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Angular Momentum Conservation

In a central force field (like gravity), the angular momentum (L⃗=r⃗×p⃗\vec{L} = \vec{r} \times \vec{p}) of the orbiting body is conserved. This dictates that the orbit lies in a fixed plane and determines the relationship between velocity and distance.
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The statement of the theorem

Define the specific angular momentum vector h\mathbf{h} as h=r×v\mathbf{h} = \mathbf{r} \times \mathbf{v}. For motion under a central force F=f(r)r^\mathbf{F} = f(r)\mathbf{\hat{r}}, the torque τ=r×F\mathbf{\tau} = \mathbf{r} \times \mathbf{F} is zero. Consequently, the specific angular momentum is conserved:\ndhdt=0\frac{d\mathbf{h}}{dt} = 0 \nThis implies that h\mathbf{h} is a constant vector, defining the plane of the orbit.