The Stark problem in the Weierstrassian formalism

TL;DR

This paper introduces a comprehensive closed-form solution to the Stark problem using Weierstrass elliptic functions, enabling exact analysis of system dynamics for all external force values.

Contribution

It provides the first exact, universal solution to the Stark problem in Weierstrass formalism, applicable to all initial conditions and parameters.

Findings

Identification of quasi-periodic and periodic orbits

Analytical criterion for trajectory boundness

Characterization of equilibrium points

Abstract

We present a new general, complete closed-form solution of the Stark problem in terms of Weierstrass elliptic and related functions. With respect to previous treatments of the problem, our analysis is exact and valid for all values of the external force field, and it is expressed via unique formul\ae{} valid for all initial conditions and parameters of the system. The simple form of the solution allows us to perform a thorough investigation of the properties of the dynamical system, including the identification of quasi-periodic and periodic orbits, the formulation of a simple analytical criterion to determine the boundness of the trajectory, and the characterisation of the equilibrium points.