# Mechanical similarity in Landau

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?

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.