Corrigé TD semaine 3 Fichier - umtice

(r, φ, z)
r=r(z)z=z(r)
T=1
2 (1 + dr dz
2
) ˙z2+r2(z)˙ φ2!
T=1
2 (1 + dz dr
2
) ˙r2+r2˙ φ2!
˙x2+ ˙y2= ˙r2+r2˙ φ2
x=rcos φ
y=rsin φ
~r =r~er+z ~ez
˙x= ˙rcos φ−r˙ φsin φ
˙y= ˙rcos φ+r˙ φcos φ
d
dt~r = ˙r~er+r˙
φ ~eφ+ ˙z ~ez
T=1
2~v2
T=1
2( ˙r2+r2˙
φ2+ ˙z2)
dr
dt =dr dz
dz
dt

T=1
2 (1 + dr dz
2
) ˙z2+r2(z)˙ φ2!
φ
L=T
d
dt
∂L
∂˙ φ−∂L
∂φ = 0
r2¨ φ= 0
pφ=∂L
∂˙ φ
pφ=r2˙ φ
pz=∂L
∂˙z
pz= (1 + (dr
dz )2) ˙z
H=pφ˙ φ+pz˙z−L
H=1
2 (1 + dr dz
2
) ˙z2+r2(z)˙ φ2!
H=1
2 p2
z
1+(dr
dz )2+p2
φ
r2!
φ
z
˙pφ=−∂H
∂φ = 0
pφ=r2˙
φ

~
L=~r ∧~p
~
L=m~r ∧~v
~
L=~r ∧d
dt~r
~
L=r~er∧( ˙r~er+r˙ φ ~eφ+ ˙z ~ez)
~
L=r2˙ φ~ez−r˙z ~eφ
pφ
~ez
α
rsin α
~v
(~er, ~ez)~eφ
~v. ~eφ= sin α|~v|
~v =d
dt~r = ˙r~er+r˙
φ ~eφ+ ˙z ~ez
r˙
φ= sin α|~v|
φ
Oxy α =π/2r
z
|~v|
r˙ φ=Ksin α
K
pφ=r2˙ φ
Kr sin α=pφ
rsin α

r r
rsin α
α
r
r
m
r O Oz
ω Ox φ =ωt
θ
(rsin θcos ωt, r sin θsin ωt, r cos θ)
L=m
2(ω2r2sin2θ+r2˙
theta2)−mgr cos θ
x=rcos φsin θ
y=rsin φsin θ
z=rcos θ
~r =r ~er
~v =d
dt~r = ˙r~er+rsin θ˙
φ ~eφ+r˙ θ ~eθ
L=T−V=1
2m~v2−mgz
V=mgr cos θ−m
2ω2r2sin2θ

V=Acos θ−Bsin2θ
V, q A/B θ =π
A= 0
B= 0 A
B A
B= 0 θ=π
B
˙q, q
B
α
ω
m
l
T=1
2m˙
l2+1
2mω2r2
r=lsin α