Dargush G F, Kim J
Department of Mechanical and Aerospace Engineering University at Buffalo, State University of New York, Buffalo, New York 14260, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Jun;85(6 Pt 2):066606. doi: 10.1103/PhysRevE.85.066606. Epub 2012 Jun 19.
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.
通过构建混合卷积作用的概念，利用时间卷积和分数阶导数，基于位移和力变量来表述，为动力学系统建立了一个平稳原理。经典的单自由度规范动力学系统被视为初始应用。通过这种新的表述，一个单一的实标量泛函作为混合卷积作用平稳性产生的欧拉 - 拉格朗日方程，提供了控制微分方程以及所有相关的初始条件。保守和非保守过程都可以在一个共同的框架内进行考虑，从而解决了动力学系统变分方法长期存在的局限性。还得出了分数阶微积分中的几个结果。
