Lie groups and continuum mechanics: where do we stand today?

The geometric methods have experienced a fast growth in the past few decades. In this survey, we discuss the use of Lie groups in continuum mechanics. We address both the theoretical and numerical aspects. We explore the classical symmetry groups of the mechanics, the covariant form of the equations and the symmetry group of constitutive laws. We consider the Lie symmetry group of the equations of a mechanical problem and investigate how to take advantage of them in developping analytical models (self-similar solutions, conservation laws, turbulence, ...) of the physical phenomena encoded in these equations. Lastly, we present a method of constructing robust numerical integrators from the knowledge of the Lie symmetry group of the equations.