A Geometric Reinterpretation of Energy Dynamics via Real–Imaginary Orthogonal Projections

This paper introduces a geometric reinterpretation of energy dynamics by representing kinetic and potential energy components as orthogonal projections on real and imaginary axes of an abstract energy-state space. The system energy is modeled as a rotating unit vector confined to a quarter-circle, providing an intuitive geometric picture of energy exchange, oscillatory behavior, equilibrium, and stability. The framework preserves the scalar Hamiltonian while revealing a nontrivial angular structure underlying energy redistribution, offering a transparent bridge between Lagrangian and Hamiltonian mechanics with potential applications in dynamical systems, control theory, and energy-based modeling.