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Applied Calculus of Variations for Engineers, 2nd Edition
book

Applied Calculus of Variations for Engineers, 2nd Edition

by Louis Komzsik
June 2014
Intermediate to advanced content levelIntermediate to advanced
233 pages
5h 42m
English
CRC Press
Content preview from Applied Calculus of Variations for Engineers, 2nd Edition
148 Applied calculus of variations for engineers
Differentiating and applying Lagrange’s equation we obtain
˙p
i
=
d
dt
p
i
=
d
dt
∂L
˙q
i
=
∂L
∂q
i
.
Hence the total differential of the Lagrangian becomes
dL =
∂L
∂t
dt +
n
i=1
p
i
dq
i
+ p
i
d ˙q
i
).
Exploiting that
d(p
i
˙q
i
)=dp
i
˙q
i
+ p
i
d ˙q
i
,
and reordering yields
d(
n
i=1
p
i
˙q
i
L)=
∂L
∂t
dt
n
i=1
p
i
dq
i
˙q
i
dp
i
).
The left-hand side term is called the Hamiltonian
H =
n
i=1
(p
i
˙q
i
L)=f(p
i
,q
i
,t),
which is now only a function of the new and old generalized displacement
variables and time. Its total differential is
dH =
∂H
∂t
dt +
n
i=1
(
∂H
∂p
i
dp
i
+
∂H
∂q
i
dq
i
).
Matching terms between the dH and dL differentials produces the relationship
∂H
∂t
=
∂L
∂t
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Publisher Resources

ISBN: 9781482253597