128
DOC.
1
MECHANICS LECTURE NOTES
[47]The
factor of
2 is
incorrect here but should
be
kept
in
the second
equation
after this
one,
where
it
has been crossed
out.
[48]A
and
B
are
the reaction
forces
on a
rigid beam;
the
Pi
are
the
mutually parallel applied
forces (normal to
the
beam),
and
l is
the beam's
length.
[49]An
=
sign is missing
between
Yv
and
Pv.
a, ß
and
y
are
the cosines of the
angles
between
the
parallel forces
Pv
and the coordinate
axes.
[50]The origin
of coordinates
is
placed
at
the
primed
attachment
point,
so
that
only
the
double-primed force
of
constraint
exerts
a
moment.
[51]In
the
equations
below,
M
is
the total
mass
of the
disk,
and
£,
rj
are
the coordinates of
its
center
of
mass.
The
object
is
symmetric
about the axis
through
the
origin,
which
is
not
the
center
of
mass.
[52]A0
is
the distance of
the center
of
mass
from the
origin
in
equilibrium.
The
flexible
axis
to
which the disk
rigidly
attaches
is subject
to
a
harmonic
restoring
force
proportional
to the
bending
£
of
the
axis
when the disk
is
rotating.
[53]In
this
equation
cp
should
be 3.
l
is
the distance of the
center
of
mass
from the
axis.
[54]See
Helmholtz
1898,
p.
191,
for
a
more
detailed discussion of
this
method.
[55]In
the time around the
turn
of the
century
the
equations
as
given
above
were
sometimes
denoted
as
Lagrange's
equations
of the
first kind,
as
distinct from the
equations
of the second
kind,
which
correspond to
Lagrange's
equations
as
they
are
known
today
(see
Voss
1901,
p.
81,
fn.
220,
and Helmholtz
1898,
p. 316).
[56]Although
earlier
on,
the
generalized
coordinates
are
denoted
by
q,
the
use
of
p
was
not
uncommon
at
the
time.
See,
for
instance,
Einstein
1902b
(Vol.
2,
Doc.
3),
p.
417,
and Boltzmann
1898, §25.
[57]In the
equations
below M
is
the
mass
of the
separate
rods and
l
their distance. The
angle
(p
specifies
the orientation of the
upper
rod with
respect to
the vertical. For the
meaning
of
k,
see
the
discussion of inertial
momenta
on
[pp. 60-62].
[58]In the last
term
of this
equation,
a
factor of
3'cp'
and
a
closing parenthesis
are missing.
[59]The
left-hand
side
of
this
equation
should read
-
alco2A1
-
(fca2co2 +
gaj)l2.
[60]The
L,
M, N
here
represent
the
components
of
torque.
[61]The
preceding
five
words
are
interlineated
in
the
original.
[62]The
main
text
of
the
first notebook ends
here.
The
remaining
two
pages
(which
are upside
down and
start at
the
end)
are
printed
as
[p. 116]
and
[p.
117]
at
the
very
end of
these
lecture
notes.
[63]"endlich" should
be
"unendlich."
[64]The
preceding eight
words
are
interlineated
in
the
original.
[65]The
factor of
sin
ß
in
the denominator should have been
deleted,
as was
done
in
the
expressions
above.
[66]These
equations
and
the words
"T berechenbar"
are
added
in
pencil.
[67]oy
should
be
ox.
[68]The
term
"kinetic
potential"
is used
by Helmholtz;
see
Helmholtz
1898,
pp.
359-361. In
the
following,
L is
the kinetic
energy
and
II
the
potential.
[69]See
Helmholtz
1898, p. 360,
where Helmholtz
emphasizes
the
applicability
of Hamilton's
principle
to fields
other than
dynamics.
The
word
"Kraftäusserungen"
is
interlineated
in
the
original.
[70]This
is
the
Lagrangian
for
a
pair
of
interacting
circuits with
currents
n1,
n2. L1, L2,
and
M
are
the
coefficients
of self-induction and mutual
induction,
respectively.
The
pv
are
the
generalized
coordinates of the circuits.
[71]Einstein did
in fact
include
an
introductory
discussion of the
Lagrangian
and Hamil-
tonian formulation of classical mechanics
in
his lectures
on
statistical
mechanics,
as is
clear
from the student
notes
by
Dällenbach
(1913/14),
Zabel
(1917),
and Reichenbach
(1917/18)
discussed
in
the editorial
note,
"Einstein's Lecture
Notes,"
pp.
3-10.
[72]This
argument is
based
on
Helmholtz's
attempt
to
establish
an
analogy
between thermo–