Types of quantum field thories
However, the rotational equation is somewhat more complicated than the linear one: firstly because and are not naturally vectors but bivectors; and secondly because they are not necessarily proportional, so that cannot be a scalar. In general, the moment of inertia is a linear function
so that the above equation becomes simply
Similarly, differentiating this equation once with respect to time (and assuming that is constant as it is for a rigid body), we have
In low dimensions, the situation can be (and usually is) simplified.
In three dimensions, bivectors form a three-dimensional vector space, so that the moment of inertia can be represented by a symmetric matrix. Additionally, in three dimensions, there is an isomorphism between bivectors and vectors (once we choose an orientation to go with our inner product); so that angular velocity and momentum can be (and usually are) identified with vectors, and the moment of inertia with a symmetric rank-2 tensor.
In terms of the discussion at Hamiltonian dynamics on Lie groups, the rigid body dynamics in is given by Hamiltonian motion on the special orthogonal group . It is defined by any left invariant? Riemannian metric
This bilinear form is the moment of inertia. (For instance AbrahamMarsden, section 4.6.)
over all space, where is the vector from the origin to the point of integration, denotes the interior product? of a vector with a bivector (yielding a vector), and denotes the exterior product of two vectors (yielding a bivector).
When is the same everywhere (as for a rigid body), then we may view this as a function from to ; this function is the moment of inertia.
A classical textbook discussion is for instance section 4.6 of
or around page 56 of
and around slide 6 in