Balance of angular momentum

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.

[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] Swiss mathematician Jakob I Bernoulli applied the balance of angular momentum in 1703 – without explicitly formulating it – to find the center of oscillation of a pendulum, which he had already done in a first, somewhat incorrect manner in 1686.

The balance of angular momentum thus preceded Newton's laws, which were first published in 1687.

In 1750, in his treatise "Discovery of a new principle of mechanics"[3] he published the Euler's equations of rigid body dynamics, which today are derived from the balance of angular momentum, which Euler, however, could deduce for the rigid body from Newton's second law.

[2] In 1822, French mathematician Augustin-Louis Cauchy introduced the stress tensor whose symmetry in combination with the balance of linear momentum made sure the fulfillment of the balance of angular momentum in the general case of the deformable body.

According to Newton's second law, an external force leads to a change in velocity (acceleration) of a body.

The inertia relating to rotation depends not only on the mass of a body but also on its spatial distribution.

Here the moment of inertia is not only dependent on the position of the axis of rotation (see Steiner Theorem) but also on its direction.

With the two-dimensional special case, a torque only results in an acceleration or slowing down of a rotation.

In 1905, Austrian physicist Ludwig Boltzmann pointed out that with reduction of a body into infinitesimally smaller volume elements, the inner reactions have to meet all static conditions for mechanical equilibrium.

For the analogous statement in terms of torque, German mathematician Georg Hamel coined the name Boltzmann Axiom.

The line of action of the inertia forces and the normal stress resultants σxx·dy and σyy·dx lead through the center of the volume element.

In order that the shear stress resultants τxy·dy and τyx·dx lead through the center of the volume element must hold.

[9] If these fluxes are treated as usual in continuum mechanics, field equations arise in which the skew-symmetric part of the stress tensor has no energetic significance.

American mathematician Clifford Truesdell saw in this the "true basic sense of Euler's second law".

[2] The area rule is a corollary of the angular momentum law in the form: The resulting moment is equal to the product of twice the mass and the time derivative of the areal velocity.