Energy content in linear mechanical systems with arbitrary time dependence

A novel complementary variables formulation, permits the obtainment of invariants of a mechanical system following the rationale of Abel's differential equation identity. This approach allows for the decomposition of the total energy of the system into the energy of the object and the dynamic energy of the field. The force acting on the object must be linear in the spatial variable but is arbitrary in the time variable. Several examples, such as the time dependent harmonic oscillator and a swing are described with this complementary variables formulation. The exact solution for the energy of a Lorentz pendulum with uniformly varying length is presented.