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).
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.