Chapter 1: Mechanics
7
the equations of Lagrange can be derived:
d
dt
∂
L
∂
˙
q
i
=
∂
L
∂
q
i
When there are additional conditions applying to the variational problem
δ
J
(
u
)=0
of the type
K
(
u
)=
constant, the new problem becomes:
δ
J
(
u
)

λδ
K
(
u
.
1.7.2
Hamilton mechanics
The
Lagrangian
is given by:
L
=
∑
T
(˙
q
i
)

V
(
q
i
)
. The
Hamiltonian
is given by:
H
=
∑
˙
q
i
p
i

L
. In 2
dimensions holds:
L
=
T

U
=
1
2
m
r
2
+
r
2
˙
φ
2
)

U
(
r,
φ
)
.
If the used coordinates are
canonical
the Hamilton equations are the equations of motion for the system:
dq
i
dt
=
∂
H
∂
p
i
;
dp
i
dt
=

∂
H
∂
q
i
Coordinates are canonical if the following holds:
{
q
i
,q
j
}
=0
,
{
p
i
,p
j
}
,
{
q
i
j
}
=
δ
ij
where
{
,
}
is the
Poisson bracket
:
{
A,B
}
=
±
i
²
∂
A
∂
q
i
∂
B
∂
p
i

∂
A
∂
p
i
∂
B
∂
q
i
³
The Hamiltonian of a Harmonic oscillator is given by
H
(
x,p
p
2
/
2
m
+
1
2
m
ω
2
x
2
. With new coordinates
(
θ
,I
)
, obtained by the canonical transformation
x
=
´
2
I/m
ω
cos(
θ
)
and
p
=

√
2
Im
ω
sin(
θ
)
, with inverse
θ
=arctan(

p/m
ω
x
)
and
I
=
p
2
/
2
m
ω
+
1
2
m
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This note was uploaded on 11/16/2010 for the course ENGR 201 taught by Professor Elder during the Spring '10 term at Blinn College.
 Spring '10
 elder

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