Hamilton's Principle for Material and Nonmaterial Control Volumes Using Lagrangian and Eulerian Description of Motion

The standard form of Hamilton's principle is only applicable to material control volumes. There exist specialized formulations of Hamilton's principle that are tailored to nonmaterial (open) control volumes. In case of continuous mechanical systems, these formulations contain extra terms for the virtual shift of kinetic energy and the net transport of a product of the virtual displacement and the momentum across the system boundaries. This raises the theoretically and practically relevant question whether there is also a virtual shift of potential energy across the boundary of open systems. To answer this question from a theoretical perspective, we derive various formulations of Hamilton's principle applicable to material and nonmaterial control volumes. We explore the roots and consequences of (virtual) transport terms if nonmaterial control volumes are considered and show that these transport terms can be derived by Reynolds transport theorem. The equations are deduced for both the Lagrangian and the Eulerian description of the particle motion. This reveals that the (virtual) transport terms have a different form depending on the respective description of the particle motion. To demonstrate the practical relevance of these results, we solve an example problem where the obtained formulations of Hamilton's principle are used to deduce the equations of motion of an axially moving elastic tension bar.