Action integral classical, variation

Noether s theorem will be proved here for a classical relativistic theory defined by a generic field , which may have spinor or tensor indices. The Lagrangian density (, 9/x) is assumed to be Lorentz invariant and to depend only on scalar forms defined by spinor or tensor fields. It is assumed that coordinate displacements are described by Jacobi s theorem S(d4x) = d4x 9/xgeneral variation of the action integral, evaluated over a closed space-time region 2, is... 

The variation of the Lagrangian ojjerator is formally identical to the variation of the classical action integral as developed in eqns (8.45)-(8.48). We may take over the final result in eqn (8.48) completely in its corresponding operator form, retaining in this case the terms involving Sq at the time endpoints as these variations are no longer required to vanish. The addition of this result to the end-point variations in eqn (8.82) yields, for the general... 

But no fine structure - yet - until in 1915 Bohr considered the effect of relativistic variation of mass with velocity in elliptical orbits under the inverse square law of binding, and pointed out that the consequential precessional motion of the ellipses would introduce new periodicities into the motion of the electron, whose consequences would be satellite lines in the spectra. The details of the dynamics were worked out independently by SOMMERFELD [38] and WILSON [39] in 1915/16 based on a generalisation of Bohr s quantization, namely, the quantization of action the values of the phase integrals Jf = fpj.d, - of classical mechanics should be constrained to assume only integral multiples of h. 

The boundary conditions of (9) are the same as of the usual classical action in (2) (fixed end points and total time). However, separation of variables is now self-evident. The second term on the right hand side of the equation Et) is independent of the coordinates (the energy is a constant). Similarly the first term is independent of time. In fact, it is not necessary to write dQ = Qdr. Any parameterization of the trajectory with respect to a scalar variable s which is monotonic between the points Q (0) and Q t) will do dQ = dQ/ds) ds). It means that variation of the integral part of the right hand side of (9) with respect to the coordinates can be done with no reference to time. We therefore consider a variation with the abbreviated action [5]... 

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