Revista Tinerilor Economiºti (The Young Economists Journal)

Economic Theories – International Economic Relations
I have to remark that in the case u
obtain a degenerate cost (metric)
2
0,
u = 0 it result F = 0 that is we
The distribution D is generated by the vectors X 1 , X 2 and we can write
D={ X 1 , X 2 }. We observe that
⎧2
rankD = ⎨
⎩1
if
if
x ≠ 0
x = 0
2
∂
In the canonical base of R we have X 1 = ,
∂x
given by
∂
X 2 = x and the Lie brackets are
∂y
∂
[ X 1, X 2 ] = = X 3 ∉ D .
∂y
It results that the distribution is nonholonomic, but is bracket generating, because the
vector fields { X 1,
X 2 , X 3
2
= [ X 1,
X 2 ] } generate the entire space R .
From (6) we obtain
⎧ dx not
1 1
⎪
= u = s
dt
⎨
dy not
2 2
⎪ = u x = s
⎩ dt
The cost function can be written in the form ( x ≠ 0 )
( s )
F =
1 2 2 2
( u ) + ( u )
1
+ u =
1 2
( s ) +
2 2
2
x
1
+ s =
3
i j
g ij s s + ∑
i=
1
i , j = 1,
1 2
) where b = 1,
b = 0,
and
119
1 ≤
i
b s
(Einstein’s summation, 2
⎛1
0 ⎞
gij = ⎜ ⎟ 2
⎝1
1/
x ⎠
1
The Lagrangian has the form F
2
form
ij
2
1 ⎛ g p ⎞ i p j
H = ⎜ ⎟ , (8)
2 ⎜ i
2 ⎟
⎝ b pi
⎠
⎛1
0 ⎞
where g = ⎜ ⎟ 2
⎝1
x ⎠
ij
, is the inverse of the matrix g ij .
In these conditions we obtain that
2 2 2
2
1 ⎛ p1
+ x p ⎞ 2
H =
2 ⎜
2 ⎟ ,
⎝ p1
⎠
or, in the equivalent form
2
L = and using [4] we obtain the Hamiltonian in the
i