Resolving Absorbed Work and Generalized Inertia Forces from System Energy Equation - A Hamiltonian and Phase-Space Kinematics Approach

This paper develops a theoretical basis and a systematic process for resolving all inertia forces along generalized coordinates from the overall energy equation of a dynamical system. The theory is developed for natural systems with scleronomic constraints, where the potential energy is independent of generalized velocities. The process involves expansion of the energy equation, and specifically a special expansion of the kinetic energy term, from which the inertia forces emerge. The expansion uses fundamental kinematic identities of the phase space. It is also guided by insights drawn from the structure of the Hamiltonian function. The resulting equation has the structure of the D'Alembert's equation but involving generalized coordinates, from which the Lagrange's equations of motion are obtained. The expansion process elucidates how certain inertia forces manifest in the energy equation as composite terms that must be accurately resolved along different generalized coordinates. The process uses only the system energy equation, and neither the Hamiltonian nor the Lagrangian function are required. Extension of this theory to non-autonomous and nonholonomic systems is an area of future research.