EulerGleichung

EulerGleichung is a term used in German to refer several equations named after Leonhard Euler. In common scientific usage, it most often denotes two related but distinct concepts: Euler equations in fluid dynamics and the Euler–Lagrange equation in the calculus of variations. The name reflects Euler’s foundational contributions to both topics.

In fluid dynamics, the Euler equations describe the motion of an inviscid (zero viscosity) fluid. They express conservation of mass and momentum. The compressible form consists of the continuity equation ∂t ρ + ∇·(ρ v) = 0 and the momentum equation ∂t(ρ v) + ∇·(ρ v ⊗ v) + ∇p = ρ f, where ρ is density, v is velocity, p is pressure, and f represents body forces. For incompressible flow, the equations reduce to ∇·v = 0 together with a momentum equation. These equations are fundamental in aerodynamics, meteorology, and astrophysical fluid flows, and they form the basis for more advanced models like the Navier–Stokes equations when viscosity is included.

In the calculus of variations, the Euler–Lagrange equation determines stationary points of a functional. Given a Lagrangian L(y, y′, x), the equation is d/dx (∂L/∂y′) − ∂L/∂y = 0. This form is often referred to in German as the Euler-Gleichung or Euler-Lagrange-Gleichung, depending on context, and underpins classical mechanics, optics, and field theory.

See also: Navier–Stokes equations, continuity equation, conservation laws, variational principles.