GEOMETRIC OPTIMAL CONTROL OF SIMPLE QUANTUM SYSTEMS - Advances in Chemical Physics - page 173

and

r cos 2 ϕ ) ∂
∂r

r sin 2 ϕ F 0 = ( − + γ cos ϕ − γ

− +

1
r sin ϕγ

∂
∂ϕ

+ ( − sin ϕ cos ϕ −

− +

γ cos ϕ sin ϕ )

+

∂
∂ϕ −

cot ϕ cos θ ∂
∂θ

F 1 =− sin θ

∂
∂ϕ −

cot ϕ sin θ ∂
∂θ

F 2 = cos θ

Hence, one deduces that the system can be written as:

r cos 2 ϕ r sin 2 ϕ r = γ cos ϕ − γ −

− +

1
r sin ϕγ

ϕ =− sin ϕ cos ϕ −

− +

γ cos ϕ sin ϕ − sin θu 1 + cos θu 2

+

θ

=− cot ϕ cos θu 1 − cot ϕ sin θu 2

Using the following rotation on the control
v 1
v 2

cos θ u 1

u 2

sin θ

=

− sin θ cos θ

the system takes the form

r cos 2 ϕ r sin 2 ϕ r = γ cos ϕ − γ −

− +

1
r sin ϕγ

ϕ =− sin ϕ cos ϕ −

− +

cos ϕ sin ϕγ

+ +

v 2

θ

=− cot ϕv 1

The quantities P i can be written as follows:

cos 2 ϕ sin 2 ϕ )] p r + P 0 = [ γ cos ϕ − r ( γ +

− +

1
r sin ϕγ

[ −

− +

( γ

+ −

) cos ϕ sin ϕ ] p ϕ

P 1 =− p θ cot ϕ

P 2 = p ϕ

and we obtain the following Hamiltonian by replacing v 1 and v 2 by the extremal
controls.

cos 2 ϕ sin 2 ϕ )] p r H = [ γ cos ϕ − r ( γ +

− +

p θ cot 2 ϕ

1
r sin ϕγ

p ϕ + [ −

− +

( γ

+ −

) sin ϕ cos ϕ ] p ϕ + +