Proof of angular momentum
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:
- L = r × p
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 =
ma, 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}