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Tops are dynamical systems describing the motion of a rigid body in n dimensions
possibly exposed to an external field.
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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.
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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)
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d/dt L = [L,I(L)], d/dt R=R I(L),
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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
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d/dt L = [L,I(L)] + R-1 n R, d/dt R=R I(L)
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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).
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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.
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