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Review Article

Hamilton’s Principle for Material and Non-Material Control Volumes Using Lagrangian and Eulerian Description of Motion

[+] Author and Article Information
Andreas Steinboeck

Automation and Control Institute, Vienna University of Technology, Gußhausstraße 27-29/376, 1040 Vienna, Austria
andreas.steinboeck@tuwien.ac.at

Martin Saxinger

Christian Doppler Laboratory for Model-Based Process Control in the Steel Industry, Automation and Control Institute, Vienna University of Technology, Gußhausstraße 27-29/376, 1040 Vienna, Austria
saxinger@acin.tuwien.ac.at

Andreas Kugi

Professor Christian Doppler Laboratory for Model-Based, Process Control in the Steel Industry, Automation and Control Institute, Vienna University of Technology, Gußhausstraße 27-29/376, 1040 Vienna, Austria
kugi@acin.tuwien.ac.at

1Corresponding author.

ASME doi:10.1115/1.4042434 History: Received September 21, 2018; Revised December 27, 2018

Abstract

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 non-material (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 non-material control volumes. We explore the roots and consequences of (virtual) transport terms if non-material control volumes are considered and show that these transport terms can be derived by Reynold's 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.

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