On Hamilton's Principle for Discrete and Continuous Media

Abstract

In an attempt to generalize the Hamilton's principle, an action functional is proposed which, unlike the standard version of the principle, accounts properly for all initial data and the possible presence of dissipation. To this end, the convolution is used instead of the $L^2$ inner product so as to eliminate the undesireble end temporal condition of Hamilton's principle. Also, fractional derivatives are used to account for dissipation and the Dirac delta function is exploited so as the initial velocity to be inherently set into the variational setting. The proposed approach apllies in both finite and infinite dimensional systems.