This question has occurred to me many times and I have always been too lazy to try to solve the pertinent equations. I hate reinventing wheels which have been in use for a long time.

Imigine a sphere with a rod through the poles. The rod is one spoke of a wheel. Now spin the sphere with the polar rod as the axis and roll the wheel.

Given the result of the above, what is the effect of a third rotation?

Surely this problem has been analyzed and solved.

I think maybe that your wording of the problem is bad, so I will answer the problem for how I believe it was posed.

In general this body has 3 rotational degrees of freedom, one for each of the Euler angles. This is not so hard to visualise, imagine the 3 axis of a 3 dimensional cartesian space that intersect at the centre of the sphere. These will define 3 axis around which the sphere can turn.

Now take the normal vector to a sphere, which is proportional to xi+yj+zk, where x, y and z are the coordinates of any point on the shere, and i, j and k are 3 orthonormal vectors, or a basis for R3. Now there are 2 ways to proceed here, both equivalent, but one more enlightening.

If you introduce the concept of infinitesimal rotations about these 3 axis, you can introduce a matrix lie group representation of rotations, which gives you SO(3), A set of 3 x 3 orthogonal matrices with determinant one. The short of the long is that this group is non-commutative (rotations do not commute), has 4 elements (One identity I, and 3 elements do to rotations about the 3 axis) Knowing the algebra of this group, you can calculate the effect of rotating a body n-times about any axis in any order.

SO(3) ... has 4 elements (One identity I, and 3 elements do to rotations about the 3 axis)
SO(3) has a nondenumerable infinity of elements (as does any real manifold of positive dimension). it is generated by the Lie algebra which also has a nondenumerable infinity of elements (as does any real vector space with positive dimension). however, the Lie algebra is finite dimensional; it has dimension 3, so there are 3 independent elements that generate SO(3)

Sorry, you are correct. A continuous group has an infinite set of elements. I knew what I was trying to say, but it came out incorrectly.


And yes, the Lie algebra has an infinite number of elements, although each element does not have a unique representation.

Sorry, you are correct. A continuous group has an infinite set of elements. I knew what I was trying to say, but it came out incorrectly.


And yes, the Lie algebra has an infinite number of elements, although each element does not have a unique representation.
maybe you were referring to the fact that in the group SU(2), any element can be written as a sum of the identity and the 3 Pauli matrices? this is true, but this does not make the dimension of SU(2) 4, because not just any linear combination of these four guys is allowable.

there is a constraint:

aI+bσ_x+cσ_y+dσ_z

is in SU(2) iff a²+b²+c²+d¹=1

in other words, SU(2) is the three sphere.

the situation with SO(3) is a bit more complicated.

also, you seem to be saying that the Lie algebras generate the rotations. of course, this is true, but they do not generate the rotations about the Euler angles that you mention above. the Lie algebra generates members of the Lie group with 1 parameter subgroups, along which the exponential mapping is a homomorphism. the Euler angles do not parametrize any 1 parameter subgroups. there are many valid choices of basis for the Lie algebra, the most common one (that is use above) are the generators of rotation about the x-, y-, and z-axes

Now why are we using group theory to solve what seems to me to be a relatively simple mechanics problem? I would imagine that one could just use conservation of angular momentum and such to figure out the motion of the wheel.

Now why are we using group theory to solve what seems to me to be a relatively simple mechanics problem? I would imagine that one could just use conservation of angular momentum and such to figure out the motion of the wheel.
be my guest

Yes, but Lie groups are so beautiful. And any way, conservation of angular momentum is already given by Noether's theorem, as you have found a symmetrical property of the problem

I'm not clear about what you're actually asking, Dinosaur.

You have two rotations already - one about the spoke axis, and one about the axis of the wheel (which is at right angles to the other one). These two add together to give the sphere a net angular momentum.

What is the third rotation you're talking about, and what do you wish to know about it?