Abstract
This work, relative to the principle of virtual power, is composed of three distinct but nevertheless complementary parts. The first part follows the line of thought developed by professor Maugin and his students on complex continuous media subject to the objectivity requirement (translational and rotational invariances). The second part shows that this principle is extensible to other types of invariance such as gauge and scale invariances. Gauge invariance allows to express Maxwell equations, usually derived through a vector approach, by use of a scalar principle having the same formal structure as the principle of virtual power. As to scale invariance, it allows to deal, in a general and unified way whatever the underlying physics, with the passage from a continuous medium to a discontinuous one (singular surfaces, lines or points). The third part concerns the foundations of dynamics where the principle of virtual power appears as a theorem, like other analytical principles, each corresponding to one point of view, deductible from a general intrinsic (viewpoint independent) dynamical framework. The attention will be focused on the origin of the duality notion, at the basis of the principle of virtual power.
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Acknowledgements
This study relative to the principle of virtual power, based on the notions of duality and invariance, owes much to the scientific formation that one of us received directly, in Paris, from Professor Maugin and his first students and collaborators, mainly B. Collet and J. Pouget. As for its extension to gauge and scale invariances and the search for a more solid conceptual basis likely to go back to the origin of the notion of duality, they would not have been possible without the contributions, remarks and criticism of the members (epistemologists, physicists and mathematicians) of the “Epiphymaths” group (an interdisciplinary seminar held weekly at Besançon), especially, J. Merker who presented, in the nineties of the last century, recent studies concerning an autonomous dynamical framework, dealt with through group theory, and C. A. Risset who, later on, accompanied this research over the years, bringing different suggestions and improvements.
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Hirsinger, L., Daher, N., Devel, M., Lecoutre, G. (2018). The Principle of Virtual Power (PVP): Application to Complex Media, Extension to Gauge and Scale Invariances, and Fundamental Aspects. In: Altenbach, H., Pouget, J., Rousseau, M., Collet, B., Michelitsch, T. (eds) Generalized Models and Non-classical Approaches in Complex Materials 2. Advanced Structured Materials, vol 90. Springer, Cham. https://doi.org/10.1007/978-3-319-77504-3_2
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