A vector-space implementation of Hamilton's Law of Varying Action (HLVA) is used to generate direct solutions to linear and nonlinear initial value response problems in the time domain. This vector-space perspective utilizes unconstrained temporal-basis-functions while preserving HLVA in its original form. For linear systems, the general response solution may be obtained without reference to initial conditions or forcing functions. By subsequently using the initial conditions and forcing functions, the unique response may be generated. The vector-space approach not only generates the response for linear and nonlinear systems, but as a matter of course, generates the set of spatiotemporal functions spanning the solution space of the response. These spatiotemporal functions are descriptively denoted as fundamental-time-modes (FTMs). The FTM composition of the response of linear systems, and nonlinear systems exhibiting nonlinear normal mode response are illustrated. The FTMs composing the nonlinear normal mode response of the examples presented, are also the eigenfunctions of HLVA. Thus, the vector-space approach also provides the eigenfunction composition of the nonlinear normal modes for the examples considered. The vector-space perspective takes advantage of classical model reduction techniques by eliminating non-dominant FTMs from the solution process, resulting in model reduction in the time domain. Model reduction for linear and nonlinear systems are illustrated.; The vector-space perspective of HLVA. has unique capabilities not offered by the traditional viewpoint of HLVA. One of these unique capabilities manifests itself in the scalable superposition of FTMs While this is expected and demonstrated for a linear system, scalable superposition of FTMs for a nonlinear system, as an extraordinary feature uncovered via the vector-space approach to HLVA, has been demonstrated herein. |