VC2.9

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We have \frac{d}{dt}[r\times \frac{dr}{dt}]=\frac{dr}{dt}\times \frac{dr}{dt}+r\times \frac{d^2 r}{dt^2}\,

=r\times\frac{d^2 r}{dt^2}\, [Since \frac{dr}{dt}\times \frac{dr}{dt}=0\,]

Applying integral bothsides,

\int a\cdot [r\times \frac{d^2 r}{dt^2}]\,dt=r\times \frac{dr}{dt}+c\, where c is an arbitrary constant vector.

Now \int a\cdot [r\cdot \frac{d^2 r}{dt^2}]\,dt=a\cdot\int r\times \frac{d^2 r}{dt^2}\,dt=a\cdot [r\times \frac{dr}{dt}+c]=a\cdot r\times \frac{dr}{dt}+d\,

where d(=a.c) is an arbitrary constant scalar.

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