Department of Mathematics & Statistics, Monash University
In this paper we show how the equations of motion for the particle method SPH may be derived from a variational principle for both non relativistic and relativistic motion when there is no dissipation. Because the SPH density is a function of the coordinates the derivation of the equations of motion through variational principles is simpler than in the continuum case where the density is defined through the continuity equation. In particular, the derivation of the General Relativistic equations is more direct and simpler than that of Fock. The symmetry properties of the Lagrangian lead immediately to the familiar additive conservation laws of linear and angular momentum and energy. In addition, we show that there is an approximately conserved quantity which, in the continuum limit, is the circulation.
*The pdf version is an electronic version of an article published in Monthly Notices of the Royal Astronomical Society: complete citation information for the final version of the paper, as published in the print edition of Monthly Notices of the Royal Astronomical Society, is available on the Blackwell Science Synergy online delivery service, accessible via the journal's Website at:http://www.blacksci.co.uk/MNR