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The Method of Averaging
CDS140B Lecturer: Wang Sang Koon
Winter, 2005
1
Introduction
Remarks:
The method leads generally to asymtotic series as opposed to convergent series. It is
not restriced to periodic solutions.
Averaging Method.
Put the equation
¨
x
+
x
=
±f
(
x,
˙
x
)
into Lagrange stardard form and do the averaging.
Example 11.1
¨
x
+
x
=
±
(

˙
x
+
x
2
)
.
1
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The Lagrange standard form
Unperturbed Equation is Linear.
˙
x
=
A
(
t
)
x
+
±g
(
t, x
)
,
x
(0) =
x
0
.
Remarks:
(1) The procedure is called “introducing comoving coordinates”. (2) If the unper
turbed equation is nonlinear, the variation of paramters techniques still applies. In practice, how
ever, there are usually many technical obstructions while carrying out the procedure
3
Avaraging in the Periodic Case
Asymptotic Validity of Averaging Method.
Consider equation (11.17)
˙
x
=
±f
(
t, x
) +
±
2
g
(
t, x, ±
)
,
x
(0) =
x
0
.
We assume that
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