Smoothed Particle Magnetohydrodynamics - II. Variational principles and variable smoothing-length terms

In this paper we show how a Lagrangian variational principle can be used to derive the Smoothed Particle Magnetohydrodynamics (SPMHD) equations for ideal Magnetohydrodynamics (MHD). We also consider the effect of a variable smoothing length in the Smoothed Particle Hydrodynamics (SPH) kernels, after which we demonstrate by numerical tests that the consistent treatment of terms relating to the gradient of the smoothing length in the SPMHD equations significantly improves the accuracy of the algorithm. Our results complement those obtained in the companion paper for non-ideal MHD where artificial dissipative terms were included to handle shocks.