Problems
4
and
5
are concerned with systems that have two degrees of freedom. The generalization of Eq.
(
1
)
to a functional with two degrees of freedom is
J
(
x
,
y
)
=
∫
a
b
F
(
t
,
x
,
y
,
x
′
,
y
′
)
d
t
,
(
7
)
where
t
is a parameter, not necessarily time. Necessary conditions, stated as differential equations, for a vector-valued function
t
→
〈
x
¯
(
t
)
,
y
¯
(
t
)
〉
to be a stationary function of Eq.
(
7
)
are found by an analysis that is analogous to that leading up to Eq.
(
6
)
. Consider the perturbed functional
ϕ
(
ε
)
=
J
(
x
¯
+
ε
ξ
,
y
¯
+
ε
η
)
=
∫
a
b
F
(
t
,
x
¯
+
ε
ξ
,
y
¯
+
ε
η
′
,
x
¯
′
+
ε
ξ
,
y
¯
′
+
ε
η
′
)
d
t
,
(
8
)
where the perturbation vector
〈
ξ
(
t
)
,
η
(
t
)
〉
satisfies
〈
ξ
(
a
)
,
η
(
a
)
〉
=
〈
0
,
0
〉
and
〈
ξ
(
b
)
,
η
(
b
)
〉
=
〈
0
,
0
〉
.
(
9
)
Thus, for
ϕ
(
ε
)
defined by Eq.
(
8
)
, we compute
ϕ
(
0
)
=
0
, integrate by parts, and use the endpoint conditions
(
9
)
to arrive at
∫
a
b
{
[
F
x
−
F
x
′
t
]
ξ
+
[
F
y
−
F
y
′
t
]
η
}
d
t
=
0
.
(
10
)
Since Eq.
(
10
)
must hold for any
ξ
and
η
satisfying the conditions
(
9
)
, we conclude that each factor multiplying
ξ
and
η
in the integrand must equal zero,
F
x
−
F
x
′
t
=
0
F
y
−
F
y
′
t
=
0
.
(
11
)
Thus the Euler-Lagrange equations for the function
(
7
)
are given by the system
(
11
)
.
J
(
y
)
=
∫
a
b
F
(
x
,
y
,
y
′
)
d
x
,
(
1
)
∂
F
∂
y
(
x
,
y
,
y
′
)
−
∂
2
F
∂
x
∂
y
′
(
x
,
y
,
y
′
)
=
0.
(
6
)
The Ray Equations. In two dimensions, we consider a point sound source located at
(
x
,
y
)
=
(
x
0
,
y
0
)
in a fluid medium where the sound speed
c
(
x
,
y
)
can vary with location (see Figure
3
.P
.5
and Project
3
in Chapter
3
for a discussion of ray theory). The initial value problem that describes the path of a ray launched from the point
(
x
0
,
y
0
)
with a launch angle
α
, measured from the horizontal, is
d
d
s
(
1
c
d
x
d
s
)
=
−
c
x
c
2
,
x
(
0
)
=
x
0
,
x
′
(
0
)
=
cos
α
d
d
s
(
1
c
d
y
d
s
)
=
−
c
y
c
2
,
y
(
0
)
=
y
0
,
y
′
(
0
)
=
sin
α
(
i
)
where
c
x
=
∂
c
/
∂
x
and
c
y
=
∂
c
/
∂
y
. The independent variable
s
in Eq.
(
i
)
is the arc length of the ray measured from the source. Derive the system
(
i
)
from Fermat’s principle: rays, is physical space, are paths along which the transit time is minimum. If
P
:
t
→
〈
x
(
t
)
,
y
(
t
)
〉
,
a
≤
t
≤
b
, is a parametric representation of a ray path from
(
x
0
,
y
0
)
to
(
x
1
,
y
1
)
in the
x
y
-plane and sound speed is prescribed by
c
(
x
,
y
)
, then the transit time function is
T
=
∫
P
d
s
c
=
∫
a
b
x
′
2
+
y
′
2
c
d
t
,
(
ii
)
where we have used the differential arc length relation
d
s
=
x
′
2
+
y
′
2
d
t
,
(
iii
)
Find the Euler-Lagrange equations from the functional representation on the right in Eq.
(
ii
)
and use Eq.
(
iii
)
to rewrite the resulting system using arc length
s
as the independent variable. s