Derivation of Euler’s Equations

We first take the equation derived in the relationship between vectors in inertial and rotating frames page, and plug in the angular momentum vector L.


dL dt

From here, given the following conditions:

1.L,T, and omega are each 3-component vectors (with x,y and z components)
2.T=dL dt (The torque about an axis on an object  is equal to the rate of change of its angular momentum about that axis.)
3.L=I omega (The angular momentum about a principal axis is
equal to the component of the moment of inertia tensor corresponding to that principal axis multiplied by the angular velocity about that axis.)

The above equation becomes the following:


      T in rotating and stationary frame

Evaluating the cross product and derivative gives the following three equations:


eulers equations

These equations are known as the Euler equations for the dynamics of a rigid body.


Introduction

Fixed Frames and Body Frames

Moment of Inertia, Principal Moments of Inertia, Inertia Tensor

Relationships Between Vectors in Inertial and Rotating Frames

Bibliography