Coordinateintegrable

Coordinateintegrable is a term used in mathematical discussion to describe a property of equations, systems, or functions that can be integrated in closed form when expressed in an appropriate coordinate system. The emphasis is on the role of coordinates in enabling integration, rather than on a coordinate-free formulation.

In differential equations and dynamical systems, a coordinateintegrable problem is one that becomes separable, triangular, or linear after a suitable change of variables, allowing solution by quadrature or standard special functions. In Hamiltonian mechanics, a system may be called coordinateintegrable if a canonical transformation brings the equations of motion into a form solvable by explicit integrals.

Coordinateintegrability is related to, but not identical with, classical notions such as separation of variables, change of variables, and Liouville integrability. It is often a local property: a problem may be coordinateintegrable in a neighborhood where a convenient chart exists, without guaranteeing a global solution on the entire manifold.

Examples include the simple harmonic oscillator, which separates in Cartesian coordinates, and certain variable-separable nonlinear systems. The term is mainly informal and appears in discussions of solvability via coordinate transformations rather than as a formal axiom in standard textbooks.

See also: separation of variables, quadrature, Liouville integrability, integrable systems, change of variables.