From: parendt@nmt.edu (Paul Arendt) Subject: Re: Spinning Things and Hamiltonians Date: 6 Sep 2000 13:19:51 GMT Newsgroups: sci.physics.research Summary: [missing] A while ago, Michael Weiss wrote about configuration space for a spinning object: -I could use the three Euler angles to specify it, or better yet, I -could use an element of SO(3). Looks like phase space is 6 -dimensional. ^^^^^^^^^^^ Toby Bartels responded: >Nope. > >The space of angular momenta is (classically) so(3)*, a 3D vector space. ^^^^^^^^^^^^^^^ They're both right! (Notice the difference in wording, though.) Let's fill in the missing steps... but first, let's back up just a bit. The classical "rigid body" is a collection of a bunch of particles, but they're all stuck together: the distance between any two of them remains fixed. These constraints take us from what would be a 6N-dimensional phase space (3 position and 3 momentum components for each of the N particles) down to a much simpler system. In fact, the motion "factors" (usually) into that of two separate mechanical systems, which can be solved independently. These separate systems are the motion of the center of mass, and the orientation of the body (around the center of mass). The second of these subsystems is where spin and angular momentum live. What does the phase space of a rigid body look like (ignoring any motion of its center of mass)? Barring bizarre cases where the particles comprising the body are all in a line, we can specify the orientation as follows: pick a three-dimensional set of axes fixed in space, and a second three-dimensional set fixed on the body in question, with the origin of both at the center of mass. Then the orientation of the body can be specified by giving the *rotation* needed to bring one set of axes in agreement with the other. If that sounds bizarre, it shouldn't! Just think of how an ordinary coordinate system is set up, to specify *where* something is in space. We pick an origin, and a set of directions eminating from it (called the x, y, and z directions usually). Then the location of something is uniquely specified by how it has to move in order to get back to the origin. We usually think of this motion as the x, y, and z coordinates of the object, but we could equally think of it as the motion itself: which *translations* in space are needed to bring the thing back to the origin. In that sense, the location of an object (given a choice of origin) can be thought of as the element of a Lie group: the group of translations in space. Similarly, given a set of reference axes, we can think of the orientation of a body as an element of the Lie group of rotations in 3-space. This group is known as SO(3), for the "Special Orthogonal group in 3 dimensions." But there's a very important difference between this Lie group and the Lie group of translations mentioned before: translations commute with one another, while not all rotations do. In other words, it doesn't matter whether you translate right by 3 units, then up by 5, or do so in the reverse order... either way, you wind up at a point we can unambiguously call "(3,5)". But it matters greatly for rotations, so special care needs to be taken in specifying the *order* in which a set of "basis" rotations is carried out; one such choice is the Euler angles Michael mentioned above. But we're not finished yet: doing mechanics problems involves knowing not only the position of something, but also its velocity! After all, there's a big difference between how two gyroscopes will behave in the future if they're both initially leaning to the side, but one of them is spinning rapidly and the other isn't spinning at all, even if they have the same initial orientation. So the rotational degrees of freedom of a classical rigid body can be fully specified by giving both an element of the group SO(3), specifying the body's orientation, and giving an angular velocity, specifying how the body is moving at some time. But an angular velocity is really just a change from one orientation to another some very short time away, divided by that very short time. In other words, it's just an element of the tangent space of SO(3) -- the Lie algebra denoted by so(3) (note the lowercase letters for the algebra; that's a fairly standard convention). (For those out there trying to use dimensional analysis to check this: we really need to choose a standard of time to identify an angular velocity with an element of so(3) -- so that the Lie algebra elements represent something like radians per second.) So, phase space is 6 dimensional at the outset, like Michael Weiss stated above! Three numbers are needed to specify an orientation, and three more are needed to specify an initial angular velocity. The latter three numbers can be replaced by three giving the angular *momentum* instead, just like momentum can be used instead of velocity in ordinary mechanics. And just like in mechanics, the momentum counterpart lives in the dual of the vector space where velocity lives. The dual of a vector space can be thought of as the space of linear maps from the vector space to the real numbers, and it's referred to by attaching an asterisk to the name of the vector space. So angular *momenta* live in the vector space called so(3)*. The analogous construction for translations in ordinary space depends on what we call the Lie group of translations (in 3 dimensions). I've seen it called T(3), but since it's isomorphic (= the same for all practical purposes) to R^3 (under addition) let's just call it R(3). Then velocities live in r(3), and momenta live in r(3)*, but books don't usually mention this, since the Lie group construction isn't as important as it is for rotations. We'll see why in a followup post. This post grew until it was too big for its britches, so I'm splitting it in half.