9c6c8fc5bd-p-80.pdf

# 9c6c8fc5bd-p-80.pdf - 7.8 APENDICE 2 M LAGRANGIANA Y...

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7.8. AP ´ ENDICE 2: M. LAGRANGIANA Y HAMILTONIANA 147 N´otese que fijado cualquier entorno coordenado ( U, q 1 , . . . , q n ) de p 0 la matriz de la diferencial de Leg ( p 0 ,t 0 ) en cada v p 0 T p 0 Q resulta ser 2 L ˙ q i ˙ q j ( v p 0 , t 0 ) i,j , por lo que L es regular en ( p 0 , t 0 ) si y s´olo si esta matriz (para unas coordenadas alrededor de p 0 y, por tanto, para cualesquiera) es regular en todo vector v p 0 tangente a p 0 . Diremos que L es regular si es regular en todo ( p 0 , t 0 ) Q × R . En adelante, supondremos por simplicidad la condici´on algo m´as fuerte de que L sea hiper-regular , esto es, que Leg ( p 0 ,t 0 ) sea un difeomorfismo (no s´olo local, sino global) ( p 0 , t 0 ) Q × R . Por tanto, en este caso se tiene el difeomorfismo: Leg : TQ × R TQ * × R ( v p , t ) v p = Leg ( p,t ) ( v p ) , t ) . (7.6) Ejercicio. Sea ( Q, g ) una variedad semi-riemanniana y consideremos la lagrangiana L : TQ × R R ( v p , t ) 1 2 g p ( v p , v p ) - V ( p, t ) (7.7) para cierta funci´on V : Q × R R ( potencial ). Demu´ estrese que, independientemente del valor de t

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