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In-Depth Information
S
2
R
+
onto
P
2
O
, degenerate Euler-Lagrange ellipse/hyperbola
Fig. 1.17.
Orthogonal projection
C
r
(
r
)=
r
2
,
G
r
(
r
)=
r
2
0
.
0
(1.126)
R
2
R
2
−
01
0
r
2
Right Euler-Lagrange matrix in polar coordinates:
2E
r
=G
r
−
C
r
,
E
r
=
1
2
00
0
.
(1.127)
r
2
R
2
−r
2
−
Right eigenvalues:
R
2
R
2
r
2
− r
2
, λ
2
=1
,κ
1
=
1
2
κ
i
=
λ
i
−
, λ
1
=
1
∀
i
∈{
1
,
2
}
− r
2
>
0
,
2
R
2
κ
2
=0
.
(1.128)
Right Euler-Lagrange tensor:
r
2
1
2
g
2
E
r
=
−
⊗
g
2
r
2
.
(1.129)
R
2
−
Box 1.20 (Orthogonal projection
, polar coordinates, the transformations from
the right Euler-Lagrange matrix to the left Euler-Lagrange matrix).
S
2
R
+
onto
P
2
O
E
r
→
E
l
:
E
l
=J
l
E
r
J
l
,
2
=
R
2
cos
2
Φ,
J
l
=
1
r
2
,
E
l
=
00
=
2
R
2
00
.
0
1
−
√
R
2
−
(1.130)
r
2
/
2
0cos
2
Φ
0
−
0
−
Left eigenvalues:
1=
1
1=
1
1
2
K
1
=
Λ
1
−
1
,
2
K
2
=
Λ
2
−
2
cos
2
Φ, K
2
=0
.
λ
1
−
λ
2
−
1
,K
1
=
−
(1.131)
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