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508d894fJack Halford small oscillations 1 year, 1 month ago

#title: mechanical similarity type: theory

\begin{equation} U(\alpha\mathbf{r}_1,\alpha\mathbf{r}_2,\ldots,\alpha\mathbf{r}_n)=\alpha^kU(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_n) \end{equation} \begin{equation} t'/t=(l/l)^{1-k/2} \end{equation}

\begin{equation} v'/v=(l/l)^{k/2},\quad E'/E=(l/l)^{k},\quad M'/M=(l/l)^{1+k/2}\quad \end{equation}

\begin{equation} 2T=\sum_a\mathbf{p}_a\mathbf{v}_a=\frac{\dd{}}{\dd{t}}(\sum_a\mathbf{p}_a\mathbf{r}_a)-\sum_a\mathbf{r}_a\dot{\mathbf{p}}_a \end{equation} \begin{equation} 2\overline{T}=\overline{\sum_a\mathbf{r}_a\cdot\partial U/\partial \mathbf{r}_a} \end{equation} \begin{equation} 2\overline{T}=k\overline{U} \end{equation} \begin{equation} \overline{U}=2E/(k+2),\qquad \overline{T}=kE/(k+2) \end{equation}

#Assumptions