Eulerkraft

Eulerkraft, or Euler force, is a fictitious force that appears in non-inertial reference frames, particularly those rotating with an angular velocity that changes in time. It is named after the Swiss mathematician Leonhard Euler and is one of the standard fictitious forces used to describe motion in rotating frames.

In a frame rotating with angular velocity ω(t) about a fixed origin, the motion of a particle of mass m with position r relative to the origin includes three fictitious forces: the centrifugal force F_cent = - m ω × (ω × r), the Coriolis force F_cor = - 2 m ω × v_rel, and the Euler force F_Euler = - m (dω/dt) × r, where v_rel is the velocity relative to the rotating frame. The Euler force vanishes if the rotation rate is constant (dω/dt = 0). The complete equation of motion in the rotating frame is m a_rel = F_real + F_cent + F_cor + F_Euler, with a_rel the observed acceleration in that frame.

Physically, Eulerkraft arises from the time dependence of the rotating frame itself: as the rotation rate increases or decreases, points at a distance r from the axis experience a sideways tendency that manifests as this force. It is generally smaller than the centrifugal and Coriolis effects unless the angular acceleration is large or the distance from the rotation axis is significant.

Applications of Eulerkraft appear in the analysis of motion within accelerating rotors, spinning machinery, and atmospheric or fluid flows in rapidly changing rotating systems. See also: Coriolis effect, centrifugal force, fictitious forces, rotating reference frame.