Mathematical Physics, Analysis and Geometry 15 3 , , Journal of Dynamics and Differential Equations 30 4 , , Automatic Control and Computer Sciences 51 7 , , Automatic Control and Computer Sciences 47 7 , , Journal of Difference Equations and Applications, , Differential Equations and Dynamical Systems, , Also, the more recent "sum over histories" formulation of quantum mechanics by Richard Feynman is arguably based on the same variational principles.
Pierre-Louis Moreau de Maupertuis. Maupertuis' Principle is thus also called the Least Action Principle.
The time variable t has thus disappeared from the integral to be minimized, which is now a purely static function of the spatial path from A to B. Note from the darker shading that the other saddlepoint is at higher elevation.
This, then, is the decaying wavefunction solution of the Airy equation that we are looking for, and it is clear that it goes smoothly from this exponential to the cosine form. Incidentally, Landau gives a quick way to see how these formulas connect.
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The exponential factor at first increases in modulus, then becomes pure imaginary, exactly equal to one of the two terms in the cosine form for positive x. If one takes the cosine formula and continues it into the upper half complex plane, one term grows exponentially, the other goes to zero. This will give.
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Taking the Potential Near the Turning Points to be Linear… However, for a reasonably smooth potential it may be an adequate approximation to treat a turning point region as one where the potential is increasing linearly with distance over a sufficient range that beyond this point the WKB approximation can be used in both directions.