4.5 Extremal Curves with Cuspidal Points
303
Substituting the expressions into the expression (5), we obtain
⎡
⎣
1 + c 2
1 + (−1 − c 1 )
c 1
1 + c 2
1
⎤
⎦
x=x c −0
=
⎡
⎣
1 + d 2
1 + (−1 − d 1 )
d 1
1 + d 2
1
⎤
⎦
x=x c +0
(6)
The result from the reduction is
1 − c 1
1 + c
2
1
=
1 − d 1
1 + d
2
1
(7)
According to the boundary condition y(0) = 1.5, we get 1.5 = 0 + c 2 , c 2 = 1.5.
From the boundary condition y(1.5) = 0, we give 0 = 1.5d 1 + d 2 , thus there is
d 2 = −1.5d 1
(8)
Substituting the constraint equations of corner point ϕ(x c ) = −x c +2 and c 2 = 1.5
into Eq. (3) and Eq. (4), we obtain
c 1 x c + 1.5 = −x c + 2
( 9 )
d 1 x c + d 2 = −x c + 2
( 1 0 )
Solving simultaneously Eqs. (7)–(10), we obtain
c 1 = −0.5, d 1 = −2, d 2 = 3, x c = 1
Therefore, the extremal curve is
y 1 = −0.5x + 1.5 x ∈ [0, x c ]
(11)
y 2 = −2x + 3 x ∈ [x c , 1.5]
(12)
The extremal curve is the route that the angle of incidence is equal to the angle of
reflection.
4.6 One-Sided Variational Problems
If the extremal function in the variational problem is subject to a certain inequality,
then the variational problem constrained by this inequality is called the one-sided
variational problem. Consider the extremal problem of the simplest functional
303
Substituting the expressions into the expression (5), we obtain
⎡
⎣
1 + c 2
1 + (−1 − c 1 )
c 1
1 + c 2
1
⎤
⎦
x=x c −0
=
⎡
⎣
1 + d 2
1 + (−1 − d 1 )
d 1
1 + d 2
1
⎤
⎦
x=x c +0
(6)
The result from the reduction is
1 − c 1
1 + c
2
1
=
1 − d 1
1 + d
2
1
(7)
According to the boundary condition y(0) = 1.5, we get 1.5 = 0 + c 2 , c 2 = 1.5.
From the boundary condition y(1.5) = 0, we give 0 = 1.5d 1 + d 2 , thus there is
d 2 = −1.5d 1
(8)
Substituting the constraint equations of corner point ϕ(x c ) = −x c +2 and c 2 = 1.5
into Eq. (3) and Eq. (4), we obtain
c 1 x c + 1.5 = −x c + 2
( 9 )
d 1 x c + d 2 = −x c + 2
( 1 0 )
Solving simultaneously Eqs. (7)–(10), we obtain
c 1 = −0.5, d 1 = −2, d 2 = 3, x c = 1
Therefore, the extremal curve is
y 1 = −0.5x + 1.5 x ∈ [0, x c ]
(11)
y 2 = −2x + 3 x ∈ [x c , 1.5]
(12)
The extremal curve is the route that the angle of incidence is equal to the angle of
reflection.
4.6 One-Sided Variational Problems
If the extremal function in the variational problem is subject to a certain inequality,
then the variational problem constrained by this inequality is called the one-sided
variational problem. Consider the extremal problem of the simplest functional