Classical Equations

Abstract

We begin with nonrelativistic equations that govern a matter density field ρ(t,r) and a velocity field vector v(t, r),taken in any number of dimensions. The equations of motion comprise a continuity equation,

$$\frac{\partial }{{\partial t}}\rho \left( {t,r} \right) + \nabla \cdot \left( {\rho \left( {t,r} \right)v\left( {t,r} \right)} \right) = 0,$$

(2.1)

which ensures matter conservation, that is, time independence, of N = ∫ dr ρ, and Euler’s equation, which is the expression of a nonrelativistic force law

$$\frac{\partial }{{\partial t}}v\left( {t,r} \right) + v\left( {t,r} \right) \cdot \nabla v\left( {t,r} \right) = f\left( {t,r} \right).$$

(2.2)