Accelerated Particles in an AVF Cyclotron with a One or a Two Dee System
A general analytical theory is given, describing the motion of accelerated particles in an AVF cyclotron with a one or a two Dee-system. A Hamiltonian is formulated for the horizontal motion of these particles. Especially on the first turns, where the energy gain per turn is not small with respect to the particle energy, the structure of the Dees has much influence. For high harmonic acceleration, the motion in the radial phase space may even become unstable. A coupling exists between radial and longitudinal motion. In a few examples the applicability of the formulated theory is illustrated.
In this paper we study the multivariate ANOVA decomposition for $1$-periodic functions on the torus. In particular we use the integral projection operator that leads to the classical ANOVA decomposition. Relationships between the Fourier coefficients of the function and its ANOVA terms lead to special frequency index sets and give an understanding of the decomposition working in the frequency domain. Moreover, we consider the truncated ANOVA decomposition and provide error bounds for approximation in $\mathrm{L}_\infty$ and $\mathrm{L}_2$. We present an approximation method based on the truncated decomposition with regard to a superposition dimension $d_s$.