Similarly, one can define a discrete gradient operator to approximate the pressure gradient at the position of particle formula_20:  where formula_45 denote the set of particle pressures. There are several ways to define discrete operators in SPH; the above divergence and gradient formulae have the property to be skew-adjoint, leading to nice conservation properties. On the other hand, while the divergence operator formula_46 is zero-order consistent, it can be seen that the approximate gradient formula_47 is not so. Several techniques have been proposed to circumvent this issue, leading to renormalised operators (see e.g.). The SPH operators can be used to discretize numbers of partial differential equations. For a compressible inviscid fluid, the Euler equations of mass conservation and momentum balance read:  All kinds of SPH divergence and gradient operators can practically be used for discretization purposes. Nevertheless, some perform better regarding physical and numerical effects. A frequently used form of the balance equations is based on the symmetric divergence operator and antisymmetric gradient:  Although there are several ways of discretizing the pressure gradient in the Euler equations, the above antisymmetric form is the most acknowledged one. It supports strict conservation of linear and angular momentum. This means that a force that is exerted on particle formula_52 by particle formula_53 equals the one that is exerted on particle formula_53 by particle formula_52 including the sign change of the effective direction, thanks to the antisymmetry property formula_33. The above SPH governing equations can be derived from a Least action principle, starting from the Lagrangian of a particle system:  where formula_58 is the particle specific internal energy. The Euler–Lagrange equation of variational mechanics reads, for each particle:  When applied to the above Lagrangian, it gives the following momentum equation:  where we used the thermodynamical property formula_61. Pluging the SPH density interpolation and differentiating explicitly formula_62 leads to  which is the SPH momentum equation already mentioned, where we recognize the formula_47 operator. This explains why linear momentum is conserved, and allows conservation of angular momentum and energy to be conserved as well. From the work done in the 80's and 90's on numerical integration of point-like particles in large accelators, appropriate time integrators have been developed with accurate conservation properties on the long term; they are called symplectic integrators.