S.
Widnall
16.07
Dynamics
Fall
2009
Version
3.0
Lecture
L20
 Energy
Methods:
Lagrange’s
Equations
The
motion
of
particles
and
rigid
bodies
is
governed
by
Newton’s
law.
In
this
section,
we
will
derive
an
alternate
approach,
placing
Newton’s
law
into
a
form
particularly
convenient
for
multiple
degree
of
freedom
systems
or
systems
in
complex
coordinate
systems.
This
approach
results
in
a
set
of
equations
called
Lagrange’s
equations.
They
are
the
beginning
of
a
complex,
more
mathematical
approach
to
mechanics
called
analytical
dynamics.
In
this
course
we
will
only
deal
with
this
method
at
an
elementary
level.
Even
at
this
simplified
level,
it
is
clear
that
considerable
simplification
occurs
in
deriving
the
equations
of
motion
for
complex
systems.
These
two
approaches–Newton’s
Law
and
Lagrange’s
Equations–are
totally
compatible.
No
new
physical
laws
result
for
one
approach
vs.
the
other.
Many
have
argued
that
Lagrange’s
Equations,
based
upon
conservation
of
energy,
are
a
more
fundamental
statement
of
the
laws
governing
the
motion
of
particles
and
rigid
bodies.
We
shall
not
enter
into
this
debate.
Derivation
of
Lagrange’s
Equations
in
Cartesian
Coordinates
We
begin
by
considering
the
conservation
equations
for
a
large
number
(N)
of
particles
in
a
conservative
force
field
using
cartesian
coordinates
of
position
x
i
.
For
this
system,
we
write
the
total
kinetic
energy
as
M
1
2
T
=
m
i
x
˙
(1)
2
i
.
n
=1
where
M
is
the
number
of
degrees
of
freedom
of
the
system.
For
particles
traveling
only
in
one
direction,
only
one
x
i
is
required
to
define
the
position
of
each
particle,
so
that
the
number
of
degrees
of
freedom
M
=
N
.
For
particles
traveling
in
three
dimensions,
each
particle
requires
3
x
i
coordinates,
so
that
M
= 3
∗
N
.
The
momentum
of
a
given
particle
in
a
given
direction
can
be
obtained
by
di
ff
erentiating
this
expression
with
respect
to
the
appropriate
x
i
coordinate.
This
gives
the
momentum
p
i
for
this
particular
particle
in
this
coordinate
direction.
∂
T
=
p
i
(2)
∂
x
˙
i
The
time
derivative
of
the
momentum
is
d
∂
T
=
m
i
x
¨
i
(3)
dt
∂
x
˙
i
1
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For
a
conservative
force
field,
the
force
on
a
particle
is
given
by
the
derivative
of
the
potential
at
the
particle
position
in
the
desired
direction.
∂
V
F
i
=
−
∂
x
i
(4)
From
Newton’s
law
we
have
F
i
=
dp
i
(5)
dt
Equating
like
terms
from
our
manipulations
on
kinetic
energy
and
the
potential
of
a
conservative
force
field,
we
write
d
∂
T
∂
V
dt
∂
x
˙
i
=
−
∂
x
i
(6)
Now
we
make
use
of
the
fact
that
∂
T
=
0
(7)
∂
x
i
and
∂
V
=
0
(8)
∂
x
˙
i
Using
these
results,
we
can
rewrite
Equation
(6)
as
dt
d
∂
(
T
∂
x
−
˙
i
V
)
−
∂
(
T
∂
x
−
i
V
)
= 0
(9)
We
now
define
L
=
T
−
V
:
L
is
called
the
Lagrangian.
Equation
(9)
takes
the
final
form:
Lagrange’s
equations
in
cartesian
coordinates.
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 Polar coordinate system, Lagrangian mechanics, Coordinate systems, Generalized coordinates

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