9c6c8fc5bd-p-79.pdf

# 9c6c8fc5bd-p-79.pdf - 7.8 APENDICE 2 M LAGRANGIANA Y...

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7.8. AP ´ ENDICE 2: M. LAGRANGIANA Y HAMILTONIANA 145 conveniente introducir una nueva funci´on, la transformada de Legendre de L , que admite una interpretaci´ on geom´ etrica natural. Aplicaci´on entre V y V * asociada a una funci´on sobre V Sea V ( R ) un espacio vectorial, B = ( v 1 , . . . , v n ) una base suya, B * = ( φ 1 , . . . , φ n ) su correspondiente base dual y ( x 1 , . . . , x n ) (resp. ( p 1 , . . . , p n )) las coordenadas sobre todo V (resp V * ) inducidas por B (resp. B * ). Sea f : V R una aplicaci´on diferenciable (no necesariamente lineal) y consideremos su diferencial como una aplicaci´on entre V y V * . Es decir, Df : V V * u Df u donde Df u : V R coincide con la diferencial df u : T u V T f ( u ) R R salvo por la identificaci´on natural de T u V con V . Por tanto, Df u = n i =1 ∂f ∂x i ( u ) φ i , o, directamente en las coordenadas introducidas, Df ( x 1 , . . . , x n ) = ( p 1 ( x 1 , . . . , x n ) , . . . , p n ( x 1 , . . . , x n )) , siendo p i ( x 1 , . . . , x n ) = ∂f ∂x i ( x 1 , . . . , x n ) , i = 1 , . . . , n. Si calculamos la diferencial de Df en u 0 V , se obtiene que su
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