Re: Langrange strange
From: David McAnally (D.McAnally_at_i'm_a_gnu.uq.net.au)
Date: 01/31/05
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Date: Mon, 31 Jan 2005 03:00:57 +0000 (UTC)
D.McAnally@i'm_a_gnu.uq.net.au (David McAnally) writes:
<snip>
>An important feature of the Lagrangian is the corresponding variational
>principle. The action is defined as the integral of the Lagrangian with
>respect to time, and so is a function of the time evolution of the
>coordinate variables q_k over the appropriate time interval. For specific
>values of the coordinate variables q_k at initial time t_i and specific
>values of the same coordinate variables at final time t_f, then the
>natural evolution of the coordinate variables from t_i to t_f is
>determined as that evolution at which the action takes a stationary value
>(originally, it was specified that the trajectories make the action the
>minimum possible, thus the Principle of Least Action). This variational
>principle is of a similar nature to Fermat's Principle of Least Time
>(that, in geometric optics, light takes the path between two points that
>takes the least time to traverse).
Fermat's Principle of Least Time is the geometric optics limit to the laws
of physical optics (i.e. as the wavelength approaches zero and the
frequency approaches infinity, or alternatively, the scale of the
apparatus is very large compared to the wavelength).
In the same way, the Principle of Least Action (or, more appropriately,
the Principle of Stationary Action) is the limit of Feynman's Path
Integral approach to Quantum Mechanics. The path integral approach uses
the integral of A exp(iS/\hbar), for a common modulus A, and where S is
the action along the path, over all possible paths (so the modulus of the
contributions of all paths are equal, but the argument of the contribution
depends on the action along the path). In the macroscopic case, since
\hbar is small, the action changes so quickly as the path changes,
compared to \hbar, that the phases of the contributions are distributed
throughout the entire unit circle, and so the contributions of
neighbouring paths cancel each other. The exception is when the path is
a stationary path for the action (i.e. a path at which the action has a
stationary point), as in that case, the action changes negligibly in a
region about the path in question, and so the phase also changes
negligibly in the same region, and the contributions reinforce. This
means that those paths which are stationary points of the action will be
singled out as the paths to be taken by the dynamical system in the
classical limit.
David
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