Resonant normal form and asymptotic normal form behavior in magnetic bottle Hamiltonians

TL;DR

This paper introduces a new method for constructing normal forms in magnetic bottle Hamiltonians, analyzes their asymptotic behavior, and demonstrates how these forms can predict invariant structures and chaos thresholds.

Contribution

It presents a novel approach to normal form construction in resonant cases and studies their asymptotic properties, with applications to magnetic bottle Hamiltonians.

Findings

Normal form series have an optimal truncation order with exponentially small remainders.

The method accurately describes invariant curves inside resonance islands.

Normal forms can estimate the chaos threshold in magnetic bottle systems.

Abstract

We consider normal forms in `magnetic bottle' type Hamiltonians of the form $H=\frac{1}{2}(\rho^2_\rho+\omega^2_1\rho^2) +\frac{1}{2}p^2_z+hot$ (second frequency $\omega_2$ equal to zero in the lowest order). Our main results are: i) a novel method to construct the normal form in cases of resonance, and ii) a study of the asymptotic behavior of both the non-resonant and the resonant series. We find that, if we truncate the normal form series at order $r$, the series remainder in both constructions decreases with increasing $r$ down to a minimum, and then it increases with $r$. The computed minimum remainder turns to be exponentially small in $\frac{1}{\Delta E}$, where $\Delta E$ is the mirror oscillation energy, while the optimal order scales as an inverse power of $\Delta E$. We estimate numerically the exponents associated with the optimal order and the remainder's exponential asymptotic behavior. In the resonant case, our novel method allows to compute a `quasi-integral' (i.e. truncated formal integral) valid both for each particular resonance as well as away from all resonances. We applied these results to a specific magnetic bottle Hamiltonian. The non resonant normal form yields theorerical invariant curves on a surface of section which fit well the empirical curves away from resonances. On the other hand the resonant normal form fits very well both the invariant curves inside the islands of a particular resonance as well as the non-resonant invariant curves. Finally, we discuss how normal forms allow to compute a critical threshold for the onset of global chaos in the magnetic bottle.