Time modes and nonlinear systems

A vector-space implementation of the equation that has been known as Hamiltonʹs Law of Varying Action is used with a non-virtual perspective to generate direct solutions to nonlinear initial value response problems in the time domain. This novel vector-space perspective unleashes the power of linear algebra to provide new capabilities with Hamiltonʹs Law of Varying Action and new insights into nonlinear dynamics. In particular, this perspective permits the use of unconstrained temporal-basis-functions without the need to augment Hamiltonʹs Law of Varying Action with Lagrange multipliers, and provides a linearly independent set of spatiotemporal functions spanning the solution space of the response. These spatiotemporal functions are descriptively denoted as fundamental-time-modes. By providing this basis of fundamental-time-modes, the concepts of linear algebra can be taken advantage of to accomplish model reduction and superposition for nonlinear systems. In particular, classical model reduction techniques are used to eliminate non-dominant fundamental-time-modes from the solution process, resulting in model reduction in the time domain. In addition, using the novel concept of eigen-direction iteration, linearly independent fundamental-time-mode response trajectories along eigen-directions are superposed to yield new response trajectories for nonlinear systems. These capabilities of the vector-space approach are demonstrated for nonlinear systems exhibiting nonlinear normal mode response. The article concludes with an Appendix A entitled “The Law of Evolutionary Energy, Hamiltonʹs Law of Varying Action and the Principle of Virtual Work”, providing essential insight to the non-virtual working equation of the authors’ previous articles and the current article, referred to as Hamiltonʹs Law of Varying Action, to distinguish it from virtual concepts and virtual variational principles of classical dynamics. It is shown that this particular equation referred to as Hamiltonʹs Law of Varying Action is a special case of the Law of Evolutionary Energy which inherently encompasses the First Law of Thermodynamics involving quantifiable real changes (alterations) along the real dynamic path, the application of which to mechanical systems yields both the current non-virtual working form of Hamiltonʹs Law of Varying Action and the original form of the only enunciated (non-virtual) “law of varying action” by Hamilton which are not the same expressions.