# Motivation for using Euler angles instead of other coordinates

Why are Euler angles used when describing the rotation of a rigid body instead of angles around the principal axes of inertia? What exactly is the benefit of introducing Euler angles? They seem not like a "natural" choice but they are the standard way to handle rigid body rotation in classical and quantum mechanics.

I have never seen a discussion why we use them and not some other set of coordinates. What makes Euler angles so particularly useful? What is the motivation to introduce them and what problems would we run into if we tried to use other coordinates? Are they truly the only set of coordinates that lead to "manageable" equations when it comes to rigid body rotation?

• This article discusses the advantages and disadvantages of Euler angles. The usual suspect will inform you about other manageable sets of coordinates. Mar 27, 2023 at 10:41
• @KurtG. Thank you for the link. The article did greatly improve my understanding of the problem of describing the orientation of a body in 3D. It also made me realize that I never consciously made the distinction between direction and orientation. Mar 28, 2023 at 9:11

Why are Euler angles used when describing the rotation of a rigid body instead of angles around the principal axes of inertia?

How do you propose to use angles around the principal axes, as they need to be applied with a specific sequence, and once you have applied the first one, the principal axes aren't aligned anymore? Besides, using three rotations about the principal directions is still considered Euler angles. Just a different scheme of angles.

If you use a sequence of three orthogonal rotations, regardless if they start from a principal axis or not, you have Euler angles. Some Euler angle schemes use a sequence of body-centered axis (like you propose) and some use inertially centered axis.

When describing the orientation of a body, you can either use an arbitrary angle about an arbitrary direction (1+2 quantities defined), or 3 arbitrary angles about three fixed directions (3 quantities defined).

An alternative, which I recommend is to use quaternions that encode the axis-angle system but without gimbal lock that might happen with spherical coordinates for the direction of rotation.

In robotics, you might use Euler angles as each rotation would correspond to a physical joint between parts. But for free rigid bodies, go with quaternions instead.

See my post in SO about how to use quaternions.

• Can you express kinetic energy or a Hamiltonian in terms of quaternion parameters? Mar 31, 2023 at 8:34
• @HansWurst - you can express $\vec{\omega}$ in terms of quaternion parameters and their derivatives as the parameters along only represent orientation. For $q = (\boldsymbol{v},\;s)$ you recover rotational speed by $$\pmatrix{ \boldsymbol{\omega} \\ 0} =2 \pmatrix{ s \boldsymbol{\dot{v}} - \dot{s} \boldsymbol{v} + \boldsymbol{v} \times \boldsymbol{\dot{v}} \\ s \dot{s} + \boldsymbol{v} \cdot \boldsymbol{\dot{v}} }$$ Mar 31, 2023 at 13:16
• @HansWurst - but usually the state of velocity-related quantities are tracked on their own. Quaternions only offer a convenient way to store the orientation and deriving the rotation matrix at any instant. Mar 31, 2023 at 13:19