Revista Tinerilor Economiºti (The Young Economists Journal)

Revista Tinerilor Economişti (The Young Economists Journal)
p p
H +
2 4
2
=
1
8
2
+ 2
8p1
4
x
2 2
x .
4
The Hamilton’s equations (5) lead to the following differential equations
4 4
dx ∂H
p1
p2
x
= = − , (9)
3
dt ∂p1
4 4 p1
3 4 2
dy ∂H
p2
x p2
x
= = + , (10)
2
dt ∂p2
2 p1
2
4 3 2
dp1
∂H
p2
x p2
x
= − = − − , (11)
2
dt ∂x
2 p1
2
dp2
∂H
= − = 0 ⇒ p2
= a = const.
dt ∂y
and we obtain
120
p
4 2 4
dx p1
− a x
= ,
dt 4 p
3
1
3 4 2
dy a x + ap1
2
dt 2 p1
dp
dt
1
2
x
= ,
2 4 3
a x a x
= − − ,
2 2 p
In these conditions the expresion of the Hamiltonian leads to the following change of
variables:
r(
t)
sin Aθ
( t)
x(
t)
= , p1( t)
= r(
t)
cos Aθ
( t)
(12)
a
We obtain
4
4
dx r(cos
Aθ
− sin Aθ
)
=
3
dt 4cos
Aθ
and
3
dp1
ar sin Aθ
ar sin Aθ
= − − 2
dt 2 2cos
Aθ
But on the other hand
.
.
dx r rAθ
= sin Aθ
+ cos Aθ
dt a a
dp .
.
1
= r cos Aθ
− rAθ
sin Aθ
dt
and it follows
2
1