KINEMATICAL CONSERVATION LAWS IN A SPACE OF ARBITRARY DIMENSIONS

In a large number of physical phenomena, we find propagating surfaces which need mathematical treatment. In this paper, we present the theory of kinematical conservation laws (KCL) in a space of arbitrary dimensions, i.e., d-D KCL, which are equations of evolution of a moving surface Omega(t) in d-dimensional x-space, where x = (x(1), x(2),..., x(d)) is an element of R-d. The KCL are derived in a specially defined ray coordinates (xi = (xi(1), xi(2),..., xi(d-1)), t), where xi(1), xi(2),..., xi(d-1) are surface coordinates on Omega(t) and t is time. KCL are the most general equations in conservation form, governing the evolution of Omega(t) with physically realistic singularities. A very special type of singularity is a kink, which is a point on Omega(t) when Omega(t) is a curve in R-2 and is a curve on Omega(t) when it is a surface in R-3. Across a kink the normal n to Omega(t) and normal velocity m on Omega(t) are discontinuous