Spatial Stark-Zeeman Systems and Their Regularizations

Seongchan Kim; Kevin Ruck

Home/Archive/arXiv

arXiv · 2507.15172

Spatial Stark-Zeeman Systems and Their Regularizations

Seongchan Kim·Kevin Ruck

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this article, we study spatial Stark-Zeeman systems which describe the dynamics of a charged particle moving in three-dimensional space under the influence of a Coulomb potential, a magnetic field, and an electric field, possibly time-dependent. Such systems are modeled by Hamiltonian flows on the cotangent bundle of an open subset of $\mathbb{R}^3, $ equipped with a twisted symplectic structure. The presence of the Coulomb singularity leads to the study of collision orbits, and hence understanding the regularization of these orbits is essential for global dynamical properties. We investigate regularization techniques for spatial Stark-Zeeman systems, both in time-independent and time-dependent cases. In particular, in the time-dependent case, following a new regularization method developed by Barutello, Ortega, and Verzini, we formulate the corresponding regularized variational principles and carefully analyze the effects of magnetic and electric terms under the Kustaanheimo-Stiefel transformation. The resulting regularized action functional yields a variational characterization of collision orbits and facilitates further analysis of periodic solutions. Our results provide a general scheme for regularizing spatial Stark-Zeeman systems, opening the door for further applications in symplectic geometry, Floer theory, and celestial mechanics.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Seongchan Kim, Kevin Ruck. 2025-09-25. Spatial Stark-Zeeman Systems and Their Regularizations. https://arxiv.org/abs/2507.15172

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Pivoting technique for the circle homeomorphism group

We adapt Gou{ë}zel's pivoting technique to the circle homeomorphism group. As an application, we give different proofs of Gilabert Vio's probabilistic Tits alternative and Malicet's exponential synchronization.

math.DS↗

arXiv2026

Asymmetry of a class of Mellin transforms

We introduce the quantity $μ_η$, defined for every complex $s$ in the critical strip, as a transformation of the Mellin transform associated to the functions $η$. We establish a sufficient condition on $η$ under which $μ_η(s)$ and $μ_η(1-s)$ cannot both vanish outside the critical line. An application is given to the case in which $η$ is the fractional part function, and the zeros of $μ_η$ coincide with the zeros of the Riemann zeta function.

math.DS↗

arXiv2026

Infinite Set of Resonances in the Linear Damped Oscillator Subject to Harmonic Forcing with Non-standard Frequency Modulation

It is shown that harmonic signals incorporating a type of weak non-standard frequency modulation (wNSFM) have interesting spectral properties, namely, time-dependent bandwidths that become increasingly broader with increasing time. As such, they represent a class of signals with frequency-time coupling in their spectra. Specifically, the weakly damped oscillator exhibits always two transient resonance captures involving two distinct harmonics possessing relatively high amplitudes over finite time intervals, while the overall response decays as $~t^{-1/2}$ as $t\rightarrow\infty$. Considering the undamped oscillator, it possesses two types of resonances, referred to as simple and non-simple resonances. Simple resonances correspond to finite-amplitude steady-state responses caused by two sustained resonance captures, in the form of two distinct modulated quasi-periodic responses, which, however are "activated" at different time instances. The necessary and sufficient conditions for non-simple resonances are given in the form of a theorem which predicts the existence of resonant harmonics and specifies the special phase conditions that the resonant harmonics must satisfy for constructive interference; the resulting undamped non-simple resonance grows unboundedly as $~t^{-1/2}$ as $t\rightarrow\infty$, in contrast to the classical resonance growth of the linear resonator with unmodulated harmonic excitation whose response grows as $~t$ as $t\rightarrow\infty$. These resonant responses are persistent to changes in the parameters of the wNSFM. Our results reveal interesting infinite sets of resonances in linear SDOF resonators under frequency-modulated excitations.

math.DS↗