{"id":126,"date":"2018-11-02T07:01:13","date_gmt":"2018-11-02T07:01:13","guid":{"rendered":"http:\/\/phyp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=126"},"modified":"2022-01-07T05:32:52","modified_gmt":"2022-01-07T05:32:52","slug":"calculus-of-variations","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/chapter\/calculus-of-variations\/","title":{"rendered":"Calculus of Variations"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
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1. Introduction:<\/strong><\/p>\r\n \r\n

Newton\u2019s equation of motion can be restated in terms of Lagrange\u2019s Equations by using d\u2019Alembert\u2019s Principle <\/strong>as seen in the previous module. The Euler-Lagrange\u2019s equation can be derived in an elegant manner by using a Variational Principle<\/strong> called Hamilton\u2019s Principle<\/strong>. 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.<\/p>\r\n \r\n

2. Function and Functional<\/strong><\/p>\r\n \r\n

A function \u00a0of \u00a0is defined by a rule that maps the input set of numbers \u00a0to an output set of numbers. It may or may not be expressed in terms of an analytic relationship.<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

A functional \"\"\u00a0where \u00a0is some function of \"\"\u00a0and \u00a0and \u00a0is an independent variable defines a rule that maps a function on a set of functions on to an output set of numbers, for example<\/p>\r\n\"\"\r\n

Is a functional and \u00a0\"\"is a simple function that accepts three arguments. For example,we can have<\/p>\r\n \r\n\r\n \r\n\r\n \r\n

The functional \u00a0is only one simple specific example of a functional. Every functional need not be in an integral form. For a normal function of n variables<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n

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  1. Calculus of Variations<\/strong>:<\/li>\r\n<\/ol>\r\n \r\n

    The problem in the calculus of variations is to determine the function \u00a0such that the integral .<\/p>\r\n \r\n\r\n\"\"\r\n\r\nIs an extremum i.e. either a minimum or maximum.\u00a0 In the above equation \u00a0is the same function of x <\/em>\u00a0and \u00a0is an independent variable.\u00a0 The \u00a0is a functional and is given and the limits are fixed.\u00a0 The function \u00a0is then varied until an extremum value of J\u00a0is found. This means that if a function \u00a0gives integral J a minimum value, then any\r\n

    neighbouring function however close to \u00a0must result in an increase in J.<\/p>\r\n \r\n

    Let us represent all possible functions \u00a0be a parameteric representation \u00a0such that when \u00a0is the function that extremizes J.\u00a0 We can then write.<\/p>\r\n \r\n\r\n\"\"\r\n\r\nWhere \u00a0is some function of \u00a0which has continuous first derivative and which vanishes at the ends points and\u00a0\u00a0Since \u00a0at the end points of the path.\r\n\r\n \r\n\r\n\"\"\r\n

    The integral J now becomes a function of\u00a0 . For the integral J to have stationary (extremum) value<\/p>\r\n\"\"\r\n\r\nFor all functions .\u00a0 Now\r\n\r\n\"\"\r\n\r\n\"\"\r\n

    This is Euler\u2019s equation and is a necessary condition for J to be extremum. The equation was derived by Euler in the year 1744.<\/p>\r\n \r\n\r\nApplications:<\/strong>\r\n\r\n \r\n\r\na) Shortest distance between two points on a plane.<\/strong>\r\n\r\n \r\n

    The infinitesimal distance ds between two neighbouring points in the x-y plane is<\/p>\r\n\"\"\r\n\r\nThis distance s between the two points is\r\n\r\n\"\"\r\n\r\nWe have to extremize S\u00a0 given the functional\r\n\r\n\"\"\r\n

    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.<\/p>\r\n \r\n\r\nb) Geodesic on a sphere<\/strong>\r\n\r\n \r\n

    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<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n

    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 \u2018great circle\u2019. Thus the shortest distance (geodesic) between two points of a sphere lie on a great circle passing through these points.<\/p>\r\n \r\n\r\nc) The Brachistochrone Problem<\/strong>\r\n\r\n \r\n

    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.<\/p>\r\n \r\n

    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.<\/p>\r\n \r\n\r\n\"\"\r\n\r\nAnd since\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nThus the curve taken by the particle lies as a \u2018cycloid\u2019.\r\n

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    1. Summary:<\/strong><\/li>\r\n<\/ol>\r\n