9c6c8fc5bd-p-81.pdf - 7.8 APENDICE 2 M LAGRANGIANA Y...

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7.8. AP ´ ENDICE 2: M. LAGRANGIANA Y HAMILTONIANA 149 ser´a un hiperplano { ( u, φ ( u )) : u V } que pasa por el origen. Despla- zando paralelamente este hiperplano podemos construir un hiperplano tangente a la gr´afica de la funci´on f : V R en un (´unico) punto ( u φ , f ( u φ )). La ordenada en el origen de este hiperplano coincide con L [ f ]( φ ). El principal inter´ es geom´ etrico de la transformada de Legendre aparece cuando, por alguna raz´on, resulta conveniente usar como coor- denadas las derivadas parciales de una funci´on 4 f . Si conocemos la fun- ci´on f ( p 1 , . . . , p n ) pero no el difeomorfismo Df entonces, en principio, no podemos recuperar la funci´on original f ( x 1 , . . . , x n ). Sin embargo, si conocemos L [ f ] podemos recuperar la gr´afica de f como la hipersu- perficie envolvente a todos los hiperplanos H φ = { ( u, φ ( u ) + L [ f ]( φ )) : u V } obtenidos variando φ V * . Se dice entonces que la transfor- mada de Legendre de f “conserva la informaci´on” de f (al menos bajo la hip´otesis de que Df sea un difeomorfismo).
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  • Winter '15
  • Derivada, Teorema de Noether, Mecánica hamiltoniana, Hiperplano, Lagrangiano, Forma diferencial

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