108
DOC.
1
MECHANICS LECTURE NOTES
Helmholtz found that these
general equations are
suitable
for
representing
the
dynamical
properties
of
physical
systems
far
beyond
the domain of
mechanics.[69]
However,
it
can
happen
that
we
know not to conceive
of H
as
n
-
L,
& also do not
want
to
be
so
constrained.
[p. 108]
We
ask
therefore whether the
energy principle is
maintained
if
H
takes
an
arbitrary
form.
To
this
end
we
multiply
the
generalized
Lagrange
equation
by
dp
=
p'dt
and
sum
over
all
coordinates
dOll
_
311
_
Pv+dPv+~doiP'Stt
0
A'(dH
A
-
SJL "•
dtWp'J
w,
.(oH,\
_
__
__
_
P;dp~+d~
0 '
V
' dH
thus
d(H
v
E
From
this
we see
that
the
generalized
Lagrange
equations (Hamilton's principle)
involve
the
energy
principle.
We
also show
that
in
the
special case
of
ordinary
mechanics
one
arrives
back
at
the
customary
expression
for the
energy.
Here H
=
J
-
L
2L
=
AnPi +
^iiPiPi
+
^nP'iPi
E E
a^PIPI
IM
v
+
AlA2
+
where
Auv
=
Avu.
We
obtain