Newtonian Central Force System

[itev07]

I have a problem and can’t seem to work it out! Ok, here goes:

X/|X|^3 = grad U(X)

which, when integrated gives

U(X)= -1/|X|

But I can’t seem to integrate to get the correct answer. Also, if

U(X)= -1/(|X|^v )

where v is a constant, then what is grad U(X) now? Thanks for reading and any help will be much appreciated!

 

[matematikawan]

I have a problem and can’t seem to work it out! Ok, here goes:

X/|X|^3 = grad U(X)

which, when integrated gives

U(X)= -1/|X|

Probably X on the LHS is a radial vector $$\vec{r}$$.

$$\frac{\vec{r}}{r^3}=\nabla U(r)$$

$$\nabla U(r)=\frac{\hat{r}}{r^2}$$

Integrate, U(r)=-1/r.