PH101

Lecture 10

Variational Calculus History of Variational Calculus (Wikipedia/Rana&Joag)

Pierre de Fermat Isaac Newton Jacob Bernoulli (1607 –1665) (1642 – 1727) Johann (Jean or John) (1655 – 1705) Bernoulli ( 1667 –1748) Algebra Variational calculus

Leonhard Euler Joseph-Louis Daniel Bernoulli ( 1700 –1782) (1707-1783) Lagrange Bernoulli's principle on fluids Fermat’s principle of least (extremum) time ~1662

speed of light in vacuum n = speed of light in medium

www.physicsforums.com Jean Bernoulli’s challenge! “Brachistochrone”

 What should be the shape of a stone’s trajectory (or, of a roller coaster track) so that released from point 1 it reaches point 2 in the shortest possible time? Brachistochrone problem! ~1696 1 X Brachisto ~ shortest Chrone ~ time dx ds dy = = = v= -Y 2

Time (from 1 to 2) I = = =

Cycloid

Wiki Extremums!

Can we prove mathematically  Shortest distance between two points is a straight line?

 Shortest path between two points on the surface of a sphere is along the great-circle ?

 To answer these questions, one need to know necessary condition

that the integral where = = = is stationary (ie, an extremum ! – either a maximum or a minimum !) .

 Interestingly we are already familiar with the solution ! Extremum of a function

To locate the minimum, x0 →
= f(x+
x)- f(x) → 0 (for small
x) f(x+
x)

f(x)

x
x 0 We say the function is stationary at, x0 (Meaning, for small steps,
x, at x0 the value of the function does not change.
=
x = 0 Possible integration paths

 Out of the infinite number of possible paths, , which path makes the integral, ()

-stationary ? = Y B

A X

We need to find the condition for an integral to be stationary, where the variable is a “function” itself [ ] -the integration path! = () Smart choice of varied paths  Step 1: Let’s assume as the path for which integral, () = is stationary . =  Step2: can represent all possible paths between and for() different = (). () Can you choose suitable mathematical form of such that () should represent all varied paths but must not (i) have() variations = at () A( and (fixed points). ) () goes to zero in the limiting case when the varied paths are very (ii) ()close to . () Let’s check this choice () = ()  where is any arbitrary function of such that () . [condition (i) satisfied] = =  is a parameter which can vary from 0 to higher value continuously. If we take limit , then condition (ii) satisfied . → and are indeed smart choice ()

Typical choice of arbitrary function Y = .9() , B () . = = = ()  By varying , different strength of () = .5() can be added to to generate range A of possible paths() between A and B. () X  gives us the true path → .

For another choice of to generate Y = .() another series of possible paths between A B and B by varying . = () Thus arbitrary and can produce = .3() A all possible paths. () X

Stationary condition of integral

Step 3 : Variation of the integral value for different paths nearby to the stationary path ], is negligibly small. () [ → → The meaning of the statement is

along stationary path ntegration = y(x) = () and along the nearby paths [ = = → ] () = ] = must be equal . i.e, , () = →

This can be achieved by, () = → For stationary path () = →

Where, () = = → = = =

Integration by parts = − =0 As = − − = = = This equation is true for any possible choice of , thus () =0, Euler-Lagrange equation! − This is necessary condition for to be stationary! = Application of Variational principle: Example1

 Given two points in a plane, what is the shortest path between them? We certainly know the answer: Straight line. Let’s prove it using variation method B( ) A( )

 Consider an arbitrary path elementary length  = = =  Total path length =  Necessary condition for this integral to be stationary (maximum)

=0; Here − = Application of variational principle: Example1

= = = A const. = → Thus = ( − ) = = ( ) = = Equation of straight line.

 Shortest distance between two points in a plane is straight line . Questions?