Euler equations in rigid body: Taylor VS Kleppner - Kolenkow

In summary: I hope this helps.In summary, Kleppner - Kolenkow say that the Euler equations relate all the quantities in the inertial frame at time t, while Taylor says that the Euler equations determine the motion of a spinning body as seen in a frame fixed in the body.
  • #1
almarpa
94
3
Hello all.

After reading both chapters on rigid body motion both in Kleppner - Kolenkow and Taylor books, I still do not undertand the physical meaning of Euler equations. Let me explain:

In Kleppner - Kolenkow, they claim (page 321 - 322) that in Euler equations, Γ1, Γ2 and Γ3 are the components of the torque as viewed in the inertial (space) frame at some time t, ω1, ω2, ω3 are the components of angular velocity in that same frame, and dω1/dt, dω2/dt, dω3/dt the instantaneous rate of change of those components. Thus, Euler equations relate all these quantities in the inertial space frame at time t.

On the other hand, in page 396, Taylor says that the Euler equations determine the motion of a spinning body as seen in a frame fixed in the body (so I guess he means, as seen in the body frame). So Γ1, Γ2 and Γ3, and ω1, ω2, ω3, are the componnetes of torque and angular velocity in the rotating body frame.

As you see, I am really confused. Besides, to derive Euler equations, Kleppner - Kolenkow use a vector approach and the small angle approximation, while Taylor uses a relation between inertial and non inertial frames, which I have nor studied yet, and maybe this is the source of my confusion.

Are they the same statement, but explained in diffrents ways? If so, I do not understand it.

Is any of the explanations wrong? If so, which is the correct one?

Thanks a lot for your help.
 
Last edited:
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  • #2
Hi
I'm not sure, but i guess they mean the inertial frame of referece. If they meant frame of the body, ω would be zero as well as ∝. But you should wait for an answer of someone more experienced :)
 
  • #3
The Euler equations are in the non-inertial frame comoving with the top (body frame).
 
  • #4
Thanks for your reply. That is what I thought, too.

Then, there must be a mistake in Kleppner - Kolenkow's book (or, at least, they do not explain it properly, because that is what I understood after reading that section).
 
  • #5
Unfortunately my manuscript on mechanics is in German. There I worked out the relations between the body and inertial frame in detail. Perhaps you can understand it, because there are many formulae:

http://theory.gsi.de/~vanhees/faq/mech/node73.html

or in pdf

http://theory.gsi.de/~vanhees/faq-pdf/mech.pdf
 
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FAQ: Euler equations in rigid body: Taylor VS Kleppner - Kolenkow

What are Euler equations in rigid body?

The Euler equations in rigid body refer to a set of equations used to describe the motion of a rigid body in three-dimensional space. They are based on the principles of conservation of angular momentum and energy, and are named after the mathematician Leonhard Euler.

What is the difference between Taylor and Kleppner-Kolenkow's approach to Euler equations?

Taylor's approach to Euler equations involves using vector and matrix algebra to solve for the angular velocity and angular momentum of a rigid body. On the other hand, Kleppner-Kolenkow's approach uses the Lagrangian formulation and calculus of variations to derive the equations of motion.

Which approach is more commonly used in physics and engineering?

Both approaches are commonly used in physics and engineering, but Taylor's approach is more widely used in introductory courses and textbooks. Kleppner-Kolenkow's approach is often used in more advanced courses and research.

What are some real-world applications of Euler equations in rigid body?

Euler equations in rigid body have various applications in fields such as aerospace engineering, robotics, and biomechanics. They are used to model the motion of satellites, aircrafts, and other rigid bodies in space and on earth. They are also used in the design and control of robots and in understanding the mechanics of human movement.

Are there any limitations to the use of Euler equations in rigid body?

While Euler equations are useful for describing the motion of rigid bodies, they have limitations when applied to systems with non-rigid components or when dealing with complex forces such as friction. In these cases, more advanced equations and techniques may be needed.

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