forminvariance

Forminvariance, often written as form invariance or form-invariance, is a concept in mathematics and physics describing when the governing equations of a system retain their structural form under a specified set of transformations. In practice, a model is form-invariant if, after transforming the variables according to a defined group (for example, coordinate changes, scaling, or boosts), the equations governing the system have the same functional form in the transformed variables. Forminvariance is closely linked to symmetry and is a foundational idea behind identifying conserved quantities and guiding model construction.

Mathematically, consider a model described by an equation E(x, u, du/dx, t, ...)=0. If a transformation T maps variables as x’=g(x), u’=h(u), t’=φ(t), etc., and the transformed equation E’(x’, u’, du’/dx’, t’, ...)=0 has the same structure as E (possibly with relabeled constants), the model is form-invariant under T. Exact form invariance requires the same functional form to hold without approximation; approximate invariance acknowledges near-symmetries that hold within a tolerance. Forminvariance is distinct from, but related to, covariance and broader notions of symmetry.

Examples include scaling invariance in differential equations, Galilean invariance in non-relativistic mechanics, Lorentz invariance in electromagnetism, and general covariance in general relativity. In each case, the laws retain their form under the relevant transformation group, which informs solution methods, interpretation, and consistency checks. Forminvariance thus serves as a diagnostic and design principle across physics and applied mathematics.