Chemistry Reference
In-Depth Information
ceases to be optimal. Some tools have to be introduced to compute the position of
this conjugate point.
Locally, we can define a smooth function
u
(
x, p
) and plugging this function into
the pseudo-Hamiltonian
H
gives the normal Hamiltonian
H
r
(
x, p
). The reference
extremal is then solution of
∂H
r
∂p
∂H
r
∂x
x
(
t
)
=
(
x
(
t
)
,p
(
t
))
, p
(
t
)
=−
(
x
(
t
)
,p
(
t
))
Note:
These equations are derived by using the fact that
∂H
r
∂x
∂H
∂x
∂H
∂u
∂u
∂x
∂H
∂x
=
+
=
and
∂H
r
/∂p
=
∂H/∂p
. We introduce Eq. (8)
d
dt
δz
=
d
∇
H
(
z
(
t
))
δz
(8)
which is called the Jacobi equation. Explicitly, this means that
δz
=
(
δx, δp
)is
solution of the system
∂
2
H
r
∂x∂p
(
x, p
)
δx
∂
2
H
r
∂p
2
δ x
=
+
(
x, p
)
δp
∂
2
H
r
∂x
2
∂
2
H
r
∂x∂p
(
x, p
)
δp
δ p
=−
(
x, p
)
δx
−
with
δx
(0)
p
0
. A Jacobi field is a nontrivial solution of this equa-
tion. It is said to be vertical at time
t
if
δx
(
t
)
=
0 and
δp
(0)
=
=
0.
Next, we fix
x
0
=
x
(0) and introduce the exponential mapping
exp
x
0
(
t, p
0
)
=
x
(
t, x
0
,p
0
)
Let
L
0
be the space of points of the form
{
x
0
,p
}
. The tangent space of
L
0
is
spanned by the
n
vertical Jacobi fields at time
t
=
0, since this space is locally
n
. If the final time is not fixed, the PMP yields the additional
diffeomorphic to
R
condition
H
1 (see below
for a concrete computation of the basis of this tangent space). We then introduce
the space
L
t
that is the image of
L
0
at time
t
by the map exp
x
0
. The tangent
space at time
t
of
L
t
is spanned by the Jacobi fields image by the linearized sys-
tem (8) of the initial vertical Jacobi fields. A time
t
c
is said to be conjugate if
there exists a Jacobi field that is vertical at time
t
c
. This is a precise formula-
tion of the physical intuition of accumulation of trajectories at a conjugate point
since for a vertical Jacobi field one has
δx
(
t
c
)
=
0 and the dimension of this space is at most
n
−
0. This behavior is analogous
to the concept of caustics in optics. Under suitable assumptions, it can then be
shown that the trajectory is locally optimal up to the first conjugate point [30]. A
=
