Can you explain the derivation of mr^2 [n X (dn/dt)] in orbital theory?

[knockout_artist]

Hi,

I need help with the following equations.

n = r/r (unit vector in r direction)


v =d r/dt = dr/dt n + r dn/dt



L = r X (mv) = mr^2 [n X (dn/dt)]

how did this
mr^2 [n X (dn/dt)]
came about?

I would expect some thing from
r X (mv)
to this
m (v X r)

 

[DrClaude]

Start from $\mathbf{L} = \mathbf{r} \times m \mathbf{v}$ and substitute for $\mathbf{r}$ and $\mathbf{v}$ their values expressed in terms of $\mathbf{\hat{n}}$.

 

[knockout_artist]

n = r/r
r=rn

v = r

L = r X mv

v =d r/dt = dr/dt n + r dn/dt
L = m . rn X[ dr/dt n + r dn/dt ]
= m [rn X dr/dt n + rn X r dn/dt]
(red disappear since in same direction)

= m [rn X rdn/dt]

is this right?

 

[DrClaude]

Correct. And you can take out the $r^2$ from the bracket as it is a scalar.

 

[knockout_artist]

ohh

took me 1 year to figure that our. :)