SUB-RIEMANNIAN METRICS 9
mannian geodesies, reformulated in the Hamiltonian language that gen-
eralizes naturally to the sub-Riemannian case. The second one is the
simplest possible sub-Riemannian but not Riemannian situation, cor-
responding to the Heisenberg Lie algebra. In this case, the "natural"
extension of the Riemannian theory of geodesies turns out to work, in
the sense that the length-minimizing arcs are characteristic curves of
the Hamiltonian associated to the metric. Finally, the third example
—first discussed in — exhibits a case where the natural extension
does not work: we explicitly show that a certain arc is a minimizer but
is not a characteristic of the Hamiltonian.
2.1 Riemannian geodesies
If M is a Riemannian manifold, with metric tensor G, then the curves
that locally minimize length and are parametrized by arc-length are
exactly those that satisfy, in local coordinates, the geodesic equation
x1 + Yljk T^k&x1* — 0, where the Tljk are the Christoffel symbols of
the Riemannian connection associated to the metric, together with the
constraint J2i(x1)2 = 1. Alternatively, one can avoid the Christoffel
symbols by using the Legendre transformation to pass to the Hamilto-
nian formulation. This amounts to working on the cotangent bundle
T*M, and introducing a "momentum" covector A whose components A^
are given by A ^ = — Y,j
(This is precisely —up to a sign*— the
familiar "lowering indices" operation.) We then define the Hamiltonian
H : T*M - R by letting H = - \ £ZJ glj(x)XlXJ1 i.e.
# ( M ) = ^I|A|| ^ , (1)
is the norm of the covector A, regarded as a linear func-
tional on the tangent space T^M of M at x, endowed with the inner
^Our somewhat nonstandard sign conventions are chosen so as to be consistent
with those commonly used in Optimal Control Theory.