Mixed convolved action.

A stationary principle is developed for dynamical systems by formulating the concept of mixed convolved action, which is written in terms of displacement and force variables, using temporal convolutions and fractional derivatives. The classical canonical single-degree-of-freedom dynamical system is considered as an initial application. With this new formulation, a single real scalar functional provides the governing differential equations, along with all the pertinent initial conditions, as the Euler-Lagrange equations emanating from the stationarity of the mixed convolved action. Both conservative and non-conservative processes can be considered within a common framework, thus resolving a long-standing limitation of variational approaches for dynamical systems. Several results in fractional calculus also are developed.