Question

If ur^•• = - k/r^2 and prove that the first integral of the equation of motion is 1/2ur^•2 = - c[1/R - 1/r]

'u' is meu

Answer 1

We need to find the first integral of the equation

$$\mu \ddot r = - \frac{k}{{{r^2}}}$$

We can do it by two different ways, we choose the direct integration (the second way - using energy conservation)

Let's multiply the equation by ${\dot r}$

$$\mu \ddot r\dot r = - \frac{k}{{{r^2}}}\dot r$$

and notice that

$$\mu \ddot r\dot r = \frac{\mu }{2}\frac{d}{{dt}}({{\dot r}^2})$$

then notice that

$$- \frac{k}{{{r^2}}}\dot r = \frac{d}{{dt}}(\frac{k}{r})$$

It means, that our equation can be rewritten as

$$\frac{d}{{dt}}[\frac{\mu }{2}{{\dot r}^2} - (\frac{k}{r})] = 0$$

And you can simply integrate it

$$\frac{\mu }{2}{{\dot r}^2} - (\frac{k}{r}) = C$$

Now $C$ is some constant. We can use that in some point $r = R$ (colled turning point) $\dot r = 0$ , we can use it to find $C$

$$C = - \frac{k}{R}$$

Then just rewrite equation as

$$\frac{\mu }{2}{{\dot r}^2} = k(\frac{1}{r} - \frac{1}{R})$$

We got what we wanted - the first integral.