The motion of a continuous medium in a force field with a rooted singularity
The Newton's equation \(\ddot x= G(u,w,t)+ \sqrt{u}H(\sqrt{u},w,t)\), where \(x\in R^ n\), \(\{u=u(x,t)=0\}\) is the temporal front of the (root) singularity of force, \(x=(u,w)\), \(w\in R^{n-1}\) and G, H are analytic functions is considered. The evolution of the field of velocities of some medium of particles, smooth at initial moment, is analyzed. It turns out that the instantenous velocity field has a singularity 3/2 on the front. Namely, by means of a smooth local change of coordinates \(<x,y>\to <q(t,x,v)\), \(p(t,x,v)>\) it can be reduced to the form \(p_ 1=q_ 1^{3/2}\) if \(q_ 1>0\); 0 if \(q_ 1\leq 0\) and \(p_ 2=...=p_ n=0\). The model has an application in the gravity theory of media.
scientific article; zbMATH DE number 4056058