k \u03b4 ik f g f qi g p i f p i g q i i 1 \u2113 Crochets de Poisson k 1 \u2113 p k H q k qk

K δ ik f g f qi g p i f p i g q i i 1 ℓ crochets

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k { } = δ ik f , g { } = f q i g p i f p i g q i i = 1
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Crochets de Poisson k = 1,..., ! p k = H q k ! q k = + H p k ! p k = p k , H { } ! q k = q k , H { } p k , f { } = p k q k f p k p k p k f q k = f q k q k , f { } = q k q k f p k q k p k f q k = f p k
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Mécanique analytique - Formalisme lagrangien L - Formalisme hamiltonien - Crochets de Poisson - Équation fondamentale de la dynamique H } {
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! f = df dt = f q k ! q k + f p k ! p k k = 1 + f t Évolution temporelle f q 1 ,..., q , p 1 ,... p , t ( )
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Évolution temporelle ! f = df dt = f q k ! q k + f p k ! p k k = 1 + f t f q 1 ,..., q , p 1 ,... p , t ( ) H q k + H p k = =
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Évolution temporelle = f q k H p k f p k H q k k = 1 + f t ! f = df dt = f q k ! q k + f p k ! p k k = 1 + f t f q 1 ,..., q , p 1 ,... p , t ( ) = f , H { } + f t = ! f = f , H { } + f t
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Évolution temporelle ! f = f , H { } + f t f q 1 ,..., q , p 1 ,... p , t ( ) Équation fondamentale de la dynamique
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Évolution temporelle ! f = f , H { } f q 1 ,..., q , p 1 ,... p ( ) Équation fondamentale de la dynamique
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MOOC Introduction à la Physique Quantique Davide Boschetto Transparents de la vidéo en appendice
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Formalisme lagrangien T = 1 2 m i d ! r i dt 2 i = 1 N Énergie cinétique totale : k 1, T q k = m i d ! r i dt d dt ! r i q k i = 1 N T q k = 1 2 m i q k d ! r i dt 2 i = 1 N = m i d ! r i dt q k d ! r i dt i = 1 N = m i d ! r i dt d dt ! r i q k i = 1 N
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Formalisme lagrangien T = 1 2 m i d ! r i dt 2 i = 1 N Énergie cinétique totale : k 1, T q k = m i d ! r i dt d dt ! r i q k i = 1 N T ! q k = m i d " r i dt ! q k d " r i dt i = 1 N
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Formalisme lagrangien ! r i = ! r i q 1 ,..., q ( ) d ! r i dt = ! r i q j dq j dt = j = 1 ! r i q j # q j j = 1
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Formalisme lagrangien ! q k d " r i dt = ! q k " r i q j j = 1 ! q j = " r i q j j = 1 ! q j ! q k = " r i q j j = 1 δ jk = " r i q k T ! q k = m i d " r i dt " r i q k i = 1 N d dt T ! q k = d dt m i d ! r i dt ! r i q k i = 1 N = m i d 2 ! r i dt 2 ! r i q k i = 1 N + m i d ! r i dt d dt ! r i q k i = 1 N
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Formalisme lagrangien ! q k d " r i dt = ! q k " r i q j j = 1 ! q j = " r i q j j = 1 !
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  • mécanique quantique, Lois du mouvement de Newton, Principe de moindre action, Théorie quantique des champs, Mécanique analytique

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