I've read this very short paragraph from Landau & Lifshitz's Mechanics (Chap.2, Par.10) (that you can find here) about Mechanical similarity.

I was looking for some more detailed explanations of the matter, at a level like the one in the first chapters of Landau's book. I've been able to find this article but it's a little too much for me.

Can you give me some references that do not go into quantum mechanics, i.e. that refer only to classical mechanics and Lagrangian formalism, about the subject?

Thank you in advance.

Content of the cited paragraph As requested in the comments, I will summarize the paragraph's content; I will give my personal understanding of Landau's explanation, so I can be corrected if I'm wrong.

Suppose a system of particles is described by a Lagrangian $\cal L(\mathbf{r_1},\mathbf{\dot{r_1}},\mathbf{r_2},\mathbf{\dot{r_2}},...,t)$ and suppose the potential energy is such that $U(\alpha \mathbf{r_1},\alpha \mathbf{r_2},...)=\alpha ^k ·U(\mathbf{r_1},\mathbf{r_2},...)$. Since multiplying the Lagrangian by a constant leaves the equations of motions unaltered, we may multiply $\cal L$ by $\alpha ^k$. In that case, the kinetical energy becomes (let's look at the kinetic energy of a generic particle): $$\alpha ^ k · T = \frac{m}{2} (\alpha ^{k/2}\dfrac{ \text d \mathbf r}{\text d t})^2=\frac{m}{2}(\dfrac{\text d \,\alpha \mathbf{r}}{\text d \, \alpha^{1-k/2}t})^2,$$ so letting $\mathbf{r'}=\alpha \mathbf{r}$ and $t'=\alpha^{1-k/2}t$ we get $$\alpha ^k \cal L (\mathbf{r_1},\mathbf{\dot{r_1}},\mathbf{r_2},\mathbf{\dot{r_2}},...,t)=\cal L (\mathbf{r_1'},\mathbf{\dot{r_1'}},\mathbf{r_2'},\mathbf{\dot{r_2'}},...,t').$$ In conclusion, if lenght and times are scaled respectively by a factor $\alpha$ and $\alpha ^{1-k/2}$, the resulting equations of motion are identical and the paths followed by the system of particles are similar. From this we may infer the ratios of times in similar paths from the ratios of lenghts beetween the two paths. For example: for gravitational potential $k=-1$, so:$$t/t'=(l/l')^{3/2},$$ which is Kepler's Third Law.

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    $\begingroup$ Copy the quote here, already! Why ask every possible reader to download a fat PDF? And then hunt through it to find the paragraph you want. Why risk link-rot leaving the question incomplete in the future? Likewise, it is better to link to the abstract page for arXiv articles so that you reader can check the abstract before downloading a potentially very large file. $\endgroup$ Jun 29 '13 at 17:59
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    $\begingroup$ I agree with you about link-rot, but I point out that the pdf he links to is four pages alone and contains only the relevant section (section 10 of chapter 2) of the book. $\endgroup$
    – Mark Allen
    Jun 29 '13 at 18:18
  • $\begingroup$ Thank you @dmckee for your suggestions. I've added an explanation of the cited paragraph. I've preferred not to copy-paste Landau's derivation, but to include my own explanation, so I can be corrected if I'm wrong in interpreting Landau's. PS: first link is surely worth a reading, so I will leave it there. $\endgroup$
    – pppqqq
    Jun 29 '13 at 18:37
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    $\begingroup$ Sedov has a book on Similarity. Worth checking your library for a copy, if you cannot purchase one. $\endgroup$
    – Kyle Kanos
    Jun 29 '13 at 19:13
  • $\begingroup$ Your interpretation of the derivation is perfectly rigth, I would say. So does there still remain anything of your question? :) $\endgroup$
    – Ilja
    Apr 4 '16 at 22:11

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