newtonian mechanics - The two causes for the factor 2 in Coriolis effect

While reading this document on Coriolis effect http://empslocal.ex.ac.uk/people/staff/gv219/classics.d/Persson98.pdf, I saw the followig sentence

Two kinematic effects each contribute half of the Coriolis acceleration: relative velocity and the turning of the frame of reference.

And this is the reason why Coriolis term has that factor $2$. Unfortunately it does not specify anything about this two causes. Does anyone have some further explanation for how "relative velocity" and "turning of the frame" actually give rise to the Coriolis term?

Answer

Here is one way of looking at it via a velocity-dependent potential.$^1$ The Coriolis potential is

$$\tag{1} U_{\rm cor} ~=~ -m({\bf v} \times {\bf \Omega})\cdot{\bf r} ~=~-{\bf v}\cdot ({\bf \Omega}\times{\bf r} ),$$

cf. Ref. 1. The factor $2$ comes from two different terms in the corresponding force formula

$$\tag{2} {\bf F}~=~\frac{\mathrm d}{\mathrm dt} \frac{\partial U_{\rm cor}}{\partial {\bf v}} - \frac{\partial U_{\rm cor}}{\partial {\bf r}} ~\stackrel{(1)}{=}~m\frac{\mathrm d}{\mathrm dt} ({\bf r}\times {\bf \Omega}) + m{\bf v} \times {\bf \Omega} ~\stackrel{\begin{matrix}\text{Leibniz}\\ \text{rule}\end{matrix}}{=}~\underbrace{2m {\bf v} \times {\bf \Omega}}_{\text{Coriolis force}} + \underbrace{m{\bf r} \times \dot{{\bf \Omega}}}_{\text{Euler force}} .$$

References:

1. L.D. Landau & E.M. Lifshitz, Mechanics, vol. 1, (1976); $\S$39.

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$^1$ Alternatively, an elementary derivation of the Coriolis acceleration is given in this Phys.SE post, where the factor 2 appears from a binomial coefficient $\begin{pmatrix}2 \\1 \end{pmatrix}=2$ in a cross-term.