2. The Principle of Virtual Power (PVP): Application to Complex Media, Extension to Gauge and Scale Invariances, and Fundamental Aspects

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.