= Equation of motion in a rotating frame
= Equations of motion in a rotating frame
{synonym}
Suppose a <rotating reference frame> shares its origin with an <inertial frame> and rotates at constant <angular velocity> $\boldsymbol\Omega$. Let $\mathbf x$ be the relative <position>, with dots denoting derivatives of its rotating components. The two <accelerations>, expressed in the same axes, satisfy
$$
\mathbf a_{\rm inertial}=\ddot{\mathbf x}+2\boldsymbol\Omega\times\dot{\mathbf x}+\boldsymbol\Omega\times(\boldsymbol\Omega\times\mathbf x).
$$
Apply the derivative rule $(d\mathbf A/dt)_{\rm inertial}=(d\mathbf A/dt)_{\rm rotating}+\boldsymbol\Omega\times\mathbf A$ twice to the <position> vector to derive this relation. <Newton's second law> therefore gives the rotating-frame equation
$$
m\ddot{\mathbf x}=\mathbf F-2m\boldsymbol\Omega\times\dot{\mathbf x}-m\boldsymbol\Omega\times(\boldsymbol\Omega\times\mathbf x).
$$
Here $\mathbf F$ contains the actual interaction <forces>; the extra terms are the <Coriolis force> and <centrifugal force>.<Introduction to Celestial Mechanics by Richard Fitzpatrick>[Online Rotating reference frames, equations 6.3–6.9]{ref}
For example, an object at rest on a rotating table still needs an inward interaction <force> to maintain its <circular motion>; in rotating coordinates it balances the outward <centrifugal force>. If the origin has <acceleration> $\mathbf a_O$, add $-m\mathbf a_O$ to the right side. If the <angular velocity> varies, also add the <Euler force> $-m\dot{\boldsymbol\Omega}\times\mathbf x$. These terms cannot be omitted merely because the object is instantaneously at rest in the rotating axes.
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