In physics, Euler's equations govern the rotation of a rigid body. We choose the body fixed axes to be principal axes of inertia. This will make the calculations easier, since we can now split the change in angular momentum into a component that describes the change of the size of and another component that compensates for the change in direction of . Since antiquity, people have tried to understand the behavior of matter: why unsupported objects drop to the ground, why different materials have different properties, and so forth. ... In physics, a rigid body is an idealisation of a solid body of finite dimension in which deformation is neglected. ... In physics, angular momentum is analogous to (linear) momentum except that it applies to rotating objects. ...

The equations are:

where is the projection of the angular momentum in the body fixed axes, the change of the angular momentum of the body with respect to the body fixed axes, the rate of change of the Euler angles of the body connected axes with respect to the space axes, and the external torque. In physics, angular momentum is analogous to (linear) momentum except that it applies to rotating objects. ... In physics, angular momentum is analogous to (linear) momentum except that it applies to rotating objects. ... Euler angles are the classical way of representing rotations in 3-dimensional Euclidean space, named after Leonhard Euler. ...

Proof

If we replace with its components we can replace with . If we choose the basis vectors to be the body fixed axes, the first three terms are equal to and the rest is

Application

In component form, the Euler equations become

For the LHSs equal to zero there are non-trivial solutions: torque-free precession. In mathematics, LHS is informal shorthand for the left-hand side of an equation. ... There are two types of precession: torque-free precession and torque-induced precession. ...

It is also possible to use these equations if the axes in which is described are not connected to the body. should then be replaced with the rotation of the axes instead of the rotation of the body. It is, however, still required that the chosen axes are still principal axes of inertia! This form of the Euler equations is handy for rotation-symmetric objects that allow some of the principal axes of rotation to be chosen freely.

See Poinsot's construction.