11ème cours de mécanique analytique (24/11/2011) · 2012-01-24 · 26 • 2.6 l’approche...

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11ème cours de Mécanique Analytique (24/11/2011)

Comète Mc Naught 2006

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3

4

5

6

7

8

9

10

•  2.3 Le principe variationnel d’Hamilton modifié

(2.11)

(2.12)

(2.13)

(2.14)

11

•  2.4 Transformations canoniques

(2.15)

(2.16)

12

•  2.4 Transformations canoniques

(2.16)

(2.17)

(2.18)

13

•  2.4 Transformations canoniques

(2.19)

(2.20)

(2.21)

(2.22)

14

•  2.4 Transformations canoniques

(2.23)

(2.24)

(2.25a)

(2.25b)

(2.25c)

(2.19) (2.19)

15

•  2.4 Transformations canoniques

(2.26)

(2.27)

(2.28a)

(2.28b)

(2.28c)

(2.19) (2.19)

16

•  2.4 Transformations canoniques

(2.29)

(2.30a)

(2.30b)

(2.30c)

17

•  2.4 Transformations canoniques

(2.31)

(2.32a)

(2.32b)

(2.32c)

18

•  2.4 Transformations canoniques

(2.19)

(2.20)

(2.21)

(2.22)

19

•  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)

24

•  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 ~

25

•  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 ~

26

•  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)

27

•  2.6 L’approche symplectique des transf. canoniques

(2.50)

(2.51) (2.52)

(2.53)

(2.54)

(2.55)

28

•  2.6 L’approche symplectique des transf. canoniques

(2.55)

(2.56)

(2.57a)

(2.57b)

29

•  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∂

∂

∂

∂−

∂

∂⋅

∂

∂=

31

•  2.7 Les crochets de Poisson

(2.61)

(2.62)

(2.63)

(2.64) (2.65)