The Equations of Motion of Particles in Smoothed Particle Hydrodynamics

Detailed attention is given to the derivation of the equations of motion in smoothed particle hydrodynamics (SPH). It is rigorously shown that the four-dimensional divergence of the momentum density can be replaced by the total derivative with respect to time of the unsmoothed particle velocity. No approximations are required other than the standard SPH ansatz of approximating integrals by sums. The result is independent of the form of the kernel and provides a sound basis for the consideration of variable smoothing length kernels in SPH. It is noted that kernels with a variable smoothing length do not conserve momentum and energy exactly and explicit expressions for the rate of change of total momentum and energy are given.