Balance of angular momentum

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The balance of angular momentum or Euler's second law in classical mechanics is a law of physics, stating that to alter the angular momentum of a body a torque must be applied to it.

An example of use is the playground merry-go-round in the picture. To put it in rotation it must be pushed. Technically one summons a torque that feeds angular momentum to the merry-go-round. The torque of frictional forces in the bearing and drag, however, make a resistive torque that will gradually lessen the angular momentum and eventually stop rotation.

The mathematical formulation states that the rate of change ${\displaystyle {\dot {\vec {L}}}_{c}}$ of angular momentum ${\displaystyle {\vec {L}}_{c}}$ about a point ${\displaystyle {\vec {c}}}$, is equal to the sum of the external torques ${\displaystyle {\vec {M}}_{c}}$ acting on that body about that point:

${\displaystyle {\vec {M}}_{c}={\dot {\vec {L}}}_{c}}$

The point ${\displaystyle {\vec {c}}}$ is a fixed point in an inertial system or the center of mass of the body. In the special case, when external torques vanish, it shows that the angular momentum is preserved. The d'Alembert force counteracting the change of angular momentum shows as a gyroscopic effect.

From the balance of angular momentum follows the equality of corresponding shear stresses or the symmetry of the Cauchy stress tensor. The same follows from the Boltzmann Axiom, according to which internal forces in a continuum are torque-free.^[1] Thus the balance of angular momentum, the symmetry of the Cauchy stress tensor, and the Boltzmann Axiom in continuum mechanics are related terms.

Especially in the theory of the spinning top the balance of angular momentum plays a crucial part. In continuum mechanics it serves to exactly determine the skew-symmetric part of the stress tensor.^[2]

The balance of angular momentum is, besides the Newtonian laws, a fundamental and independent principle and was introduced first by Swiss mathematician and physicist Leonhard Euler in 1775.^[2]