# Angular momentum

Let me start with recalling from Dzhanibekov effect – Part 4: The equations of motion.
We consider a free rigid body, observed from a laboratory that is an inertial reference system. The body is rotating with a fixed point at the center of its mass. The center of mass is at rest with respect to the laboratory.

Let be an orthonormal frame corotating with the body, and aligned with its principal axes, and let be an inertial laboratory frame, both centered at the center of mass of the body. The two frames are related by time-dependent orthogonal matrix

The inverse of is denoted by

and it is often called the attitude matrix. For a rotating body, if are coordinates of a fixed point in the body, then its coordinates in the laboratory system change in time:

Differentiating we get

The matrix is orthogonal, therefore, by differentiating, the matrix is antisymmetric. Every antisymmetric matrix can be written as

Writing

(*)

we find that

Acting with the matrix is equivalent to taking the crossproduct with vector

The vector is the angular velocity vector in the laboratory frame.

We denote by the body representative of the angular velocity vector where we skip the time dependence, with components
Thus

(**)

In a comment to this post Bjab asked: why (**)? Where is it coming from?
Good question. We have the relation

for any vectors and If is a rotation matrix with determinant 1, then it follows from the definition of the cross product (see e.g. Rotational invariance of cross product) that

Therefore

Let us set then

Since can be arbitrary, we get

Now, from Eq. (*)

(1)

where In the body frame the inertia tensor is diagonal The angular momentum vector in the body frame is then given by the formula

or

The definition of the angular momentum is thus similar to the definition of the linear momentum. Liner momentum is the product of mass and velocity Angular momentum, in the body frame, is a vector whose components are products of moments of inertia and angular velocity components The expression for the angular momentum in the body frame is very simple, but the law of conservation of the angular momentum refers to angular momentum vector in the laboratory frame. The transition from the body to the laboratory frame is implemented by the attitude matrix . Therefore what is conserved is :

(2)

From this we get:

From Eq. (1) we have that therefore Eq. (??) reduces to

or, multiplying by from the left:

But
therefore

Using the definition we arrive at

(Euler)

These are the famous Euler’s equations. At the end of Dzhanibekov effect – Part 4: The equations of motion they were written in terms of Here they are written in terms of

We need to solve them. But first we will look at them with a magnifying glass. In the next couple of posts.

## 4 thoughts on “Angular momentum”

1. Added explanation in the post. Thanks for this question and for all the errata.

1. Ronan says:

You wrote is the explanitation to Bjab
Let us set .

should it be?