D O C U M E N T 4 9 F E B R U A R Y 1 9 2 1 4 5
assumes the factor only if the ’s are substituted by . Ac-
cording to Herglotz,
But here:
This situation immediately suggests considering the quadratic formula
,
which also depends only on the ratios of the ’s, and setting the variation prob-
lem
with firm boundaries.
In the conventional scaling one finds the differential equations
(A.)
[7]
For a of constant curvature and also I constant and other than zero, when
the curvature is other than zero, the geodesic lines therefore remain the same here
too. If the curvature is zero, however, then I = 0 and the transition from to
is impossible.
For spaces of constant curvature, vanishes identically as well.
One easily verifies with the formula
that the differential equations (A.) remain unchanged if the ’s are replaced by
.
Equations (A.) thus, in any event, pose an interesting generalization of geodesic
lines, in which they gradually change to constant curvature and in which they hard-
ly deviate in a region where the differential quotients of I are small. I do not dare
to decide whether physical meaning should be attached to the additional terms
linked to I.
I hope this message reaches you in the best of health. With many thanks for the
rich stimulus in Vienna, yours very sincerely,
Wirtinger.
H K
1
n 2 –
----------- - g K g K – g K g K – + + =
1
n 1 – n 2 –
- g g g g – K –---------------------------------
1 a + g 1 a + g
g K K , = g K K. =
H g 0. =
d 2 ds2I1 /2 =
g
d 0 =
2
2
d
d xr ik
r
d
dxidxk
d
1
2
-- -
d
dxr
dlgI
d
--------- -
1
4
--I–1 - /2
logI
xl
------------glr
l
– +
ik
– =
ds2
ds2 d 2
H
ik
r
1
2
-- -
xi
a
kr
xk
a
ir
gik
xh
a
ghr – + =
g
1 a + g