30
Chapter 2. Geometrical optics
Figure 2.1: Variation of the particle trajectory for fixed endpoints
of the integrand. The integrand of the second term on the right
can be rewritten (2.14) as
∂L
∂L
d ∂L
∂L d
d ∂L
δx i +
δv i = δx i
∂x i
∂v i
dt
�
∂v i
�
+
(δx ) =
δx .
∂v i dt
i
�
i
dt ∂v i
�
(2.38)
From the third identity of (2.21) together with (2.37, 2.38),
�
t
3
t b
�
b
t b
δ
L dt = L δt
+
P i δx i .
(2.39)
ta
ta
4 �
�
t
i=1
a
We now impose the condition that the endpoints x a and x b remain
fixed. To ensure this, we require that δx i = −v i δt at the end times
t a and t b , to compensate for what would otherwise be an offset of
Chapter 2. Geometrical optics
Figure 2.1: Variation of the particle trajectory for fixed endpoints
of the integrand. The integrand of the second term on the right
can be rewritten (2.14) as
∂L
∂L
d ∂L
∂L d
d ∂L
δx i +
δv i = δx i
∂x i
∂v i
dt
�
∂v i
�
+
(δx ) =
δx .
∂v i dt
i
�
i
dt ∂v i
�
(2.38)
From the third identity of (2.21) together with (2.37, 2.38),
�
t
3
t b
�
b
t b
δ
L dt = L δt
+
P i δx i .
(2.39)
ta
ta
4 �
�
t
i=1
a
We now impose the condition that the endpoints x a and x b remain
fixed. To ensure this, we require that δx i = −v i δt at the end times
t a and t b , to compensate for what would otherwise be an offset of