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By B. E. Johnson

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Dq1 J q q f) . t f) ·) A dq2 , and it is easy to see, by considering the 2n-form dO£ = dOL A · · · A dOL , n that the 2-form dOL is non-degenerate if and only if the matrix HL is nondegenerate. D Remark. Using t he 1-form OL , the Noether integral be written ( 1 . 5 ) . 6 ) that if is an infinitesimal symmetry, then X ( 1 . 17) Definition. Let (U, 'P ) be a coordinate chart on M. Coordinates 1 n = (p 1 , · · · , P n , q , . . n x U on the cotangent bundle T* M are called standard coordinates 14 if for (p, q) E T* U and f E C 00 ( U) 8f Pi ( df) = -t.

1 7) we get = so that the G-acti weandobtai n on is Hamiltonian. Using ( 1. 1 7) and another Cartan's formula,

Give examples when for some t > 0 the corresponding projection of 9t (2) onto M is no longer a diffeomorphism. H on the phase space T* M there is an associated 1-form () - H dt = pdq - H dt on the extended phase space T* M x IR, called the Poincare- Cartan form. (to) ) = qo and n (l( t1 ) ) = q1 , where n : T* M M is the canonical projection. By definition, the lift of a path 1 to the extended phase space T* M x lR is a path u : [to , t1] ---+ T* M x lR given by u ( t) = (l ( t) , t) , and a path u in T* M x lR is called an admissible path if it is a lift of a path 1 in T* M.

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