Revista Tinerilor Economiºti (The Young Economists Journal)
Revista Tinerilor Economiºti (The Young Economists Journal)
Revista Tinerilor Economiºti (The Young Economists Journal)
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Economic <strong>The</strong>ories – International Economic Relations<br />
. ⎛<br />
r<br />
. ⎞ 4<br />
4<br />
1 ⎜<br />
⎟ cos Aθ<br />
− sin Aθ<br />
sin Aθ<br />
Aθ<br />
cos Aθ<br />
3<br />
a ⎜<br />
+ =<br />
r<br />
⎟<br />
(13)<br />
4cos<br />
Aθ<br />
⎝<br />
⎠<br />
. ⎛<br />
r<br />
. ⎞<br />
3<br />
1 ⎜<br />
⎟ sin Aθ<br />
sin Aθ<br />
Aθ<br />
Aθ<br />
Aθ<br />
2<br />
a ⎜<br />
cos − sin = − −<br />
r<br />
⎟<br />
(14)<br />
2 2cos<br />
Aθ<br />
⎝<br />
⎠<br />
<strong>The</strong> equation (13) multiplied by cos Aθ<br />
, minus equation (14) multiplied by sin Aθ<br />
lead to the equation<br />
A dθ<br />
1<br />
= , (15)<br />
2<br />
a dt 4cos<br />
Aθ<br />
or<br />
4A<br />
2<br />
t = Aθdθ<br />
a ∫ cos<br />
and it follows<br />
1<br />
t = ( sin( 2Aθ<br />
) + 2Aθ<br />
)<br />
a<br />
Moreover, the equation (13) multiplied bysin Aθ<br />
, plus equation (14) multiplied by<br />
cos Aθ<br />
lead to the equation<br />
.<br />
1 r sin Aθ<br />
= (16)<br />
3<br />
a r 4cos<br />
Aθ<br />
Using the equations (15) and (16) we obtain<br />
dr sin Aθ<br />
= −A<br />
dθ<br />
,<br />
r cos Aθ<br />
which leads to the following result<br />
+<br />
ln r + ln c1<br />
= ln cos Aθ<br />
, c 1 ∈ R ,<br />
and we get<br />
1<br />
r( t)<br />
= cos Aθ<br />
( t).<br />
(17)<br />
c1<br />
Using the change of variables (12) the Hamiltonian become<br />
2 2<br />
2 4<br />
2 2<br />
2<br />
r cos Aθ<br />
r sin Aθ<br />
r sin Aθ<br />
r<br />
H = + + =<br />
2<br />
2<br />
8 8cos<br />
Aθ<br />
4 8cos<br />
Aθ<br />
and from (17) we get<br />
1<br />
H = 2<br />
8c1<br />
Considering the integral curves parametrized by arclength, that coresponds to fix the<br />
level 1/2 of the Hamiltonian, we have c = ± 1/2<br />
and it follows<br />
r = ± 2cos Aθ.<br />
(18)<br />
which together with (12) lead to the equation<br />
121
Change language