abel's mechanical problem: The extended isochronicity condition and the superposition principle
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 (d{sup 2}X)/(dt{sup 2}) +(dU)/(dX) =0 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), A=(1/√(π))∫{sub 0}{sup E}dU/√(E−U) 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{sub A}(E) +more »
- Publication Date:
- OSTI Identifier:
- 22251210
- Resource Type:
- Journal Article
- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 2; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ACCELERATORS; CHARGED PARTICLES; DESIGN; EQUATIONS OF MOTION; FUNCTIONS; INTEGRALS; LENGTH; NONLINEAR PROBLEMS; OSCILLATIONS; POTENTIALS; SPECTROMETERS