Spatial Stark–Zeeman systems and their regularizations

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 $${\mathbb {R}}^3, $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> 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 the 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.