Grundgleichung

Grundgleichung is a term used in German-language science to denote the fundamental equation of a theory or model. It represents a primary relation that links the essential state variables or dynamical quantities of a system. The exact form depends on the discipline, but the concept is to express, in one main equation, the core dependence from which other properties can be derived.

In thermodynamics the fundamental equation is often given as a function U(S, V, N) of the extensive variables entropy S, volume V, and particle number N. The differential form dU = T dS − p dV + μ dN defines the intensive variables temperature T, pressure p and chemical potential μ as partial derivatives: T = (∂U/∂S)_{V,N}, p = −(∂U/∂V)_{S,N}, μ = (∂U/∂N)_{S,V}. From U one can obtain other thermodynamic potentials via Legendre transforms, such as H = U + pV, F = U − TS, and G = U + pV − TS. These representations yield equations of state and enable Maxwell relations, providing a consistent equilibrium framework.

Beyond thermodynamics, the concept of a Grundgleichung is used to signify the essential relation governing a theory. The exact equation depends on the field and model; in mechanics or quantum theory, the corresponding fundamental equation encapsulates the core dynamics. The term emphasizes that many analyses follow from a single foundational relation, with additional structure supplied by material properties, boundary conditions, and reference potentials.