Introduction au chaos classique et chaos quantique

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T2

x(t)∈Rdt∈R
d= 1,2,3t∈R→x(t)∈Rd
m > 0
F(x, t)∈Rd
md2x
dt2=F(x, t)
x(0) dx
dt (0)
F
F(x, t)

F=−∂V
∂x := −∂V ∂x1
,..., ∂V
∂xd
V(x, t)∈R
ξ:= mdx
dt ∈Rd
H(x, ξ, t) := 1
2m|ξ|2+V(x, t)∈R
1
2m|ξ|2=1 2mdx dt 2
dx (t)
dt =∂H ∂ξ ,dξ (t) dt =−∂H ∂x
V:= ∂H
∂ξ ,−∂H ∂x (x, ξ)∈
Rd×Rd
∂H
∂ξ =
(2.4)
1
mξ=
(2.3)
dx
dt
−∂H
∂x =
(2.4) −∂V
∂x =
(2.2) F= (2.1) md2x
dt2=
(2.3)
dξ
dt
d= 1
(x, ξ)∈R22H(x, ξ)t
rot (F)=0

m= 6.1024kg x∈R3
x= 0
F(x) = −Cu
|x|2
u=x
|x|C=G · m·mSmS= 2.1030kg
G= 6,67.10−11N.m2.kg−2
V(x) = −C1
|x|.
d2x
dt2=1 mF(x) = −G·mSu
|x|2
m= 9,31.10−31kg
F(x) = −C0u
|x|2, V (x) = −C01
|x|
C0=kCq.q q = 1.6 10−19C
kC= 9.109Nm2C−2
V(x)