A novel study on the fourth first integral for the rotatory motion of an impacted charged rigid body by external torques

Abstract

The main purpose of this study is to explore the feasibility of achieving a fourth integral for a charged rigid body (RGB) rotary movement, considering the influence of gyrostatic torque (GT) and a uniform electromagnetic field. Six nonlinear first-order differential equations (DEs) and their three first integrals, pertaining to geometry, area, and energy, regulate this problem. The analytical solution to the issue necessitates approaching a crucial fourth integral. This paper presents a pivotal criterion for a function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>F</mml:mi> </mml:mrow> </mml:math> , contingent on all RGB variables, to qualify such integral is outlined. The achieved outcomes are validated through comparing them with the renowned cases and then they can be considered a consolidation or mainstreaming of prior research efforts.