Lagrangian Hamiltonian Modeling and Orbital Stability Analysis of Constrained Particle Dynamics on Rotational Surfaces in the Pseudo-Euclidean Space E24

This paper investigates the constrained particle dynamics on rotational surfaces within the 4-dimensional pseudo-Euclidean space E24, characterized by its second-order metric signature of index 2. A comprehensive Lagrangian and Hamiltonian formulation is developed to construct the specific energy and specific angular momentum as conserved Noetherian charges along timelike geodesics. By integrating Clairaut’s theorem into the geodesic flow equations, explicit analytical expressions for these fundamental physical invariants are obtained. This work explores the structural relationship between the surface’s continuous rotational symmetries and the mechanical stability of the geodesic flow. A mathematical resolution for the signature transitions manifested via the appearance of the imaginary unit i on elliptic surfaces is provided through analytic continuation and distinct coordinate charts. Furthermore, by reducing the second-order geodesic flow to a one-dimensional energy balance equation, the exact effective potentials (Veff) are derived, and the local orbital stability zones are analytically verified via second-order radial derivatives (∂s2Veff>0). These embedded geometric configurations are shown to share qualitative features with the equatorial slices of rotating relativistic spacetimes. Consequently, they can serve as potential geometric toy-models for studying the dynamics of photon spheres, ergosphere oscillations, and innermost stable circular orbits in extreme gravitational fields.