The well-known theorem that the motion of any conservative dynamical system can be determined from the “Principle of Least Action” or “Hamilton’s Principle” was carried over into General Relativity-Theory in 1915 by Hilbert, who showed that the field-equations of gravitation can be deduced very simply from a minimum-principle. Hilbert generalised his ideas into the assertion that
all physical happenings (gravitational electrical, etc.) in the universe are determined by a scalar “world-function” H, being, in fact, such as to annul the variation of the integral
∫∫∫∫H√(−g)dx
0
dx
1
dx
2
dx
3
where (
x
0
,
x
1
,
x
2
,
x
3
) are the generalised co-ordinates which specify place and time, and
g
is (in the usual notation of the relativity-theory) the determinant of the gravitational potentials
g
v
q
, which specify the metric by means of the equation
dx
2
= ∑
p, q
g
vq
dx
v
dx
q
. In Hilbert’s work, the variation of the above integral was supposed to be due to small changes in the
g
vq
's and in the electromagnetic potentials, regarded as functions of
x
0
,
x
1
,
x
2
,
x
3
.