Two Theories of Complex Bodies, one Lagrangean, the other Kinetic, Have a Common Ground

I have found in previous works that most special models proposed to represent bodies with some type of microstructure can be classified easily under the general umbrella of a theory where each element of the continuum is thought of as a Lagrangian system. To study phenomena in ‘kinetic’ continua I proposed an apparently different approach; the outcome is again a set of evolution equations. They mimic equations familiar in continua with affine microstructure: a Cauchy’s equation and an equation of balance of tensor moment of momentum, with the addition, however, of an equation of balance for a ‘Reynolds’ tensor’, an equation which, in a sense, shifts the boundary between kinetic and thermal properties of matter. I will show that there is no contrast between the two approaches. The latter one is based on an adequate and appropriately justified expression of the kinetic energy of the continuum, comprising the trace of the quoted Reynolds’ tensor and thus importing into the mechanical energy a term usually accounted by additional heat.