Geometry of strong forces in continuum mechanics

Abstract

Consider a material point in finite dimensions moving under the influence of a potential force according to Newton's laws. Suppose the potential energy function is a generalized well, strictly convex transverse to a smooth submanifold $M$ on which it is minimal. If the potential is steep, one expects the particle will oscillate rapidly about this submanifold and, if the initial displacement is not too great, should approximately move along $M$ as if it were ideally constrained (e.g. geodesic). This expectation is true if the initial conditions are very well prepared but may fail otherwise - additional potential forces determined by how the Hessian of the potential varies along $M$ may be present. The origin of this force is that the transversal motion acts as a simple harmonic oscillator with a slowly varying frequency, which approximately conserves action, not energy. In this work, we regard continuum mechanical systems such as the elastic thread or compressible fluid as material points moving in an infinite dimensional space according to Newton's laws for appropriate potential energy functionals. We show how to arrive at ideally constrained systems such as the inextensible thread and incompressible fluid as a limit of a strong potential force, computing also corrections to the naive predictions when the data is not very well prepared. For example, for the thread we find a resistance to bending emerge from a strong resistance to compression/expansion. For the fluid, the effective incompressible dynamics may be driven by a remnant acoustical wavefield. Both of these emergent features are nonlinear and non-local. Finally, we give examples of some limits for which the naive models robustly hold because the additional force is trivial. These include the homogeneous incompressible Euler, anelastic Euler, as well as the lake and great lake equations.