Varied path

FlC. 8.1. Schematic representations of actual and varied paths linking initial and final states. In (a) the time end-points are fixed and the variations in the path vanish at Ihese points. In (b) the time end-points are varied, as is the path at these points. 

To first-order in the infinitesimals this reduces to the change in along the varied path between the unvaried time end-points and the unvaried integrand times the variations in the time at the two time end-points. 

Fig. 6.1 The minimum path joining two points, tmIx), and a varied path, j( ).

The functional form, j(x), is unknown and must be determined. It is required to find the extremum value of J subject to a variation in i which the end points, (xi,> i) and (X2,y2), remain fixed. The varied path tnust thus pass through (xi,7i) and x2,yi) (Fig. AI.I). This problem is more difficult than the determination of the stationary points of a function. 

Fig. AI.I The extremum path, full curve, and a varied path, broken curve.

It is now possible to introduce a varied path, which has the properties... 

A varied path is shown in Fig. (ATI) by the broken line. For any value of a the path y(x,a) passes through the end points, from definition (AI.4), and gives the extremum path when a = 0. 

Let the two points have Cartesian coordinates (jci,ti) and (x2,T2). The element of distance, ds, along a varied path, such as that shown in Fig. AII.l, is given by Pythagoras s theorem as... 

Euler-Lagrange equation, consequently the constants m and c can be expressed in terms of (xi. i) and (x2,y2). So the extremum path is a straight line passing through the two end points. It requires further analysis to prove that it is the path of minimum length. However by examination of the varied paths close to the extremum, or on physical grounds, it can be shown to be the path of minimum length. 

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