A four-tensor momenta equation for rolling physics

Abstract

Relativistic four-tensor equation dJμν = Mμνdt is developed to analyse linear translation with rotation processes. The postulated cause-effect four-tensor equation, a relativistic generalisation for classical angular-impulse–angular-momentum variation equation dJ = Mdt, includes the Poinsot-Euler rotation (angular-impulse–angular-momentum variation) equation, Newton's second law (linear-impulse–linear-momentum variation equation), and thermodynamics first law (work–energy equation). This four-tensor formalism is applied to describe three linear translation with rotation processes: a ring rolling on the floor by a horizontal force linear impulse and torque, fulfilling the rolling condition (mechanical energy conservation), a spinning ring placed on the ground until achieved the rolling condition (mechanical energy dissipation by friction), and a fireworks wheel ascending an incline (mechanical energy production by decreasing a thermodynamic potential).