From Special Relativity to Feynman Diagrams.pdf

Expanding the square root at order o v 2 c 2 we find

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Expanding the square root at order O (v 2 / c 2 ) we find: L ( v ) α α 2 v 2 c 2 = m 2 v 2 + const .

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8.1 Dynamical System with a Finite Number of Degrees of Freedom 213 Neglecting the inessential additive constant, we can then identify: α = − mc 2 . The relativistic free particle Lagrangian is thus given by L ( v ) = − mc 2 1 v 2 c 2 , (8.21) and for a system of non-interacting particles, we have L ( v i ) = i m i c 2 1 v 2 i c 2 . (8.22) 8.2 Conservation Laws In this section we show that, if the Lagrangian of a system of particles is invariant under a group of transformations, then the dynamic system enjoys a set of conserva- tion laws . In general we shall refer to the invariance property of a Lagrangian with respect to a group of transformations G as a symmetry under this group. We first show that if a Lagrangian is invariant under translations in time, t t + δ t , then energy is conserved. For simplicity, we assume that the invariance under time translations is due to the fact that the Lagrangian does not explicitly depend on t , namely , L t = 0 . 6 Then we may write: dL dt = L q i ˙ q i + L ˙ q i ¨ q i . We now use the equations of motion ( 8.7 ) and obtain: dL dt = d dt L ˙ q i ˙ q i + L ˙ q i ¨ q i = d dt L ˙ q i ˙ q i , that is d H dt = 0 , (8.23) where we have defined: H = − L + L ˙ q i ˙ q i . (8.24) We conclude that the quantity H ( q , ˙ q ) = − L + L ˙ q i ˙ q i is conserved. 6 The proof in a more general case is given in the following subsection.
214 8 Lagrangian and Hamiltonian Formalism It is easy to recognize that H is the energy of the system. To show this in a general way, let us consider a system of particles interacting with a potential energy U ( q 1 , . . . , q n ) : L = T ( q , ˙ q ) U ( q ). Here T ( ˙ q ) is the kinetic energy which, in Cartesian coordinates, reads: T = 1 2 k , i m ( k ) ˙ x i ( k ) ˙ x i ( k ) . (8.25) Let us now switch to the generalized (or Lagrangian) coordinates q i , writing 7 : x i ( k ) = f i ( k ) ( q 1 , . . . , q n ) ; ˙ x i ( k ) = f i ( k ) q j ˙ q j . (8.26) In terms of the Lagrangian coordinates the kinetic energy takes the form: T = n i , j = 1 a i j ( q ) ˙ q i ˙ q j . (8.27) where we have set a jl ( q ) = N k = 1 3 i = 1 m ( k ) f i ( k ) q j f i ( k ) q l This shows that the kinetic energy is homogeneous of degree two in the Lagrangian velocities ˙ q i . Applying the Euler theorem for homogeneous functions , we find: i T ˙ q i ˙ q i = 2 T . (8.28) Moreover: L ˙ q i = T ˙ q i . It follows: H = i L ˙ q i ˙ q i L = i T ˙ q i ˙ q i L = T + U . 7 With an abuse of notation, we shall use the same Latin indices i , j , k , l , . . . to label the three- dimensional Euclidean coordinates x i and the generalized coordinates q i , though the reader should bear in mind that in the latter case they run over the total number n of degrees of freedom of the system. Moreover the index k , when written within brackets, is also used to label the particle in the system. The meaning of these indices will be clear from the context.

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8.2 Conservation Laws 215 However T + U
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