Lecture 11: Calculus of Variations (28 Sep 09)
A. Calculus of variations
1. FW Sec 17; G Sec. 2.2
2. Goal is to get the Lagrange equations from a variational principle.
This will lead to a reformulation of classical mechanics that emphasizes
symmetry principles, to generalizations of quantum mechanics, and to
approximation methods in quantum mechanics.
3. Quick “review” of principles of the calculus of variations.
4. Begin: assume we are given a function
φ
=
φ
[
y
(
x
)
, dy/dx, x
], i.e., this is
a functional relation that forms
φ
from a function
y
(
x
), its derivatives,
and perhaps an explicit dependence on the independent variable
x
.
5. Consider the definite integral between specified end points:
I
=
Z
x
2
x
1
φ
[
y
(
x
)
, dy/dx, x
]
dx
and ask for the condition that this is stationary (ideally, an extremum)
under small variations of the function
y
(
x
) around an optimal function
y
0
(
x
), with set values
y
(
x
1
) and
y
(
x
2
).
6. Formally start with
Y
(
x
) =
y
(
x
) +
η
(
x
); expand
I
to first order in
:
I
=
Z
x
2
x
1
φ
[
y
(
x
) +
η
(
x
)
, y
0
(
x
) +
η
0
, x
]
dx
’
Z
x
2
x
1
φ
[
y
(
x
)
, y
0
(
x
)
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 Fall '09
 BRUCH
 mechanics, Derivative, Principle of least action, Noether's theorem, A. Calculus

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