Coriolis Field - Mathematical Expression

Mathematical Expression

Being is the angular velocity vector of the rotating frame, the speed of a test particle used to meassure the field, and using the expression of the acceleration in a rotating reference frame, it is known that the acceleration of the particle in the rotating frame is:

\mathbf{a}_{\mathrm{r}} = \mathbf{a}_{\mathrm{i}} - 2 \boldsymbol\omega \times \mathbf{v} - \boldsymbol\omega \times (\boldsymbol\omega \times \mathbf{r}) - \frac{d\boldsymbol\omega}{dt} \times \mathbf{r}

the Coriolis force is assumed to be the fictitious force that compensates the second term:

\mathbf{F}_{\mathrm{Coriolis}} = -2m ( \boldsymbol\omega \times \mathbf{v}) = -2 ( \boldsymbol\omega \times \mathbf{p})

Where denotes the linear momentum. It can be seen that for any object, the coriolis force over it is proportional to its momentum vector. As vectorial product can be expressed in a tensorial way using the Hodge dual of :

\mathbf{F}_{\mathrm{Coriolis}} = -2(\mathbf{\omega} \times \mathbf{p}) = -2(\mathbf{\omega} \times) \mathbf{p} = \begin{bmatrix}\,0&\!-2\omega_3&\,\,2\omega_2\\ \,\,2\omega_3&0&\!-2\omega_1\\-2\omega_2&\,\,2\omega_1&\,0\end{bmatrix}\begin{bmatrix}p_1\\p_2\\p_3\end{bmatrix}

This matrix can be seen as a constant tensor field, defined in the whole space, that will yield coriolis forces when multiplied by momentum vectors.

Read more about this topic:  Coriolis Field

Famous quotes containing the words mathematical and/or expression:

    “All science requires mathematics. The knowledge of mathematical things is almost innate in us.... This is the easiest of sciences, a fact which is obvious in that no one’s brain rejects it; for laymen and people who are utterly illiterate know how to count and reckon.”
    —Roger Bacon (c. 1214–c. 1294)

    “The true poem is not that which the public read. There is always a poem not printed on paper,... in the poet’s life. It is what he has become through his work. Not how is the idea expressed in stone, or on canvas or paper, is the question, but how far it has obtained form and expression in the life of the artist. His true work will not stand in any prince’s gallery.”
    —Henry David Thoreau (1817–1862)