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22
Jacobian Elliptic Functions
Applications
22.19
Physical Applications
22.19
Physical Applications
22.20
Methods of Computation
Figure 22.19.1
(See
in context
.)
Figure 22.19.1:
Jacobi’s amplitude function
am
(
x
,
k
)
for
0
≤
x
≤
10
π
and
k
=
0.5
,
0.9999
,
1.0001
,
2
. When
k
<
1
,
am
(
x
,
k
)
increases monotonically indicating that the motion of the pendulum is unbounded in
θ
, corresponding to free rotation about the fulcrum; compare Figure
22.16.1
. As
k
→
1
−
, plateaus are seen as the motion approaches the separatrix where
θ
=
n
π
,
n
=
±
1
,
±
2
,
…
, at which points the motion is time independent for
k
=
1
. This corresponds to the pendulum being “upside down” at a point of unstable equilibrium. For
k
>
1
, the motion is periodic in
x
, corresponding to bounded oscillatory motion.
ⓘ
Annotations:
Symbols:
am
(
x
,
k
)
: Jacobi’s amplitude function
,
π
: the ratio of the circumference of a circle to its diameter
,
x
: real
,
k
: modulus
and
θ
(
t
)
: angular displacement
Permalink:
http://dlmf.nist.gov/22.19.F1.mag
Encodings:
Magnified png
,
pdf
See also:
Annotations for
§22.19
and
Ch.22