momentid

Momentid is a theoretical construct used in discussions of continuum mechanics to describe the spatial distribution of momentum within a moving body. It comprises a sequence of moments of the momentum density, providing information about how momentum is arranged across space beyond the total momentum.

In a continuous medium with momentum density p(r, t) = ρ(r, t) v(r, t) over a volume V, the n-th order momentid is defined as M_n(t) = ∫_V x^n p(r, t) dV in one dimension; in multiple dimensions the definition generalizes to tensorial forms M^{(i1...in)}(t) = ∫_V r^{i1}... r^{in} p(r, t) dV. The zeroth-order momentid M_0 equals the total momentum P. The first-order momentid M_1 encodes the weight of momentum relative to the origin and relates to the momentum distribution’s center of momentum. Higher-order moments describe the spread and shape of the momentum distribution.

Momentid does not, in general, have the same transformation properties as angular momentum under rotations or momentum under Galilean boosts, and it depends on the choice of origin and coordinate system. It is distinct from angular momentum L = ∫ r × p dV.

In practice, momentids can be used as diagnostic tools in simulations to characterize momentum distribution, detect anisotropies, and study transport phenomena. They are primarily of mathematical or pedagogical interest rather than standard physical observables.