SUB-RIEMANNIAN METRICS
11
Chow's Theorem, cf. [8], that, given any two points g1? q2, there is a
finite concatenation of integral curves of / and g that goes from qi to
Q2-)
We define a Riemannian metric G on E by declaring (/, g) to be an
orthonormal basis of sections of E. We now look for the curves in C
that minimize length, i.e. the curves 7 G C such that, if a, b are such
that 7 Ca,b, then ft \\i{t)\\Gdt f% \\i*(t)\\Gdt for every (a,/?,7*)
such that 7* G Cailg, 7* (a) = 7(0) and 7*(/3) = 7(6). (Here, if f G -E'(g),
we write ||u||G = jG{q)(v,v).)
If we proceed by analogy with the case of the Riemannian geodesies,
it is natural to introduce the Hamiltonian H, defined on
IR6
(i.e. on
T*R
3
) by
H(q,\) = -\\\M\G« (3)
where ||A||G?,g denotes the norm of the linear functional E(q) 3 v
(X,v) G R with respect to the inner product G(q), that is,
||A|k, = sup{(A, v : v G E(q) , H |
G
1} . (4)
Since f(q) and g(g) form an orthonormal basis of E(q), we have
H(q,V =
-l({\f(q))2
+
(Kg(q))2).
(5)
Writing q = (x,y, 2), A = (£,77, £), we find:
H(x,y,2,e^,C) = - ^ ( ( e - ! / 0
2
+ ? +
^C)2)
(6)
The bicharacteristic equations turn out to be
£ = (v + xCK, v = (yC-ZK, C = o. (7)
Notice that the third equation simply says that z xy yi. i.e. that
our curve satisfies UJ 0. The sixth equation says that ( is a constant.
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