10
12 • 2.3 Le principe variationnel d’Hamilton modifié
(2.11) (2.12)
(2.13) (2.14)
• 2.4 Transformations canoniques
(2.15)
14
• 2.4 Transformations canoniques
(2.16) (2.17)
• 2.4 Transformations canoniques
(2.19)
(2.20)
16 • 2.4 Transformations canoniques (2.23) (2.24) (2.25a) (2.25b) (2.25c) (2.19)
• 2.4 Transformations canoniques (2.26) (2.27) (2.28a) (2.28b) (2.28c) (2.19)
18 • 2.4 Transformations canoniques (2.29) (2.30a) (2.30b) (2.30c)
• 2.4 Transformations canoniques
(2.31)
(2.32a) (2.32b) (2.32c)
20 • 2.4 Transformations canoniques (2.19) (2.20) (2.21) (2.22)
• 2.5 Exemples de transformations canoniques (2.33) (2.34a) (2.34b) (2.34c) (2.36a) (2.36b) (2.35)
22 • 2.5 Exemples de transformations canoniques
(2.37) (2.38) (2.39a) (2.39b) (2.40a) (2.40b) (2.41)
• 2.5 Exemples de transformations canoniques (2.42a) (2.42b) (2.41) (2.43) (2.44)
24 • 2.6 L’approche symplectique des transformations
canoniques
(2.45)
(2.46) (2.47) (2.48)
• 2.6 L’approche symplectique des transf. canoniques (2.47) (2.49a) (2.49b) (2.49c) (2.49d)
26
• 2.6 L’approche symplectique des transf. canoniques
J
ij= 1 δ
i,j-fΠ
[1≤i≤f,f+1≤ j≤2f]- 1 δ
i-f,jΠ
[f+1≤i≤2f,1≤ j≤f]J
ij= J
ji
=
1 δ
j,i-fΠ
[1≤j≤f,f+1≤ i≤2f]- 1 δ
j-f,iΠ
[f+1≤j≤2f,1≤ i≤f]
= - (1 δ
j-f,iΠ
[f+1≤j≤2f,1≤ i≤f]- 1 δ
j,i-fΠ
[1≤j≤f,f+1≤ i≤2f])
= - (1 δ
i,j-fΠ
[1≤i≤f,f+1≤ j≤2f]- 1 δ
i-f,jΠ
[f+1≤i≤2f,1≤ j≤f])
= - J
ij~
• 2.6 L’approche symplectique des transf. canoniques
J
ij= 1 δ
i,j-fΠ
[1≤i≤f,f+1≤ j≤2f]- 1 δ
i-f,jΠ
[f+1≤i≤2f,1≤ j≤f](J J)
ij= J
ikJ
jk=
(δ
i,k-fΠ
[1≤i≤f,f+1≤k≤2f]- δ
i-f,kΠ
[f+1≤i≤2f,1≤k≤f]) .
(δ
j,k-fΠ
[1≤j≤f,f+1≤k≤2f]- δ
j-f,kΠ
[f+1≤j≤2f,1≤k≤f])
~
=
(δ
i,k-fδ
j,k-fΠ
[1≤i≤f,f+1≤k≤2f]Π
[1≤j≤f,f+1≤k≤2f]+ δ
i-f,kδ
j-f,kΠ
[f+1≤i≤2f,1≤k≤f]Π
[f+1≤j≤2f,1≤k≤f]28
• 2.6 L’approche symplectique des transf. canoniques
c.q.f.d.
J J = 1,
~
J J = 1
~
J = -J = J
~
-1J
2= J J = -J J =-1
~
dtm(J) = +1
(2.49a) (2.49b) (2.49c) (2.49d)• 2.6 L’approche symplectique des transf. canoniques
(2.50)
(2.51) (2.52)
(2.53)
30
• 2.6 L’approche symplectique des transf. canoniques
(2.55)
(2.56)
(2.57a) (2.57b)
• 2.7 Les crochets de Poisson
[ ]
(2.58) (2.59) k k i i v et u Siu
v
η η ∂ ∂ = ∂ ∂ = ~= u
iJ
ikv
k32 • 2.7 Les crochets de Poisson
(2.58)
(2.60)
= u
i[δ
i,k-fΠ
[1≤i≤f,f+1≤k≤2f]- δ
i-f,kΠ
[f+1≤i≤2f,1≤k≤f]]v
kq
p
p
q
i i i i v u v u ∂ ∂ ∂ ∂ − ∂ ∂ ⋅ ∂ ∂ =• 2.7 Les crochets de Poisson
(2.61)
(2.62) (2.63) (2.64)
34 • 2.7 Les crochets de Poisson
(2.66)
(2.67)
(2.68) (2.69)