Dynamics of Continua of Arbitrary Dimensions
Géry de Saxcé, Claude Vallée · 2016
This chapter presents the mathematical modeling for the dynamics of media of other dimensions such as 4. The motion of a 1D material body, solid or fluid, can be described by a space-time submanifold parametrized by two coordinates, the arclength and the time, hence by a submanifold of dimension 2. The chapter also defines the group of 1D linear Galilean transformation. A group of two-dimensional (2D) Galilean transformations can be associated with a submanifold of dimension 2 representing the motion of a material surface in dynamics. The chapter then develops a general approach for the dynamics of material bodies of dimension d represented by a submanifold of dimension d+1 of the space-time. It describes the force-mass tensor, which is structured into four components: density, linear momentum, tangential linear momentum and dynamical forces. The chapter derives the free-coordinate equation of motion to continua of arbitrary dimension.