Stuck on derivation of Euler's equations in rigid body dynamics

In summary: The summary is: In summary, the derivation of Euler's equations for rotational dynamics involves considering the moment of inertia tensor and constructing a "body frame" fixed within the body to better understand the rotation. In this body frame, the angular momentum is measured as L = [I]w, where [I] is the diagonal matrix and w is the angular velocity. This may be confusing at first, but the author clarifies that "L in rotating frame" refers to the components of L along the rotating axes at a specific instant.
  • #1
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i was reading about derivation of Euler's equations for rotational dynamics (john taylor, classical mechanics, chapter 10) when i got stuck on one of the reasonings
essentially it refers to the moment of inertia tensor, since the tensor itself about a point is dependent on the position of the rigid body (hence the tensor would have to be a function of time) hence it is argued that it would be helpful to construct a "body frame" fixed within the body with its axis pointing at the principal axis of rotation of the body, this body frame would rotate along with the body (since it is fixed within the body). then came the part i didnt understand, it said the angular momentum measured in the body frame is L = w, where denotes the diagonal matrix and w the angular velocity. however since the frame is fixed inside the body, should the rigid body be stationary relative to the frame and hence w is zero? what is wrong with my reasoning?
 
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  • #2
I had the same question when I first encountered Euler Rigid body equations. What the author means by "L in rotating frame" is the components of L along the rotating axes at that instant.
 
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1. What is Euler's equation in rigid body dynamics?

Euler's equation in rigid body dynamics is a set of equations that describe the rotational motion of a rigid body in three-dimensional space. It is derived from Newton's second law of motion and describes the relationship between the angular acceleration of a rigid body and the external torque acting on it.

2. How is Euler's equation derived?

Euler's equation can be derived using the Lagrangian method, which involves finding the kinetic and potential energies of the rigid body and then using the principle of least action to obtain the equations of motion. It can also be derived using the Newton-Euler equations, which use the concepts of torque, angular momentum, and angular velocity to describe the rotational motion of a rigid body.

3. What are the assumptions made in deriving Euler's equation?

The main assumptions made in deriving Euler's equation are that the rigid body is in a state of pure rotation, there are no external forces acting on the body, and the body is perfectly rigid. These assumptions are necessary to simplify the equations and make them applicable to idealized rigid bodies.

4. What are some applications of Euler's equation in rigid body dynamics?

Euler's equation is widely used in various fields such as aerospace engineering, robotics, and mechanics. It is used to analyze the motion of rotating objects, such as gyroscopes, satellites, and spinning tops. It is also used in designing vehicles with rotating parts, such as helicopters and airplanes.

5. Are there any limitations to Euler's equation in rigid body dynamics?

Euler's equation is based on idealized assumptions and does not account for factors such as friction and deformations in the body. It is also limited to rigid bodies and cannot be applied to non-rigid objects. Additionally, it is only applicable to rotational motion and does not describe translational motion.

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