Proof of angular momentum - Definition and Overview

A proof that torque is equal to the time-derivative of angular momentum can be stated as follows:

The definition of angular momentum for a single particle is:

<math>\mathbf{L} = \mathbf{r} \times \mathbf{p}<math>

where "×" indicates the vector cross product. The time-derivative of this is:

dL/dt = r × (dp/dt) + (dr/dt) × p

This result can easily be proven by splitting the vectors into components and applying the product rule. Now using the definitions of velocity v = dr/dt, acceleration a = dv/dt and linear momentum p = mv, we can see that:

dL/dt = r × m (dv/dt) + mv × v

But the cross product of any vector with itself is zero, so the second term vanishes. Hence with the definition of force F = ma, we obtain:

dL/dt = r × F

And by definition, torque τ = r×F. Note that there is a hidden assumption that mass is constant — this is quite valid in non-relativistic mechanics. Also, total (summed) forces and torques have been used — it perhaps would have been more rigorous to write:

dL/dt = τ_tot = ∑_i τ_i