Augmented Perpetual Manifolds and Perpetual Mechanical Systems-Part II: Theorem for Dissipative Mechanical Systems Behaving as Perpetual Machines

Abstract Perpetual points in mathematics defined recently, and their significance in nonlinear dynamics and their application in mechanical systems is currently ongoing research. The perpetual points significance relevant to mechanics so far is that they form the perpetual manifolds of rigid body motions of mechanical systems. The concept of perpetual manifolds extended to the definition of augmented perpetual manifolds that an externally excited multi-degree of freedom mechanical system is moving as a rigid body. As a continuation of this work, herein the internal force’s and their associated energies, for a motion of multi-degree of freedom dissipative flexible mechanical system with solutions in the exact augmented perpetual manifolds, leads to the proof of a theorem that based on a specific decomposition with respect to their state variables dependence, all the internal forcing vectors are equal to zero. Therefore there is no energy storage as potential energy, and the process is internally isentropic. This theorem provides the conditions that a mechanical system behaves as a perpetual machine of a 2nd and a 3rd kind. The developed theory leads to a discussion for the conditions of the violation of the 2nd law of thermodynamics for mechanical systems that their motion is described in the exact augmented perpetual manifolds. Moreover, the necessity of a reversible process to violate the 2nd thermodynamics law, which is not valid, is shown. The findings of the theorem analytically and numerically are verified. The energies of a perpetual mechanical system in the exact augmented perpetual manifolds for two types of external forces have been determined. Then, in two examples of mechanical systems, all the analytical findings with numerical simulations, certified with excellent agreement. This work is essential in physics since the 2nd law of thermodynamics is not admitting internally isentropic processes in the dynamics of dissipative mechanical systems. Also, the operation of a machine in the exact augmented perpetual manifolds is of extremely high significance to avoid internal damage, and degradation of any internal part of the machine, without energy loss. This article is dedicated to my educators, in all levels of my studies from school to post-doctoral researcher, including summer schools in Greece about Nonlinear dynamics and Chaos, naming few of them that cannot be easily found: Mr. Dimitris Aliapoulios, Prof. Michael Fridakis, Prof. Ajay Kapoor. (in chronological order)