An Algebraic Methodology for Generalizing the Lorentz Transformation Through Quartic Equation Solutions

In this paper, we propose a novel methodology to construct the Lagrangian function using the radical solutions of a quartic equation. By examining the generalized coordinates $q$ and velocities $\dot{q}$ expressed through the radical solutions of the quartic equation, we delve into the relationship between the roots of the quartic equation and the Lorentz transformation. Our findings suggest a potential algebraic generalization of the Lorentz transformation, shedding light on the mathematical structures underpinning relativistic physics.