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Given
The Equation of Motion (EOM) for the meglev train is modeled by:

Find

 * Get the derivation of the 1st total time derivative.


 * Get the derivation of the 2nd total time derivative.


 * Show the similarity with the derivation of the Coriolis force.

* Get the derivation of the 1st total time derivative
Apply the chain rule to $$\displaystyle f(s,t) $$.

Given that for $$\displaystyle s $$ is evaluated at $$\displaystyle Y^1(t) $$ in the Eq. (2-1). Then the equation can be rewritten as: Defining the notation as below.

As the result, Eq. (2-1) can be simplified to :

*Get the derivation of the 2nd total time derivative
Taking the 2nd derivative of the Eq. (2.2) with respect to $$\displaystyle t $$, and applying the chain rule yields:

Defining the notations as below.

Then, the Eq. (2.6) can be rewritten as below.

* Show the similarity with the derivation of the Coriolis force
Define $$\displaystyle \mathbf{r} $$ as the position vector indicating the position of the origin of the reference frame, and define $$\displaystyle A $$ is a point fixed on an object rotating with an angular velocity of $$ \displaystyle {}^{N}\boldsymbol{\omega}^{A} $$ with respect to inertial reference frame $$\displaystyle N $$. And define the three mutual perpendicular unit vectors $$\displaystyle \mathbf{i}, \mathbf{j}, \mathbf{k} $$. Then, $$\displaystyle \mathbf{r} $$ can be defined as Then, the velocity of the object as viewed by an observer fixed to the inertial reference frame $$\displaystyle N $$ is defined as below. Also, the acceleration of the object as viewed by an observer fixed to the inertial reference frame $$\displaystyle N $$ is defined as below. Substituting $$\displaystyle {}^{N} \mathbf{v} $$ from the Eq. (2.10) into the Eq. (2.11), Continual calculates with Eq. (2.12) as below. {| style="width:100%" border="0" $$ \displaystyle \begin{align} {}^{N}\mathbf{a} &= \frac{{}^{A}d}{dt} {}^{A}\mathbf{v} + \frac{{}^{A}d}{dt} \left( {}^{N} \boldsymbol{\omega}^{A} \times \mathbf{r} \right)+{}^{N} \boldsymbol{\omega}^{A} \times {}^{A}\mathbf{v} +{}^{N}\boldsymbol{\omega}^{A} \times \left({}^{N} \boldsymbol{\omega}^{A} \times \mathbf{r} \right) \\ &= {}^{A}\mathbf{a} + \frac{{}^{A}d}{dt} {}^{N} \boldsymbol{\omega}^{A}  \times \mathbf{r} + {}^{N} \boldsymbol{\omega}^{A} \times \frac{{}^{A}d\mathbf{r}}{dt} + {}^{N} \boldsymbol{\omega}^{A} \times {}^{A}\mathbf{v} + {}^{N}\boldsymbol{\omega}^{A} \times \left({}^{N} \boldsymbol{\omega}^{A} \times \mathbf{r} \right) \\ &={}^{A}\mathbf{a}+{}^{N}\boldsymbol{\alpha}^{A} \times \mathbf{r} + 2{}^{N}\boldsymbol{\omega}^{A} \times {}^{A}\mathbf{v}+{}^{N}\boldsymbol{\omega}^{A} \times \left({}^{N} \boldsymbol{\omega}^{A} \times \mathbf{r} \right) \\ \end{align} $$ As the result, the Eq.(2.12) can be rewritten as by the definition,the expression on RHS can also be written as
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The Eq. (2-8) and the Eq. (2-13) are related as below.