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 [16]— 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

9ij(%)ij•

(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)

where

||A||G,X

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.