What is euler equations of motion?

The Euler equations of motion are a set of equations that describe the motion of a rigid body in a non-inertial frame of reference. They were first developed by the mathematician Leonhard Euler in the 18th century.

In mechanics, a rigid body is an object that does not deform under the action of external forces, and a non-inertial frame of reference is a frame that is accelerating with respect to an inertial frame. The Euler equations of motion describe the rotational motion of a rigid body in a non-inertial frame in terms of the angular velocity, angular acceleration, and the moments of inertia of the body.

The equations can be written in various forms, but the most common form is:

I1(ω1̇ - ω2ω3) + (I3 - I2)ω2ω3 = τ1

I2(ω2̇ - ω1ω3) + (I1 - I3)ω1ω3 = τ2

I3(ω3̇ - ω1ω2) + (I2 - I1)ω1ω2 = τ3

Where I1, I2, and I3 are the moments of inertia about the principal axes of the body, ω1, ω2, and ω3 are the components of the angular velocity vector, and τ1, τ2, and τ3 are the components of the torque vector.

The Euler equations are important in the study of celestial mechanics, fluid dynamics, and other fields of physics and engineering where rotational motion is important. They are also used in the design and analysis of spacecraft, aircraft, and other complex systems that involve rotational motion.