Deriving Precession from First Principles

[Hornbein]

I'm trying to figure out the effect of a force applied to a rotating body in 4+1 dimensions. A good place to start would be deriving the effect of forces in 3+1 dimensions. I have been looking around the Internet and even went to the local community college and checked out texts in the library and couldn't find an explanation. In 3D translation is easy, rotation is easy, what is left is precession. How is that derived from first principles?

I don't need the formula with the angular momentum and cross product. I need a derivation of that.

 

[mpresic]

A example of the gyroscope used to be an example in most introductory college physics books like Resnick and Halliday. I notice the example has been removed from Resnick and other introductory books lately. I feel this is OK because in my experience, I always had to gloss over the derivation. I have never found a careful derivation of the precession of a gyroscope without at least junior-senior (upper undergraduate) level involving the Hamiltonian formulation. This falls in the area of space mechanics. Rotational motion in 3d involving Euler angles etc is rarely treated in freshman-sophomore courses. Most often this is addressed in the second semester of undergraduate mechanics if the student is lucky enough to have one.

Examine Symon:Mechanics; Goldstein Mechanics: Presumably Marion/Thornton Mechanics. But these treatments tend to use Energy methods rather than force methods.

As far as extending this analysis to higher dimensions, I am not sanguine about the success. Three dimensions has rotational properties that four dimensions does not have. For example rotation in 3-D always has an invariant axis. This is not guaranteed with 4-dimensions.

 

[Hornbein]

A example of the gyroscope used to be an example in most introductory college physics books like Resnick and Halliday. I notice the example has been removed from Resnick and other introductory books lately. I feel this is OK because in my experience, I always had to gloss over the derivation. I have never found a careful derivation of the precession of a gyroscope without at least junior-senior (upper undergraduate) level involving the Hamiltonian formulation. This falls in the area of space mechanics. Rotational motion in 3d involving Euler angles etc is rarely treated in freshman-sophomore courses. Most often this is addressed in the second semester of undergraduate mechanics if the student is lucky enough to have one.

Examine Symon:Mechanics; Goldstein Mechanics: Presumably Marion/Thornton Mechanics. But these treatments tend to use Energy methods rather than force methods.

Thank you very much. The Hamiltonian it is. I'll see whether I can get those through interlibrary loan.

I used Halliday and Resnick as a textbook way back when and it had that example. So I thought, "this will be easy." Not!

I think it is odd that it was removed. The result is useful "cookbook style" without the derivation.

As far as extending this analysis to higher dimensions, I am not sanguine about the success. Three dimensions has rotational properties that four dimensions does not have. For example rotation in 3-D always has an invariant axis. This is not guaranteed with 4-dimensions.

Right. The idea is to abstract out the most basic physical principles such as conservation of momentum, see how the lead to the result in 3D, then apply them in other numbers of dimensions.

So far I've found this to be quite a useful exercise. One has to REALLY understand something to extend it to N D convincingly.

 

[Nidum]

Some older textbooks on dynamics derive the gyroscope equations by initially considering the forces acting on and the motion of a single particle within the rotating body . This approach is intuitive and analytically simple .

 

[mpresic]

I played around today along the following lines for gyroscopic motion. I wrote the Hamiltonian. Found two constants of the motion, which gave me the phi, and psi motion in terms of theta, which is still necessary. The differential equation for theta in terms of the constants of the motion reduces to an integral. This integral can be solved as stated in Goldstein, but the result is unilluminating (Goldstein's words). (The answer involves Jacobi, or Weierstrass Elliptic Functions, or elliptic integrals.) But this is a tough way if you just want precession with no nutation.
In the end, I found the "effective potential" (for theta) as done in Goldstein or Symon, maybe Marion. The force should be the minus gradient of this potential. If you end up taking the negative gradient, you will get an unilluminating mess of angular momentum components, energy, and two moments of inertia. All told a real mess.

If I remember correctly, Goldstein shows in his chapter on special relativity, that rotations in 4d correspond to rotations in 3d and "boosts". I may have been hasty in dismissing the problem. Happy hunting for the solution. Best Wishes

 

[Hornbein]

I played around today along the following lines for gyroscopic motion. I wrote the Hamiltonian. Found two constants of the motion, which gave me the phi, and psi motion in terms of theta, which is still necessary. The differential equation for theta in terms of the constants of the motion reduces to an integral. This integral can be solved as stated in Goldstein, but the result is unilluminating (Goldstein's words). (The answer involves Jacobi, or Weierstrass Elliptic Functions, or elliptic integrals.) But this is a tough way if you just want precession with no nutation.
In the end, I found the "effective potential" (for theta) as done in Goldstein or Symon, maybe Marion. The force should be the minus gradient of this potential. If you end up taking the negative gradient, you will get an unilluminating mess of angular momentum components, energy, and two moments of inertia. All told a real mess.

If I remember correctly, Goldstein shows in his chapter on special relativity, that rotations in 4d correspond to rotations in 3d and "boosts". I may have been hasty in dismissing the problem. Happy hunting for the solution. Best Wishes

I've ordered Symon and Goldstein through interlibrary loan. Couldn't find Marion.

Elliptic integrals! My God, I had no idea.

I'm working on rotations in 4+1 space. The 3+1 special relativity boost rotations are partial as they are limited to <90 degrees, so precession doesn't enter into the picture (as far as I know.)

Special relativity is basically 1+1 dimensional, so it can be expected to work the same in every space of any interest. The extra dimensions are more or less irrelevant.