variationalCalculus

# VariationalCalculus - Physics 129a Calculus of Variations 071113 Frank Porter Revision 081120 1 Introduction Many problems in physics have to do

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Physics 129a Calculus of Variations 071113 Frank Porter Revision 081120 1 Introduction Many problems in physics have to do with extrema. When the problem involves fnding a Function that satisfes some extremum criterion, we may attack it with various methods under the rubric oF “calculus oF variations”. The basic approach is analogous with that oF fnding the extremum oF a Function in ordinary calculus. 2 The Brachistochrone Problem Historically and pedagogically, the prototype problem introducing the cal- culus oF variations is the “brachistochrone”, From the Greek For “shortest time”. We suppose that a particle oF mass m moves along some curve under the inﬂuence oF gravity. We’ll assume motion in two dimensions here, and that the particle moves, starting at rest, From fxed point a to fxed point b . We could imagine that the particle is a bead that moves along a rigid wire without Friction [±ig. 1(a)]. The question is: what is the shape oF the wire Forwh ichthet imetogetFrom a to b is minimized? ±irst, it seems that such a path must exist – the two outer paths in ±ig. 2(b) presumably bracket the correct path, or at least can be made to bracket the path. ±or example, the upper path can be adjusted to take an arbitrarily long time by making the frst part more and more horizontal. The lower path can also be adjusted to take an arbitrarily long time by making the dip deeper and deeper. The straight-line path From a to b must take a shorter time than both oF these alternatives, though it may not be the shortest. It is also readily observed that the optimal path must be single-valued in x , see ±ig. 1(c). A path that wiggles back and Forth in x can be shortened in time simply by dropping a vertical path through the wiggles. Thus, we can describe path C as a Function y ( x ). 1

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C a b a b (a) (b) y x (c) a b . . . . . . Figure 1: The Brachistochrone Problem: (a) Illustration of the problem; (b) Schematic to argue that a shortest-time path must exist; (c) Schematic to argue that we needn’t worry about paths folding back on themselves. We’ll choose a coordinate system with the origin at point a and the y axis directed downward (Fig. 1). We choose the zero of potential energy so that it is given by: V ( y )= mgy. The kinetic energy is T ( y V ( y 1 2 mv 2 , for zero total energy. Thus, the speed of the particle is v ( y q 2 gy. An element of distance traversed is: ds = q ( dx ) 2 +( dy ) 2 = v u u t 1+ dy dx ! 2 dx. Thus, the element of time to traverse ds is: dt = ds v = r dy dx ± 2 2 gy dx, and the total time of descent is: T = Z x b 0 r dy dx ± 2 2 dx. 2
Diferent Functions y ( x ) will typically yield diferent values For T ;weca l l T a “Functional” oF y . Our problem is to ±nd the minimum oF this Functional with respect to possible Functions y .N o t et h a t y must be continuous – it would require an in±nite speed to generate a discontinuity. Also, the accel- eration must exist and hence the second derivative d 2 y/dx 2 . We’ll proceed to Formulate this problem as an example oF a more general class oF problems in “variational calculus”.

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## This note was uploaded on 03/21/2012 for the course PHYSICS 129 taught by Professor Johnh.schwarz during the Winter '12 term at Caltech.

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VariationalCalculus - Physics 129a Calculus of Variations 071113 Frank Porter Revision 081120 1 Introduction Many problems in physics have to do

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