Formulaire - Amocops - Université de Rouen

•⃗ XR R0
d⃗ X
dt R0
=d⃗
X
dt R
+⃗ Ω(R/R0)∧⃗ X
• R0RM
⃗
V(M/R0) =⃗ V(M/R) +⃗ V(M, R/R0)
⃗
V(M, R/R0) = ⃗ V(O/R0) + ⃗ Ω(R/R0)∧−−→ OM
OR
⃗
Γ(M/R0) =⃗ Γ(M/R)
+⃗ Γ(O/R0) + d⃗
Ω(R/R0)
dt R0
∧−−→ OM +⃗ Ω(R/R0)∧⃗ Ω(R/R0)∧−−→ OM
+ 2⃗ Ω(R/R0)∧⃗ V(M/R)
•
ψθv
zv
z
1
0
u
φu
vx
y
z
ψx
yu
v
0
0
1
0
z
θ φ
⃗x0
⃗y0
⃗z0
ψ
−−−−−−−−−−→
⃗z0
⃗u
⃗v1
⃗z0
⃗u
⃗v1
⃗z0
θ
−−−−−−−−−→
⃗u 
⃗u
⃗v
⃗z
⃗u
⃗v
⃗z
ϕ
−−−−−−−−−→
⃗z 
⃗x
⃗y
⃗z
⃗
Ω(R/R0) = ˙ ψ⃗z0+˙ θ⃗u +˙ ϕ⃗z =˙ θ⃗u +˙ ψsin θ⃗v + ( ˙ ψcos θ+˙ ϕ)⃗z
= ( ˙ θcos ψ+˙ ϕsin θsin ψ)⃗x0+ ( ˙ θsin ψ−˙ ϕsin θcos ψ)⃗y0+ ( ˙ ϕcos θ+˙ ψ)⃗z0
= ( ˙ θcos ϕ+˙ ψsin θsin ϕ)⃗x −(˙ θsin ϕ−˙ ψsin θcos ϕ)⃗y + ( ˙ ψcos θ+˙ ϕ)⃗z

•
T(A, S) = ⃗
R
−→
MAA
⃗
R−→ MAA
−→
MB=−→ MA+⃗ R∧−−→ AB (−→ MB−−→ MA)·−−→ AB = 0
•S
⃗
Ω(S/R0)
⃗
V(A, S/R0)A
⃗
Ω(S/R0)SR0⃗ V(A/R0)
A S R0
•S1S2I I
S1S2
⃗
V(I, S1/S2) = ⃗ V(I∈S1/R)−⃗ V(I∈S2/R)
S1S2
•A
⃗
R
0A
⃗
R·⃗ V(I, S1/S2) = 0
•S1S2
P(S1↔S2) = 0 P(S1→S2) + P(S2→S1) = 0
O⃗z
O O⃗z O⃗z O⃗z
⃗
R:
−→
MO=⃗ 0
⃗
R:
−→
MO·⃗z = 0
⃗
R·⃗z = 0
−→
MO·⃗z = 0
⃗
R·⃗z = 0
−→
MO:
G S m
•
I(A, S) = 
(y2+z2)dm −xydm −xzdm
−xydm (x2+z2)dm −yzdm
−xzdm −yzdm (x2+y2)dm 
I(O, S) = I(G, S) + I(O, G(m))

•
m⃗
V(G/R0)
⃗σ(A, S/R0)A
⃗σ(A, S/R0)A S
R0
A
⃗σ(A, S/R0) = ⃗σ(G, S/R0) + m−→ AG ∧⃗ V(G/R0)
P
⃗σ(A, P/R0) = m−→ AP ∧⃗ V(P/R0)
A G
⃗σ(A, S/R0) = I(A, S)·⃗ Ω(S/R0)⃗σ(G, S/R0) = I(G, S)·⃗ Ω(S/R0)
•m⃗
Γ(G/R0)
⃗
δ(A, S/R0)A
⃗
δ(A, S/R0) = d⃗σ(A, S/R0)
dt R0
+m⃗ V(A/R0)∧⃗ V(G/R0)
A
⃗
δ(A, S/R0) = d⃗σ(A, S/R0)
dt R0
G
⃗
δ(G, S/R0) = d⃗σ(G, S/R0)
dt R0
•
=
•
m⃗ Γ(G/R0) = ⃗ Fext
•
⃗
δ(A, S/R0) = −→ M⃗
Fext (A)
•Σ1Σ2
{Σ1→Σ2}=− {Σ2→Σ1}

•
2T(S/R0) = ⃗ Ω(S/R0)·I(G, S)·⃗ Ω(S/R0) + mV 2(G/R0)
2T(S/R0) = mV 2(G/R0)
O S
2T(S/R0) = ⃗ Ω(S/R0)·I(O, S)·⃗ Ω(S/R0)
•A S
P=⃗
R
−→
MAA
·⃗
Ω(S/R0)
⃗
V(A, S/R0)A
=⃗ R·⃗ V(A/R0) + −→ MA·⃗ Ω(S/R0)
•Σ
dT (Σ/R0)
dt =Pe+Pi
U
Pe+Pi=−dU
dt
T+U=
Qiqii= 1,2,· · · , n T
n
d
dt ∂T ∂˙qi−∂T ∂qi
=Qi
P∗= [F][V∗] = ΣQi˙q∗
i
d
dt(T2−T0) =
n
i=1
Qi˙qi−∂T
∂t
V(q1, q2,· · · , qn;t)
dV
dt =
n
i=1
Qi˙qi−∂T
∂t
T2−T0=V+Cte
n
i=1 Qi˙qi
Tk˙qii T2=1
2∑n
i=1 ∑n j=1 aij ˙qi˙qj, T1=1
2∑n
i=1 bi˙qi
T0=C