Variational methods in which TIME is also varied
Shina Tan
We traditionally fix the time variable of a path in the variations.
In relativity, however, time is intimately related to space.
Time and space are mixed
if we go from one reference frame to a different frame.
It is unnatural to variate spatial
coordinates only but not time.
Even in nonrelativistic physics, it is beneficial to variate both space and time.
For
instance, this more general variation offers a deeper insight into the energies of periodic
motion (see Sec. 3.2).
In the following we first consider general variations in Lagrangian dynamics, and then
study those in Hamiltonian dynamics. In the latter case, the variables
q
j
,
p
j
, and
t
are all
varied independently, yielding even greater flexibility.
The general principles outlined in Secs. 1 and 2 are applicable to
both nonrelativistic
mechanics and relativity
.
1
Lagrangian Dynamics
Consider a Lagrangian system
L
=
L
(
q
1
,
· · ·
, q
s
,
˙
q
1
,
· · ·
,
˙
q
s
, t
)
.
(1)
Let us call the (
s
+ 1)dimensional space of all possible values of (
q
1
,
· · ·
, q
s
, t
) the “QT
space”.
An arbitrary infinitesimal transformation in the QT space would transform any
actual path
(
q
j
, t
)
,
t
1
< t < t
2
to a neighboring path (called “varied path”)
(
q
j
+
δq
j
, t
+
δt
)
,
t
1
+
δt
1
< t < t
2
+
δt
2
which may or may not be a classically allowed path (namely, the varied path may or may not
satisfy classical equations of motion as the actual path does). In particular, each of the two
end points of the actual path is transformed to a neighboring point with slightly different
values of
q
j
and
t
.
For any quantity
X
that depends on
q
j
, ˙
q
j
and
t
, we define the
differential
of
X
as the
infinitesimal change of
X
ALONG a path:
dX
≡
X
(
q
j
+
dq
j
,
˙
q
j
+
d
˙
q
j
, t
+
dt
)

X
(
q
j
,
˙
q
j
, t
)
.
(2)
1