Conservation of Angular Momentum

The conservation of angular momentum is a fundamental concept of physics along with other conservation laws such as those of energy and linear momentum. It states that the angular momentum of a system remains constant unless changed through an action of external forces.

In Newtonian mechanics, the angular momentum of a point mass about a point is defined as <math>\vec H = \vec r \times \vec p</math> where <math>\vec{r}</math> is the position vector of the point mass with respect to the point of reference and <math>\vec{p}</math> is the linear momentum vector of the point mass.

The principle of angular momentum can be applied to a system of particles by summing the angular momentum of each particle about the same point. This can be represented as:

<math> \vec{H}_{sys} = \sum_i \vec{H}_i </math>

where

<math>\vec{H}_{sys}</math> is the total angular momentum of the system

<math>\vec{H}_i</math> is the angular momentum of the i^th particle

Proof of conservation

The derivative of angular momentum with respect to time is equal to the sum of the external moments (or torque <math>\vec {\tau}</math>) applied to the system. Differentiating angular momentum gives:

<math>\vec {\tau} = \vec r \times \vec F + \vec{\dot{r}} \times p</math>

For a constant radius, the second term is zero. Hence <math>\vec {\tau}= \vec r \times \vec F</math> From this, it can be concluded that in the absence of an external moment, angular momentum must be conserved.