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Why is the Coriolis acceleration often written as $-2({\bf \Omega \times v})$, (for example Wikipedia)? Why is it not written $2({\bf v \times \Omega)}$ which means the negative sign is not needed? For example, the Lorentz force on a charge moving in a magnetic field is not written as ${\bf F} = - q({\bf B \times v}).$

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    $\begingroup$ To-may-to, to-mah-to... $\endgroup$
    – Puk
    Apr 10, 2022 at 19:07

2 Answers 2

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The Coriolis acceleration is the Coriolis force divided by the mass. It has the negative sign because the Coriolis force appears as an additional force to the real forces to describe the acceleration in the non-inertial frame: $$(1) \enspace m{d^{*2}\vec r \over dt^2} = \vec F -2m\vec \Omega \times {d^*\vec r \over dt}$$ The starred time derivatives are with respect to the non-inertial frame. The left side of (1) gives the motion in the non-inertial frame; that is, the acceleration in the non-inertial frame. The right side of (1) includes all the forces that dictate this motion in the non-inertial frame: both the net external force $\vec F$ and the fictitious Coriolis force. Since the Coriolis force has the negative sign, so does the Coriolis acceleration.

Similarly, other fictitious forces, such as the centrifugal force, also appear on the right side of (1).

Your Wikipedia reference also discusses (1).

The Lorentz force is a real force and as such is part of $\vec F$ when it is present.

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It shows direction, but you're already familiar with that. It's a matter of set up. The Coriolis force/acceleration is derived from Euler's approach/philosophy to mechanics, as it occurs in a non-inertial frame of reference. As such, similarly to the derivation of the normal acceleration, it uses polar coordinates/spherical in most set ups. It's a matter of choice of a certain coordinate system which you'll use when tackling a problem. Note that if you choose such a coordinate system in which the Coriolis acceleration points in the positive direction, so will Euler force and the centrifugal force (with respects to their vectors)

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