Embedding Inertia Sequence: Hidden Variable Conservation and Invariance Principle for Periodic Motion

ith periodicity and symmetry being core theme in physics and mathematics leading to nonlinearity and complexity, and yet since the time of Newton periodic motion has been regarded in physics as forced instead of inertia motion without an inherent balancing-and-unchanging reference frame of invariants connecting force and motion, functional series from calculus as has been deployed in mathematics was developed however from an isolated view on force and motion with yet inevitable connection in between, leading to historical challenge in understanding the cause and effect relation for the evolution of periodic motion. It is recognized however that rather than the observed variable energy conservation default in analytical mechanics, a periodic motion system is essentially governed by hidden variable real and imaginary energy conservation, also the periodic motion of each individual object is in fact a two-dimension inertia motion and yet it inherently evolves to follow an embedding inertia sequence of self-similar periodic motion, with the pattern of which analogous to the mechanistically yet enigmatic cohomology sequence across mathematics deploying however hidden variable based operators, revealing from the nature of evolution the limitation of the default straight line pattern of summation in calculus as least representation for the change in periodic force and motion. As the functional series convention stands essentially for a subjective approximation to the evolution without however awareness of the energy conservation constraint which actually governs the evolution, the embedding sequence concept is expected to be a unified invariant methodology of linear representation for nonlinearity and symmetry, and shall further represent a paradigm shift of cognition on change of motion and facilitate understanding on the common pattern of evolution in natural and engineering.