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1
11ème cours de Mécanique Analytique (24/11/2011)
Comète Mc Naught 2006
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• 2.3 Le principe variationnel d’Hamilton modifié
(2.11)
(2.12)
(2.13)
(2.14)
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• 2.4 Transformations canoniques
(2.15)
(2.16)
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• 2.4 Transformations canoniques
(2.16)
(2.17)
(2.18)
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• 2.4 Transformations canoniques
(2.19)
(2.20)
(2.21)
(2.22)
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• 2.4 Transformations canoniques
(2.23)
(2.24)
(2.25a)
(2.25b)
(2.25c)
(2.19) (2.19)
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• 2.4 Transformations canoniques
(2.26)
(2.27)
(2.28a)
(2.28b)
(2.28c)
(2.19) (2.19)
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• 2.4 Transformations canoniques
(2.29)
(2.30a)
(2.30b)
(2.30c)
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• 2.4 Transformations canoniques
(2.31)
(2.32a)
(2.32b)
(2.32c)
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• 2.4 Transformations canoniques
(2.19)
(2.20)
(2.21)
(2.22)
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• 2.5 Exemples de transformations canoniques
(2.33)
(2.34a)
(2.34b)
(2.34c)
(2.36a)
(2.36b)
(2.36c)
(2.35)
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• 2.5 Exemples de transformations canoniques
(2.37)
(2.38)
(2.39a)
(2.39b)
(2.40a)
(2.40b)
(2.41)
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• 2.5 Exemples de transformations canoniques
(2.42a)
(2.42b)
(2.41)
(2.43)
(2.44)
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• 2.6 L’approche symplectique des transformations canoniques
(2.45)
(2.46)
(2.47)
(2.48)
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• 2.6 L’approche symplectique des transf. canoniques
(2.47)
(2.49a)
(2.49b)
(2.49c)
(2.49d)
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• 2.6 L’approche symplectique des transf. canoniques
Jij = 1 δi,j-f Π[1≤i≤f,f+1≤ j≤2f] - 1 δi-f,j Π[f+1≤i≤2f,1≤ j≤f]
Jij = Jji
= 1 δj,i-f Π[1≤j≤f,f+1≤ i≤2f] - 1 δj-f,i Π[f+1≤j≤2f,1≤ i≤f]
= - (1 δj-f,i Π[f+1≤j≤2f,1≤ i≤f] - 1 δj,i-f Π[1≤j≤f,f+1≤ i≤2f]) = - (1 δi,j-f Π[1≤i≤f,f+1≤ j≤2f] - 1 δi-f,j Π[f+1≤i≤2f,1≤ j≤f]) = - Jij
~
J = -J ~
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• 2.6 L’approche symplectique des transf. canoniques
Jij = 1 δi,j-f Π[1≤i≤f,f+1≤ j≤2f] - 1 δi-f,j Π[f+1≤i≤2f,1≤ j≤f]
(J J)ij = Jik Jjk = (δi,k-f Π[1≤i≤f,f+1≤k≤2f] - δi-f,k Π[f+1≤i≤2f,1≤k≤f]) . (δj,k-f Π[1≤j≤f,f+1≤k≤2f] - δj-f,k Π[f+1≤j≤2f,1≤k≤f])
~
= (δi,k-f δj,k-f Π[1≤i≤f,f+1≤k≤2f] Π[1≤j≤f,f+1≤k≤2f] + δi-f,k δj-f,k Π[f+1≤i≤2f,1≤k≤f] Π[f+1≤j≤2f,1≤k≤f]
= δij = 1ij J J = 1 ~
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• 2.6 L’approche symplectique des transf. canoniques
c.q.f.d.
J J = 1, ~ J J = 1 ~ J = -J = J-1 ~
J2 = J J = -J J =-1 ~ dtm(J) = +1
(2.49a)
(2.49b)
(2.49c)
(2.49d)
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• 2.6 L’approche symplectique des transf. canoniques
(2.50)
(2.51) (2.52)
(2.53)
(2.54)
(2.55)
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• 2.6 L’approche symplectique des transf. canoniques
(2.55)
(2.56)
(2.57a)
(2.57b)
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• 2.7 Les crochets de Poisson
[ ]
(2.58)
(2.59) k
ki
i
vetuSi vu ηη ∂
∂=
∂
∂=
~
= ui Jik vk
= ui [δi,k-f Π[1≤i≤f,f+1≤k≤2f] - δi-f,k Π[f+1≤i≤2f,1≤k≤f]]vk
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• 2.7 Les crochets de Poisson
(2.58)
(2.60)
= ui [δi,k-f Π[1≤i≤f,f+1≤k≤2f] - δi-f,k Π[f+1≤i≤2f,1≤k≤f]]vk
qppq iiii
vuvu∂
∂
∂
∂−
∂
∂⋅
∂
∂=
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• 2.7 Les crochets de Poisson
(2.61)
(2.62)
(2.63)
(2.64) (2.65)