I tried to write formulas (part 1).

＜Newton's Equation of Motion＞

　Newton's Equation of Motion is described as below

\[m\frac{d^2{\boldsymbol r}}{dt^2} = {\boldsymbol F} \]

Here, is the position vector of point mass whose mass is m, and is the force vector acting on the point mass. In general, and are defined as stated below respectively.

\[{\boldsymbol r}{\equiv}{}^t\!(r_1, r_2, \cdots, r_n) = \left( \begin{array}{c} r_1 \\ r_2 \\ \vdots \\ r_n \end{array} \right)\]

\[{\boldsymbol F}{\equiv}{}^t\!(F_1, F_2, \cdots, F_n) = \left( \begin{array}{c} F_1 \\ F_2 \\ \vdots \\ F_n \end{array} \right)\]

In 3-dimensional space, they can be expressed as stated below respectively.

\[{\boldsymbol r}{\equiv}{}^t\!(x,y,z)= \left( \begin{array}{c} x \\ y \\ z \end{array} \right)\]

\[{\boldsymbol F}{\equiv}{}^t\!(F_x,F_y,F_z)= \left( \begin{array}{c} F_x \\ F_y \\ F_z \end{array} \right)\]

When x-component is extracted from Newton's equation, one-dimensional one can be obtained as below.(For the sake of simplicity, the subscript is removed from .)

\[m\frac{d^2x}{dt^2} = F \]

＜ニュートンの運動方程式＞

ニュートンの運動方程式は下記のように記述されます。

\[m\frac{d^2{\boldsymbol r}}{dt^2} = {\boldsymbol F} \]

ここで、 は質量mの質点の位置ベクトル位置ベクトルであり、 はその質点にかかる力のベクトルです。一般的にと はそれぞれ下記のように定義されます。

\[{\boldsymbol r}{\equiv}{}^t\!(r_1, r_2, \cdots, r_n) = \left( \begin{array}{c} r_1 \\ r_2 \\ \vdots \\ r_n \end{array} \right)\]

\[{\boldsymbol F}{\equiv}{}^t\!(F_1, F_2, \cdots, F_n) = \left( \begin{array}{c} F_1 \\ F_2 \\ \vdots \\ F_n \end{array} \right)\]

3次元空間においては、それら2つのベクトルはそれぞれ下記のように表すことができます。

\[{\boldsymbol r}{\equiv}{}^t\!(x,y,z)= \left( \begin{array}{c} x \\ y \\ z \end{array} \right)\]

\[{\boldsymbol F}{\equiv}{}^t\!(F_x,F_y,F_z)= \left( \begin{array}{c} F_x \\ F_y \\ F_z \end{array} \right)\]

ニュートンの運動方程式からx成分だけを抽出すれば、以下のような一次元における方程式が得られます(簡単のためにの下付き文字を外しています)。

\[m\frac{d^2x}{dt^2} = F \]