By Askold M Perelomov, Yakov B Zeldovich

Filling the space among general textbooks and magazine articles this offers a close therapy of the scattering idea, multidimensional quasi-classical approximation, non-stationary difficulties for oscillators and the speculation of risky debris.

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6) where the latter equation has been obtained by an integration by parts; note that we have used the notation · to separate the derivative from the differential. 4). 9) the new action A serves to derive the same equations of motion since the added term is really just f (x(T )) − f (x(0)), an expression that vanishes under the variation since δx(T ) = δx(0) = 0. We can derive the same equation of motion even if we put stronger conditions on the variational paths. In particular, besides holding δx(0) = δx(T ) = 0, we could also impose δ x(0) ˙ = δ x(T ˙ ) = 0 as well.

11) page 14 January 21, 2015 11:50 BC: 9452 - Enhanced Quantization ws-book9x6 15 Selected Topics in Classical Mechanics which is fixed by design. Of course, it would be a rare problem that has a valid solution in which the values of x(T ˙ ), x(T ), x(0), ˙ and x(0) had all been specified beforehand; indeed, for most sets of such data there is no solution to the equations of motion at all. That fact serves to demonstrate the clear difference between (i) deriving the equations of motion and (ii) solving the equations of motion!

As a consequence, it follows that the quartic interaction is a discontinuous perturbation of the free ultralocal model and is, instead, a continuous perturbation of the pseudofree model for which the action functional is that of the free ultralocal model restricted to the domain D(Ag ). Stated otherwise, and despite appearances, the quartic interacting ultralocal model is not continuously connected to the free ultralocal model! Just as was the case for the discussion of a single degree of freedom, it follows that a perturbation analysis of solutions of the equations of motion with a quartic interaction should be made around solutions of the pseudofree model and not around solutions of the free model.

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