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運動方程式のガリレイ変換のメモ

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$\displaystyle xyz$ 座標から $\displaystyle x'y'z'$ 座標への変換がガリレイ変換のとき,

\begin{equation*}
t'=t\ ,\ x'=x-V_{x} t\ ,\ y'=y-V_{y} t\ ,\ z'=z-V_{z} t
\end{equation*}
\begin{equation*}
\frac{dx'}{dt'} =\frac{dx}{dt} -V_{x} \ ,\ \frac{dy'}{dt'} =\frac{dy}{dt} -V_{y} \ ,\ \frac{dz'}{dt'} =\frac{dz}{dt} -V_{z}
\end{equation*}
\begin{equation*}
\frac{d^{2} x'}{dt^{\prime 2}} =\frac{d^{2} x}{dt^{2}} \ ,\ \frac{d^{2} y'}{dt^{\prime 2}} =\frac{d^{2} y}{dt^{2}} \ ,\ \frac{d^{2} z'}{dt^{\prime 2}} \ =\frac{d^{2} z}{dt^{2}}
\end{equation*}

ここで,$\displaystyle xyz$ 座標での運動方程式は,

\begin{equation*}
m\frac{d^{2} x}{dt^{2}} =F_{x} \ ,\ m\frac{d^{2} y}{dt^{2}} =F_{y} \ ,\ m\frac{d^{2} z}{dt^{2}} =F_{z}
\end{equation*}

つまり,

\begin{equation*}
F_{x'} =F_{x} \ ,\ F_{y'} =F_{y} \ ,\ F_{z'} =F_{z}
\end{equation*}

以上より,

\begin{equation*}
m\frac{d^{2} x'}{dt^{\prime 2}} =F_{x'} \ ,\ m\frac{d^{2} y'}{dt^{\prime 2}} =F_{y'} \ ,\ m\frac{d^{2} z'}{dt^{\prime 2}} =F_{z'}
\end{equation*}

となり,$\displaystyle x'y'z' $座標における運動方程式は $\displaystyle xyz$ 座標におけるものと同じ形になる.
したがって運動方程式はガリレイ変換に関して共変性があるといえる.

hironorioka28
いつもギリギリ
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