Pendules et courbes elliptiques

m
l
h0
1
2mv2+mgh =mgh0
v h
v2= 2g(h0−h).
l
h
O
m
vh′
v
h
(h0)2
v2=l2−h2 l2
v2
(h0)2=2g
l2(h−h0)(h−l)(h+l).
(h(t), h0(t))
y2=2g
l2(x+l)(x−l)(x−h0)
(h(t), h0(t))
h
T
−2−1 0
h
h′
t1
t2
t1+t2
y2=2g
l2(x)(x−2l)(x−x0)).
(h, ˙ h)
$=T/l
2l x0
x0p2lx0
2l√2l+√x
22
$
2l x0
2λ√λ
√lpx0/2

−2−1.5−1−0.5 0 0.5 1
h
−1
0
1
h′
(h(t), h′(t))
x0≈2l
2πpλ/g
λ$ = 2πsλ
g
T=l$ =2πl
√λg
√λ√l
p(l+h0)/2
−1.5
−1.2
−0.9
−0.6
−0.3
0
0.3
0.6
0.9
1.2
1.5
h′
−1 0
h
(h(t), h′(t))
g= 9.809 /s2
h0= 0 √λ
1 1/√2
1 0.70710678
0.85355339 0.84089641
0.84722490 0.84720126
0.84721308 0.84721308
λ= 0.71777
T= 2.368
T= 2πpl/g = 2.006
g−g