ref: fe6b680fd367b85811a3d1400efa48b781ad6501
physics-notes/md/mechanical-similarity.md
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We have found that the equations of motion are unchanged by multiplication of the Lagrangian by any constant. This allows us to determine some preperties of motion without necessarily solving the equations.

Let us consider a homogeneous potential $U(q)$

```
LL1/10.1
```

where $\alpha$ is any constant and $k$ is the degree of homogeneity of the function.

Let us carry out a transformation where the co-ordinates are changed by a factor $\alpha$ and the time by a factor $\beta$

$$ \mathbf{r}_a\to\alpha\mathbf{r'}_a,\qquad t\to\beta t' $$

Then all velocities $\mathbf{v}_a=\dd{\mathbf{r}_a}/\dd{t}$ are changed by a factor $\alpha/\beta$, therefore the kinetic energy by a factor $\alpha^2/\beta^2$. The potential energy is multiplied by a factor $\alpha^k$.

$$ T\to\alpha^2/\beta^2T',\qquad U\to\alpha^kU' $$

If these factor are the same, i.e. $\beta=\alpha^{1-k/2}$, then the motion is unchanged. (todo link to paragraph 2 explanation). A change of co-ordinates of the particle by the same factor corresponds to the replacement of the paths by other paths, geometrically similar but differing in size. By definition of $\alpha$ and $\beta$, the times of the motion between corresponding points are in the ratio

```
LL1/10.2
```

where $l'/l$ is the ratio of linear dimensions of the two paths. We can find similar relations for other characteristics of motion.

```
LL1/10.3
```

For $k=2$, we find the motion is `LL1/21.11`

, which verifies that the period of oscillations is independant of the amplitude.

In a uniform field, the potential energy is a linear function of co-ordinates (see `LL1/5.8`

), $k=1$. We then have $t'/t=\sqrt{l'/l}$. Hence, for example, in fall under gravity, the distance fallen goes as the square of the time you've been falling.

the Newtonian attraction of two masses, of the Coulomb attraction of two charges, the potential energy is inversely proportional to the distance apart $k=-1$. Then $(t'/t)^2=(l'/l)^3$. Hence, the square of the time of revolution in the orbit goes as the cube of the size of the orbit.

We can see in this log-log plot from wikipedia that the slope is approximately $2/3$, as expected.

```
LL1/10.4
```

```
LL1/10.5
```

```
LL1/10.6
```

```
LL1/10.7
```