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integrated over the path between the two endpoints x a and x b . It
is given by
x b
W ab =
P · ds,
(2.33)
xa
where the integration path is assumed to be the path of physically
allowable particle motion, satisfying (2.14, 2.27). Equivalently,
t b
3
4
W ab =
P i v i dt.
(2.34)
ta
i=1
The function W ab is the integral of the action along the path. It
is also known as the eikonal function. From (2.19),
t b
t b
W ab =
(L + H) dt =
L dt + H (t b − t a ),
(2.35)
ta
ta
where, in the rightmost equality, we only consider possible motion
for which H = const. The variation is
t b
δW ab = δ
L dt + H (δt b − δt a ).
(2.36)
ta
This variation is shown schematically in Figure 2.1, where the solid
curve represents the physically allowable path, and the broken
curve represents an infinitesimally displaced path, which is not
physically allowable. The endpoints are held fixed by assumption
in the variation. In order that H = const, it is necessary to allow
the end times t a and t b to vary. This is different from Hamilton’s
principle (2.7), where the end times t a and t b are assumed to be
fixed. Consequently, in the present case,
t b
t b
∂
∂
δ
L dt = δt a
+ δt b
L dt
ta
∂t a
∂t b
ta
3
t b 4 ∂L
∂L
+
δx i +
δv i dt (2.37)
ta i=1 ∂x i
∂v i
where the first term on the right accounts for the variation of the
end times t a and t b , and the second term accounts for the variation
29
2.1. Relativistic classical mechanics