DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS Geodesics of ...

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[14] Montgomery, R., A Tour of Subriemannian Geometries, their Geodesics and Applications,. AMS, 91, 2002. [15] Strichartz, R.S., Subriemannian Geometry, ...
       

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x˙ =

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'    /   ' *                          /   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    

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σ((u(t)) =c(t)  t ∈ I t ∈ I 2

*      '     (   c(t)# 

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 I

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.

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d(a, b) = inf length(c)

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   %         %     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∗ .

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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



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         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 

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