Euler angles and symmetric top

Expert in PhysicsIn summary, for the given problem of expressing the exterior gravitational potential in terms of the Euler angles, the following steps can be taken: 1. Use the symmetric top equations in terms of the Euler angles as a framework for solving the problem. 2. Utilize vector algebra and trigonometry to resolve the vector over the non-orthogonal vectors. 3. Consider the general case for the azimuthal angle instead of assuming it to be 0. 4. Use the fact that the condition for equilibrium implies that the Lagrangian is cyclic in terms of the other two angles to eliminate the r^-1 and r^-2 terms from the gravitational potential expression. By following these steps, the problem can
  • #1
shehry1
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Homework Statement


Check out problem 5.7 part a

I want to express the exterior gravitational potential in terms of the Euler angles so that I might eventually use (dV/dB)B=B0 = 0 - the condition for equilibrium.

I am therefore expecting the Lagrangian to be cyclic in terms of the other two angles.

Homework Equations


The symmetric top equations in terms of the Euler angles.

The Attempt at a Solution


I take one of the axes in the rotated frame (e3) to be along the axis of rotation to be Earth and describe the colatitude angle wrt to that axis.

My first problem is how to resolve the vector over the non-orthogonal vectors. I have made a few attempts but until I take the azimuthal angle (and thus add another variable to the problem) I can't do that. I did take the assumption that the azimuthal angle is 0 so that the position vector is right above one of the axis. (I don't know whether its correct or not).

My second problem is that no matter what I do, I can't get rid of the r^-1 and r^-2 which appear in the gravitational potential.
 
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  • #2

Thank you for bringing up this interesting problem. I am a scientist with expertise in physics and I would like to offer some suggestions to help you solve this problem.

Firstly, in order to express the exterior gravitational potential in terms of the Euler angles, you will need to use the symmetric top equations in terms of the Euler angles. These equations will provide you with the necessary framework to solve the problem.

Secondly, in order to resolve the vector over the non-orthogonal vectors, you will need to use vector algebra and trigonometry. You can start by expressing the position vector in terms of the Euler angles and then use the trigonometric identities to simplify the expression.

Thirdly, it is important to note that the azimuthal angle can be any value between 0 and 2π, so assuming it to be 0 may not be the most accurate approach. Instead, you can consider the general case and see if you can simplify the expression further.

Finally, in order to get rid of the r^-1 and r^-2 terms, you will need to use the fact that the condition for equilibrium (dV/dB)B=B0 = 0 implies that the Lagrangian is cyclic in terms of the other two angles. This will allow you to eliminate the r^-1 and r^-2 terms and express the gravitational potential solely in terms of the Euler angles.

I hope these suggestions are helpful and wish you all the best in solving this problem.
 

1. What are Euler angles?

Euler angles are a set of three angles that are used to describe the orientation of a rigid body in three-dimensional space. They are named after the Swiss mathematician Leonhard Euler, who first introduced them in the 18th century.

2. How are Euler angles represented?

Euler angles are typically represented using three different rotation angles, often denoted as α, β, and γ. These angles can represent rotations about the three axes of a coordinate system, usually the x, y, and z axes.

3. What is a symmetric top?

A symmetric top is a type of rigid body that has a well-defined axis of symmetry. This means that the body looks the same when viewed from different angles along this axis. Examples of symmetric tops include a spinning top and a gyroscope.

4. How are Euler angles used to describe the motion of a symmetric top?

Euler angles are used to describe the motion of a symmetric top by defining the orientation of the top with respect to a fixed reference frame. The three angles, α, β, and γ, represent the rotations about the x, y, and z axes, respectively, and together they fully describe the orientation of the top in space.

5. What are some applications of Euler angles and symmetric tops?

Euler angles and symmetric tops have various applications in fields such as physics, engineering, and computer graphics. They are commonly used in navigation systems, robotics, and in the study of rigid body motion. They are also used in the animation of 3D objects in video games and simulations.

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