Perpetual points in mechanical systems defined recently. Herein are used to seek specific types of solutions of N-degrees of freedom systems, and their significance in mechanics is discussed. In discrete linear mechanical systems, is proven, that the perpetual points are forming the perpetual manifolds and they are associated with rigid body motions, and these systems are called perpetual. The definition of perpetual manifolds herein is extended to the augmented perpetual manifolds. A theorem, defining the conditions of the external forces applied in an N-degrees of freedom system lead to a solution in the exact augmented perpetual manifold of rigid body motions, is proven. In this case, the motion by only one differential equation is described, therefore forms reduced-order modelling of the original equations of motion. Further on, a corollary is proven, that in the augmented perpetual manifolds for external harmonic force the system moves in dual mode as wave-particle. The developed theory is certified in three examples and the analytical solutions are in excellent agreement with the numerical simulations. The outcome of this research is significant in several sciences, in mathematics, in physics and in mechanical engineering. In mathematics, this theory is significant for deriving particular solutions of nonlinear systems of differential equations. In physics/mechanics, the existence of wave-particle motion of flexible mechanical systems is of substantial value. Finally in mechanical engineering, the theory in all mechanical structures can be applied, e.g. cars, aeroplanes, spaceships, boats etc. targeting only the rigid body motions.

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