DOC.
462
FEBRUARY
1918 471
462. From Max Planck
Grunewald,
13 February 1918
Dear
Colleague,
It
is
how
I
suspected.
If
for
a
quasi-elastic
central motion
the
Lagrangian
function
is L
=
T
-
U
(T
kinetic,
U potential
energy)
and
m
is
the
mass
of
the
oscillating point,
then
0L/0m
is
the
“force”
that
must be
exerted
externally
so
that
m
remains constant
during
the
motion.
If,
furthermore,
a
is
the constant
for
the
quasi-elastic
force,
then
-dL/da
is
the
force
that
must
be exerted
externally so
that
a
remains constant.
(With
natural
motion,
m
and
a
obviously
are
constant because
they
have
pre-
scribed
values,
in
the
sense
of
firm
conditional
the
coordinates.)
If,
now,
m
and
a
are
simultaneously
altered such
that the
relation
m/a
remains
unchanged,
and
the
period along
with
it,
then
dm/da=m/a.
Then
the total
work exerted from outside
upon
the
system is:
A
=
dL dL
8a
-z-om
-
-8a
=--
dm da
a
(
dL dL
\
=-~•L,
because
dL/dm=T/m
and
dL/da=-U/a.
Consequently,
the
temporal
mean
value
A
for
one
period
=
0 (because
T
=
U).
From this
follows that, for
the
change
in
state
under
consideration,
the
energy
of
the
system
(T
+
U)
does not
change,
hence
neither
does its half: T. And since
the
period
also does
not
change,
the
time
integral of
T
does in
fact
remain invariant
throughout
a
period.
I
hypothesis can very
well
be
applied
to
changes
in
mass
without
encountering an
inconsistency.[1]
Good-bye
until
Saturday
the 23rd.
Cordial
regards,
yours,
Planck.
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