From Special Relativity to Feynman Diagrams.pdf

N 841 however the functional dependence of l q t q t

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, . . . , n . (8.41) However, the functional dependence of L ( q ( t ), ˙ q ( t ), t ) on its arguments q ( t ), ˙ q ( t ), t is in general different from that of L ( q ( t ), ˙ q ( t ), t ) on q ( t ), ˙ q ( t ), t . Similarly the actions S and S are different functionals of ( q i ) and ( q i ) , respectively. It follows that the equations of motion derived from them will in general have a different form. A transformation of the kind ( 8.36 ) is a symmetry of the system, namely the system is invariant under ( 8.36 ), if the equations of motion, as a system of differential equations, have the same form in the new and the old variables q i ( t ) and q i ( t ). In light of our discussion in Sect.8.1.1 we can easily convince ourselves that the Euler–Lagrange equations in the two RFs have the same form provided the functional dependence of L and L on their respective arguments is the same, modulo an additional total derivative which does not affect the equations of motion. Using the general relation ( 8.40 ), this amounts to saying that L ( q ( t ), ˙ q ( t ), t ) = L ( q ( t ), ˙ q ( t ), t ) + d f dt . (8.42) At the level of the action the above property can be stated as follows: S [ q i ; t 1 , t 2 ] = t 2 t 1 dt L ( q , ˙ q , t ) = t 2 t 1 dtL ( q , ˙ q , t ) = S [ q i ; t 1 , t 2 ] , (8.43) where we have ignored the total derivative since it yields equivalent actions.
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8.2 Conservation Laws 219 Summarizing we have seen that a transformation of the kind ( 8.36 ) is a symmetry of the system if it leaves the action invariant , that is if the actions S and S exhibit the same functional dependence on the paths described by q i and q i : S [ q i ; t 1 , t 2 ] = S [ q i ; t 1 , t 2 ] , (8.44) or, equivalently, if δ S t 2 t 1 dt L ( q , ˙ q , t ) t 2 t 1 dtL ( q , ˙ q , t ) = 0 . (8.45) After these preliminaries we may state the Noether theorem as follows: If the action of a dynamic system is invariant under a continuous group of (non singular) transformations of the generalized coordinates and time, of the form q i = q i ( q , t ), t = t ( t ) , and if the equations of motion are satisfied, then the quantity: Q i L ˙ q i δ q i + L δ t , (8.46) is conserved. We stress that the variations δ q i = q i ( t ) q i ( t ) corresponding to infinitesi- mal local transformations of the form ( 8.36 ), are not arbitrary as those used in the discussion of the Hamilton action principle, but correspond to the subclass of transformations leaving the action invariant. Let us now start from the invariance property ( 8.45 ) to derive the conserved quantities. We set δ S = t 2 t 1 dt L ( q ( t ), ˙ q ( t ), t ) t 2 t 1 dtL ( q ( t ), ˙ q ( t ), t ) = t 2 t 1 dtL ( q ( t ), ˙ q ( t ), t ) t 2 t 1 dtL ( q ( t ), ˙ q ( t ), t ), (8.47) where, on the right hand side of the above equation, we have used the fact that t is an integration variable. We now decompose the integration over ( t 2 , t 1 ) as follows: Setting t 1 = t 1 + δ t 1 , t 2 = t 2 + δ t 2 , we can write t 2 t 1 = t 2 t 1 + t 2 + δ t 2 t 2 t 1 + δ t 1 t 1 , (8.48)
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220 8 Lagrangian and Hamiltonian Formalism so that the first integral of ( 8.47 ) can be written as follows: t 2 t 1 dtL ( q ( t ), ˙ q ( t ), t ) t 2 t 1 dtL ( q ( t ), ˙ q ( t ), t ) + δ t 2 L ( q ( t 2 ), ˙ q ( t 2 ), t 2 ) δ t 1 L ( q ( t 1 ), ˙ q ( t 1 ), t 1 ) (8.49) In deriving ( 8.49 ) we have replaced, in the last two terms, L ( q ( t ), ˙ q ( t ), t ) L ( q ( t ) + δ q , ˙ q ( t ) + δ ˙ q ( t ), t ) , (8.50) with
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