4 Calculus of Variations

Ashok Goyal

1. Introduction:

Newton’s equation of motion can be restated in terms of Lagrange’s Equations by using d’Alembert’s Principle as seen in the previous module. The Euler-Lagrange’s equation can be derived in an elegant manner by using a Variational Principle called Hamilton’s Principle. The development of Calculation of variation was started by Newton in 1866 and was extended by the Bernoulli brothers, by Euler, Legendre, Langrange, Hamilton and Jacobi to name a few, during the eighteen and early nineteen century.

2. Function and Functional

A function  of  is defined by a rule that maps the input set of numbers  to an output set of numbers. It may or may not be expressed in terms of an analytic relationship.

A functional  where  is some function of  and  and  is an independent variable defines a rule that maps a function on a set of functions on to an output set of numbers, for example

Is a functional and  is a simple function that accepts three arguments. For example,we can have

The functional  is only one simple specific example of a functional. Every functional need not be in an integral form. For a normal function of n variables

3. Calculus of Variations:

The problem in the calculus of variations is to determine the function  such that the integral .

Is an extremum i.e. either a minimum or maximum.  In the above equation  is the same function of x  and  is an independent variable.  The  is a functional and is given and the limits are fixed.  The function  is then varied until an extremum value of J is found. This means that if a function  gives integral J a minimum value, then any

neighbouring function however close to  must result in an increase in J.

Let us represent all possible functions  be a parameteric representation  such that when  is the function that extremizes J.  We can then write.

Where  is some function of  which has continuous first derivative and which vanishes at the ends points and  Since  at the end points of the path.

The integral J now becomes a function of  . For the integral J to have stationary (extremum) value

For all functions .  Now

This is Euler’s equation and is a necessary condition for J to be extremum. The equation was derived by Euler in the year 1744.

Applications:

a) Shortest distance between two points on a plane.

The infinitesimal distance ds between two neighbouring points in the x-y plane is

This distance s between the two points is

We have to extremize S  given the functional

Which is the equation of a straight line. Thus we have the well known result, that the shortest distance between two points on a plane lie along the straight line joining the points.

b) Geodesic on a sphere

A geodesic is a line which represents the shortest distance between two points on a surface. The element of length on a sphere of radius r in spherical polar coordinates is given by

This is the equation of a plane that passes through the centre (0,0,0) of the sphere. This plane cuts the surface of the sphere on a circle called the ‘great circle’. Thus the shortest distance (geodesic) between two points of a sphere lie on a great circle passing through these points.

c) The Brachistochrone Problem

Oone of the classical problems in the calculus of variations is the Brachistrochrone Problem. The problem is to find the path on which a particle moves in the presence of a constant force as to make the time taken by the particle to move from an initial point to a final point minimum. The problem was first solved by Johann Bernolulli in the year 1696. We choose a coordinate system so that the initial point lies at the origin and the force field is directed along the +ve x-axis.

We assume there is no force of friction and the constant force acting on the particle is the gravitational force mg. The total energy of the system during transit is conserved i.e. T+U=cosnt. At the initial point, V is taken to be zero and the particle starts at rest. At any point on the curve.

And since

Thus the curve taken by the particle lies as a ‘cycloid’.

5. Summary:

• The extremization of a functional is achieved if the function  satisfies the Euler’s equation

• The geodesic on the surface of a sphere between two points lie on a great circle.