Tudor Ratiu, SJTU and EPFL
601 Pao Yue-Kong Library
I will address the dual formulations of the equations of motion in body and space coordinates. I will start with the rigid body and the ideal incompressible homogenous fluids. Then underlying Lie-Poisson and Euler-Poicaré formulation will be given. Then I will discuss systems that have advected quantities and additional internal structure and give the abstract geometric formulation for the equations of motion. As examples I will present ideal compressible fluids, free boundary fluids, and elasticity. I will explain why in elasticity one uses the body (or convective) representation as opposed to the spatial representation which is prevalent in fluids. I will show that a spatial representation of the equations of motion of an elastic body is possible only for isotropic materials, whereas general materials do not have a spatial representation for the equations of motion. Then I will propose a material that contains, as special cases, both fluids and solids.