The top

Tops are dynamical systems describing the motion of a rigid body in n dimensions possibly exposed to an external field.
We can assume that the center of gravity (which is conserved) is at the origin. The motion of a rigid body is then described by a path R(t) in the rotation group SO(n) which is the transformation from the fixed to the moving coordinate system. The path R(t) satisfies d/dt R(t)=a(t) R(t), where a(t) in so(n) is a sqew-symmetric matrix called the "angular velocity" in the fixed coordinate system. The matrix A(t) = R(t)-1 a(t) R(t) is the "angular velocity" in the moving coordinate system. A positiv definite (and so invertible) linear transformation I-1: A -> L on the Lie algebra so(n) is called the momentum tensor . This matrix L satisfies a Lax equation d/dt L = [L,A]. The eigenvalues of L are conserved and are angular momentum integrals. It is known that the system is integrable in any dimension. Note that the angular momentum in the fixed system l = R(t)-1 L(t) R(t) is time independent. The number tr(L2) is called the total angular momentum. The number H(L)= (tr(L),I(L)) is the kinetic energy and so the Hamiltonian of the system. The symplectic structure on so(n) is given by a space dependent almost complex structure JL K = [L,K] which puts on the tangent bundle of so(n) a Poisson algebra {K_1,K_2}L = tr(K1,JL K2). The Lax pair becomes so Hamilton equation d/dt L = JL D H(L), where D is the gradient.
In summary, the equations of motion of the top in n dimensions with momentum tensor I is given by d/dt L=[L,A], d/dt R= a R = R A, A=I(L) with initial condition L(0)=L0 and R(0)=1, which can be can be reduced to a differential equation in so(3) x S0(3)
 d/dt L = [L,I(L)], d/dt R=R I(L),
If an exterior force f(t) works on the top, and n(t) is the integral of moments of this force with respect to the origin, we denote by N(t) = R(t)-1 n(t) R(t) this moment in the moving system. The now heavy top in n dimensions satisfies then
 d/dt L = [L,I(L)] + R-1 n R, d/dt R=R I(L)
Remark. The fact that in n=3 dimensions so(n) has the same dimension 3 can be a source of confusion about the top. This is similar to vector calculus, where curl , div are rather mysterious until the theory is formulated in arbitrary dimensions. Treating the top in n dimensions entangles coincidences for n=3 as a special case. The Lie algebra so(3) with multiplication [L,A] is in three dimensions usually written as a vector product L x A. It is good to keep in mind how this Lie algebra is implemented as a matrix Lie algebra so(3).
Remark. One reason for the importance of rigid motion in three dimensions is based on the fact that for a given density distribution m of compact support and fixed angular momentum, the rigid motion is the one which minimizes the kinetic energy.
Proof: write E[f] for integration with respect to the density m. This is a variational problem over all velocity vector fields x -> u(x) constrained to a fixed angular momentum L = E[ R(x) v(x)], where R(x) is the vector orthogonal to the angular momentum vector l to x and v is the angular speed vector relativ to l. The functional is the kinetic energy T=T(u) = E[ u 2 ]/2. The moment of inertia I= E[ R(x)2 ] with respect to the angular momentum vector depends only on m and is so independent of the choice velocity. Using the Schwartz inequality, one knows that E[|v|2(x)] I = E[ |v|2(x)] E[ |R(x)|2] is bigger or equal to E[ (R(x) v(x))2 ] = L2 with equality if and only if v(x) is orthogonal to R(x) at every x. so that
Therefore 2K = E[ u(x)2 ] is bigger or equal than E[v2] which is bigger or equal than L2/I and we have equality if u(x)=v(x) and v(x) is orthogonal to R(x) for all x. This means uniform rotation.

 © 2000, Oliver Knill , dynamical-systems.org