Answer to Question #96932 in Classical Mechanics for SamDevakumar

Question #96932
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
1
Expert's answer
2019-10-23T09:50:08-0400

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.





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