The generalized Hamilton’s principle for a non-material volume

Abstract

Fundamental principles of mechanics were primarily conceived for constant mass systems. Since the pioneering works of Meshcherskii, efforts have been made in order to elaborate an adequate mathematical formalism for variable mass systems. This is a current research field in theoretical mechanics. In this paper, attention is focused on the derivation of the generalized Hamilton’s principle for a non-material volume. First studies on the subject go back at least four decades with the article of McIver (J Eng Math 7(3):249–261, 1973). However, it is curious to note that the extended form of Hamilton’s principle that is derived by McIver does not recover the Lagrange’s equation for a non-material volume which is demonstrated by Irschik and Holl (Acta Mech 153(3–4):231–248, 2002). This does suggest additional theoretical investigations. In the upcoming discussion, Reynolds’ transport theorem is consistently considered regarding the original form of the principle of virtual work, and so the generalized Hamilton’s principle for a non-material volume is properly derived. It is finally shown that the generalization of Hamilton’s principle that is here proposed is in harmony with the Lagrange’s equation which is demonstrated by Irschik and Holl.