[14] Montgomery, R., A Tour of Subriemannian Geometries, their Geodesics and Applications,. AMS, 91, 2002. [15] Strichartz, R.S., Subriemannian Geometry, ...
! " # $ $% &'
!
" #
M n (D, g) D m g D (M, D, g) M !"# $%"# $&" c : I ⊂ R → M D# c· (t) ∈ Dc(t) ⊂ T M, ' t ∈ I. ' ( c ) ' L(c) = I g(c· (t))dt# g D * a b d(a, b) = inf(L(c)), ) ( a b * ) ( ' D m ' Xi, i = 1, m# M ' ' ' M
+$,
x˙ =
m
ui (t)Xi (x),
i=1
m# u(.). * ' ' D D T M # ' - &" ' +$, # ' a b . ( +M , * ' . # ' / / ' . 0 ' # $%%% &'()*+ &',-%+ .',%& + + + # + " + +
/ 0
D ' . 1
' / ' * / 2 3 # * ' 4 2 ' 5. 3 $ M (E, σ, F)
E, π) M π. σ : E → T M
! F E F : E → [0, ∞) "
# F C ∞ E \ {0}. F(λu) = λF(u) λ > 0 u ∈ Ex , x ∈ M. y ∈ Ex \{0} $ 1 ∂F 2 (y + su + tv)s,t=0 2 ∂s∂t u, v ∈ Ex , x ∈ M " . gy (u, v) =
E = M × Rm {X1, ..., Xm } % " M
σ : E → T M
σ(x, u) =
m
ui (t)Xi (x),
i=1
F & Rm E = D, σ : D → T M F = F D 1 ' / (E, σ, F) / imσ ⊂ T M #
) 2 v ∈ (imσ)x ⊂ Tx M, x ∈ M
+!,
F (v) = inf {F(u)|
u
u ∈ Ex ,
σ(u) = v}.
! u : I → E
c : I → M π(u(t)) = c(t) ·
σ((u(t)) =c(t) t ∈ I t ∈ I 2
* ' ( c(t)#
length(c) =
I
F(u(t))dt =
1 /
I
.
F (c (t))dt
d(a, b) = inf length(c)
) ( a b *
) 6 a b # σ # -7 / ) 8
% % M
. ' % E(c) = 12 I F 2 (c (t))dt %
% % L = 12 F 2 L = 12 F 2 L = L ◦ σ * / L
$
H(p) = sup {p, v − L(v)} = v
sup p, v + sup {−L(u); σ(u) = v} = v
u
sup {p, v − L(u); σ(u) = v} = u,v
sup {p, σ(u) − L(u)} = u
sup {σ (p), u − L(u)} = H(σ (p))
+8,
u
μ = σ (p)
H(p) = H(μ),
p ∈ Tx∗ M # μ ∈ Ex∗ .
* H T ∗M Kerσ * H ' 4 (x, p) 2 ∂H . ∂H .i x= +%, , pi = − i ∂pi ∂x 9 ' 5. 2 $ ( (x(t), p(t)) ) % * '
% + x(t)
'
$
% '
x(t)
%
!
,% % % *
'
-*.-!.
1 ' +3 ,2 1
2
x˙ = u X1 + u X2 , min u(.)
T 0
F(u(t))dt,
x=
x1 x2
2
∈R ,
X1 =
1 0
,
X2 =
0 x1
F(u) = u + b, u , b = (ε, 0)t , u = (u1 , u2 )t , 0 ε < 1. x(0) = 0,
x(T ) = xT .
1 ( ' * D =< X1 , X2 >
R2 L = 12 F 2 E = D :"# $8" H E ∗ 1 H= 2
+8,
εμ1 (μ1 )2 (μ2 )2 − + (1 − ε2 )2 (1 − ε2 ) 1 − ε2
μ1 μ2
=
1 H= 2
2
1 0 p1 p2 0 x1 ∗ H T M
εp1 (p2 )2 (x1 )2 (p1 )2 − + 2 2 2 (1 − ε ) (1 − ε ) 1 − ε2
2
/ +%, · x1 =
(1 + ε)2 p1 (1 −
ε2 )2
ε − 1 − ε2
+&,
·
x2 = ·
p1 = −
a2 (x1 )2 εp21 (p1 )2 + − (1 − ε2 )2 1 − ε2 (1 − ε2 )3
(x1 )2 a εa(x1 )2 p1 − 1 − ε2 (1 − ε2 )2
x1 a2 εp1 x1 a2 + 1 − ε2 (1 − ε2 )2
(p1 (1−ε2 )2
1 ⎧ ⎨ x1 = ⎩
+&,
√
+
a2 (x1 )2 1−ε2
1 )2
(p1 (1−ε2 )2
1 )2
1 (p1 )2 (1−ε2 )2
+
a2 (x1 )2 1−ε2
+ ,
a2 (x1 )2 1−ε2
p2 = a = ct.
1−ε2 a r(t) sin Aθ(t)
p1 = (1 − ε2 )r(t) cos Aθ(t)
√ A(1 − ε2 )( 1 − ε2 dθ = (1 − ε cos Aθ)2 a dt √ (1 − ε2 )( 1 − ε2 dr = εr sin Aθ(ε cos Aθ − 1) a dt 1 , c∈R c(1 − ε cos Aθ) √ dθ A(1 − ε2 ) 1 − ε2 t= . a (1 − ε cos Aθ)2 r=
9 ( ' # ). 1/2 H=
c = ±1
r2 1 (1 − ε cos Aθ)2 = 2 2 2c r=±
1 1 − ε cos Aθ
) '
sin Aθ a(1 − ε cos Aθ) √ sin2 Aθ A(1 − ε2 ) 1 − ε2 2 dθ. x = a (1 − ε cos Aθ)3 x1 = ±
1 − ε2
8 $ '
% % ! ε = 0
/
0 1 % % $ sin at t sin 2at x1 = ± . , x2 = − a 2a 4a2
!"# ;"
/ $!"# $)+ ))$* 1&2 ( + #+ & ! '( )% ! 9 ))* ;).'.)%& 1=2 ( :+ ! (+ + $%%1*2 ( + :+ ! ? @+ * + $%%1>2 ?+ + ) + + 4 "!5--;$%%'&)%.% 1.2 + 3+ 4 + + ) * * + + 7 + ' ;)..=