The Mathematical Particle Tracing interface makes it possible to solve ordinary differential equations for the particle positions. When the Newtonian formulation is selected, the particle position is computed using Newton’s second law:

where mp is the particle mass (SI unit: kg), v is the particle velocity (SI unit: m/s), and F is the total force exerted on the particle (SI unit: N). Each force is either specified directly or using a susceptibility multiplied by a suitable field:

where χ is the susceptibility tensor and Γ is a vector field. The particle velocity is defined as:

where q is the particle position vector (SI unit: m).

When using a relativistic correction of the particle mass for particles with very high velocity, the relativistic particle mass mp is defined as

where mr is the rest mass (SI unit: kg) and c = 2.99792458 × 108 m/s is the speed of light in a vacuum, which is a built-in physical constant.

If the default Newtonian formulation is selected, Newton’s second law is expressed as a set of second-order ordinary differential equations for the particle position vector components:

The advantage of using the first-order Newtonian formulation is that it avoids mixing first- and second-order equations when any Auxiliary Dependent Variables are also solved for, since the equations for auxiliary dependent variables are always first-order. This allows efficient, high-order explicit time stepping methods to be used for a wider class of problems compared to the second-order Newtonian formulation.