ERMAT
. We shall then propose a synthesis of these
two, which, perhaps, can be disputed, but which has incontestable elegance. Moreover,
we shall find a solution to the problem we have posed.
2.2. Two principles of least action in classical dynamics
In classical dynamics, the principle of least action is introduced as follows:
The equations of dynamics can be deduced from the fact that the integral
t
2
t
1
dt,
between fixed time limits,
t
1
and
t
2
and specified by parameters
q
i
which give the state of
the system,
has a stationalry value.
By definition,
, known as Lagrange’s function, or
Lagrgian, depends on
q
i
and ˙
q
i
dq
i
dt
Thus, one has:
(2.2.1)
δ
t
2
t
1
dt
0
From this one deduces the equations of motion using the calculus of variations given
by L
AGRANGE
:
(2.2.2)
d
dt
∂
∂
˙
q
i
∂
∂
q
i
where there are as many equations as there are
q
i
.
It remains now only to define
. Classical dynamics calls for:
(2.2.3)
E
kin
E
pot
i.e., the difference in kinetic and potential energy. We shall see below that relativistic
dynamics uses a different form for
.
Let us now proceed to the principle of least action of M
AUPERTUIS
. To begin, we
note that L
AGRANGE
’s equations in the general form given above, admit a first integral
called the “system energy” which equals:
(2.2.4)
W
∑
i
∂
∂
˙
q
i
˙
q
i
under the condition that the function
does not depend explicitely on time, which we
shall take to be the case below.
dW
dt
∑
i
∂
∂
q
i
˙
q
i
∂
∂
˙
q
i
¨
q
i
∂
∂
˙
q
i
¨
q
i
d
dt
∂
∂
˙
q
i
˙
q
i
∑
i
˙
q
i
d
dt
∂
∂
˙
q
i
∂
∂
˙
q
i
(2.2.5)

2.2. TWO PRINCIPLES OF LEAST ACTION IN CLASSICAL DYNAMICS
17
which according to L
AGRANGE
, is null. Therefore:
(2.2.6)
W
const
We now apply H
AMILTON
’s principle to all “variable” trajectories constrained to
initial position
a
and final position
b
for which energy is a constant. One may write, as
W
,
t
1
and
t
2
are all constant:
(2.2.7)
δ
t
2
t
1
dt
δ
t
2
t
1
W dt
0
or else:
(2.2.8)
δ
t
2
t
1
∑
i
∂
∂
˙
q
i
˙
q
i
dt
δ
B
A
∑
i
∂
∂
˙
q
i
dq
i
0
the last integral is intended for evaluation over all values of
q
i
definitely contained be-
tween states
A
and
B
of the sort for which time does not enter; there is, therefore, no
further place here in this new form to impose any time constraints. On the contrary, all
varied trajectories correspond to the same value of energy,
W
.
1
In the following we use classical canonical equations:
p
i
∂
∂
˙
q
i
. M
AUPERTUIS
’
principle may be now be written:
(2.2.9)
δ
B
A
∑
i
p
i
dq
i
0
in classical dynamics where
E
kin
E
pot
is independent of ˙
q
i
and
E
kin
is a homoge-
neous quadratic function. By virtue of E
ULER
’s Theorem, the following holds:
(2.2.10)
∑
i
p
i
dq
i
∑
i
p
i
˙
q
i
dt
2
E
kin
For a material point body,
E
kin
mv
2
2 and the principle of least action takes its oldest
known form:
(2.2.11)
δ
B
A
mvdl
0
where
dl
is a differential element of a trajectory.
1
Footnote added to German tranlation
: To make this proof rigorous, it is necessary, as it well known, to
also vary
t
1
and
t
2
; but, because of the time independance of the result, our argument is not false.

18
2. THE PRINCIPLES OF MAUPERTUIS AND FERMAT