The Mathematical Particle Tracing Interface gives access to the underlying mathematical formalism on which the Charged Particle Tracing and Particle Tracing for Fluid Flow interfaces are built. Compared to the other interfaces, the Mathematical Particle Tracing interface has the fewest dedicated physics features for specialized phenomena (such as applied forces, Monte Carlo collision modeling, evaporation, and dedicated boundary conditions) but affords the greatest flexibility in setting up user-defined equations of motion.

A model using the Mathematical Particle Tracing interface can, in principle, do anything that the Charged Particle Tracing or Particle Tracing for Fluid Flow interface can do. For example, the Charged Particle Tracing interface has a Collisions node that can predict the collisions of model particles with a rarefied background gas. You can perform Monte Carlo collision modeling in the Mathematical Particle Tracing interface by using the Velocity Reinitialization node. Similarly, the built-in variables for particle mass and temperature, which are included in the Particle Tracing for Fluid Flow interface, are special cases of the Auxiliary Dependent Variable node, which lets you solve custom ODEs along particle paths. However, creating custom user-defined features takes more time and effort than using the dedicated physics features.

The Mathematical Particle Tracing interface also supports two Formulation options that are not available in the other particle tracing interfaces: Lagrangian and Hamiltonian. Often it is easier to write down an expression for the Lagrangian or Hamiltonian for particles rather than deriving the equations of motion, especially when there is some additional constraint on the particle positions, such as having to slide along a surface of a given functional form. The Hamiltonian formulation solves for both the particle position and momentum, so the number of degrees of freedom is doubled when the Hamiltonian formulation is activated. Table 2-1 gives a summary of all supported Formulation options for each physics interface.