4 ms·
(A line at infinity, of course, as rotations are about axes, not points, and in three dimensions an axis is a line.) As long as dimension n = 3 and so [dimensi
by mananaysiempre 4y ago
(A line at infinity, of course, as rotations are about axes, not points, and in three dimensions an axis is a line.)
As long as dimension n = 3 and so [dimension of R^n] n = n(n-1)/2 [dimension of so(n)]. Not to disregard these low-dimensional isomorphisms; there’s quite a few of them (they run out at about n = 10 IIRC), each one different from the next, and most are important in one way or another. But expressing momenta and angular momenta in those terms gives the impression that their existence is due to the (numerous and sometimes very complex) particulars of three-dimensional space, which it isn’t.
- rcme 4y agoYou can rotate around a line, it doesn't necessarily need to be an axis.
- mananaysiempre 4y agoI don’t really see your point, sorry? In arbitrary dimension, you rotate in an oriented plane (aka a bivector up to a positive constant) around a fixed point, if anything; I meant to call that an “axis” by abusing the terminology a bit (it’s an oriented line in three dimensions and a point plus an orientation, i.e. either “clockwise” or “counterclockwise”, in two). Not all isometries with a fixed point will be “rotations” in that sense, though—some will be products of them, as in / cos u sin u 0 0 \ | -sin u cos u 0 0 | | 0 0 cos v sin v | \ 0 0 -sin v cos v / with non-boring values of u and v.
- deleted 4y ago[deleted]
- danbruc 4y agoRotations of a point happen in a plane around some point in the plane, not around a line or axis. In two dimensions there is no line perpendicular to the plane to rotate around, in four and higher dimensions there is no unique plane perpendicular to any given line. It is only in three dimensions that rotating in a plane and around a line coincide.
- 411111111111111 4y agoDisclaimer: I'm not a physicist and am must approaching this with a very basic understanding of the matter Isn't it generally said that the earth is rotating around the sun, or is that scientifically inaccurate?
- leephillips 4y agoThe planets all orbit in approximately the same plane, the ecliptic. So it’s fine to say that they rotate around the Sun—except that, really, they orbit in (approximate) ellipses with the Sun at one focus. But these ellipses are pretty circular (low eccentricity), so, again, we can say that we’re orbiting around the Sun without hurting anyone.
- User23 4y agoWe can also say the Sun is orbiting the Earth thanks to being able to choose our frame of reference. GPS makes good practical use of the Earth as the preferred frame.
- Ancapistani 4y ago> really, they orbit in (approximate) ellipses with the Sun at one focus Don't they technically orbit the center of mass, assuming a simplified two-body system? In the case of Sol and a planet, given the massive (pun intended) disparities, the center of mass is very, very close to the center of Sol - but not coincident.
- topaz0 4y agoThe word axis can be used to mean various things. One of those is the line that a rotation happens around.
- coldtea 4y ago>rotations are about axes, not points Well, in 2D it's about the same. In 3D it can be the same, as long as the rotation happens on a 2D plane.
- nimish 4y agoRotations always happen in a plane, in n-dimensions. In 3D there's a coincidence where planes and lines can be put in exact correspondence. This is the root of the whole "axial" and "pseudo" vectors that plague physics -- there's a (Hodge) duality being hidden, and we're stuck with Gibbs & Heaviside's vector calculus. But a plane (bivector) will always be the natural way to define a rotation.