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:
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 = τtot = ∑i τi