A Systematic Approach to Deriving Relativistic Momentum and Energy Relations

Classical Newtonian mechanics provides an exceptionally accurate framework for describing macroscopic motion at non-relativistic speeds. However, as particle velocities approach the speed of light, the classical definitions of linear momentum and kinetic energy break down, failing to satisfy the fundamental law of conservation of momentum across all inertial frames. To preserve the universality of physical laws under Lorentz transformations, the definitions of dynamical quantities must be modified. This paper presents a rigorous, step-by-step mathematical derivation of relativistic dynamical principles starting from proper-time invariance. First, relativistic linear momentum is derived using the invariant proper-time element. Second, employing the work-energy theorem alongside the calculus of the Lorentz factor, the paper derives relativistic kinetic energy and demonstrates the emergence of rest energy. Third, it formulates the fundamental energy-momentum invariant relation and conversely verifies the consistency of total energy starting directly from the hyperbolic energy-momentum relation.