{"id":160,"date":"2018-11-05T04:30:57","date_gmt":"2018-11-05T04:30:57","guid":{"rendered":"http:\/\/phyp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=160"},"modified":"2019-04-29T05:54:47","modified_gmt":"2019-04-29T05:54:47","slug":"hamiltons-principle-and-lagranges-equation","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/chapter\/hamiltons-principle-and-lagranges-equation\/","title":{"rendered":"Hamilton\u2019s Principle and Lagrange\u2019s Equation"},"content":{"raw":"
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  1. Introduction:<\/strong><\/li>\r\n<\/ol>\r\n \r\n

    Non-relativistic dynamics in an inertial frame are described by Newton\u2019s equation \u00a0. In actual physical situation the dynamical system in general, is constrained by a prior unknown forces. A particle may be constrained to move on a given surface, the motion may be restricted within certain boundaries and so on. The constraint forces may be quite complicated in which case one may have to look for alternative formalism. This alternative formalism of course, cannot go beyond Newton\u2019s Laws but may result in the simplification of the problem and may even have wider application. Historically a minimum principle based on the notion that<\/strong> nature always act in a way that during the development of a dynamical system, certain quantities are minimized, <\/strong>has been used in an alternative formulation of mechanics. We will discuss in this unit a very powerful principle due to Hamilton for the formulation of what is known Hamiltonian dynamics<\/strong>.<\/p>\r\n\r\n

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    1. Hamilton\u2019s Principle.<\/strong><\/li>\r\n<\/ol>\r\n \r\n

      The principle states that \u201cof all the possible paths consistent with the constraints along<\/strong> which a dynamical system can evolve from one point to another, the actual path followed is the one which minimizes the action\u201d. <\/strong>Where action is defined as the time integral of the Langrangian of the system i.e.<\/p>\r\n \r\n\r\n\"\"\r\n\r\nLagrange\u2019s Equation of motion<\/strong>\r\n\r\n \r\n

      Consider a multiparticle system characterized by a Lagrangian function.<\/p>\r\n \r\n\r\n\"\"\r\n

      Where \u00a0and \u00a0are the sets of generalized coordinate and velocities. The action for this Lagrangian is given as<\/p>\r\n\"\"\r\n\r\nHamilton\u2019s Principle states that of the various paths given by\r\n\r\n \r\n

      The path followed by the system is the one for which the corresponding action is minimum.Consider the path \u00a0specified by the set \u00a0and a nearby path \u00a0characterized by the set<\/p>\r\n \r\n\r\nThe change \u00a0in action is then given by\r\n\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n

      The Lagrangian as defined is not unique, we can add to it a term which is a total time derivative of any function \u00a0with no change in the equations of motion. The new action we get is<\/p>\r\n \r\n\r\n\"\"\r\n

      The last term being constant has no effect on\u00a0 \u00a0\u00a0and therefore, on Lagrange\u2019 equations.<\/p>\r\n \r\n

      Lagrange\u2019s Equation with Undetermined Multipliers:<\/strong><\/p>\r\n \r\n

      In the above derivation we had assumed that the constraints are holonomic and can be expressed in terms of algebraic relations. If same of the constraints are expressed in terms of velocities and are in the form of non-integrable equations like<\/p>\r\n\"\"\r\n

      For arbitrary \u2019s, \u2019s and \u2019s , it is possible to incorporate them in the Lagrange\u2019s eqations by means of the Lagrangian undermined multipliers. Take for example, non-holonomic constraints of the force<\/p>\r\n\"\"\r\n

      For constraints expressible as<\/p>\r\n\"\"\r\n

      Where \u2019s are the undetermined multipliers.<\/p>\r\n\r\n

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      1. Solved Examples:<\/strong><\/li>\r\n<\/ol>\r\n \r\n\r\n1.Disc rolling down an inclined plane<\/strong>\r\n\r\n \r\n

        A disc of mass M and radius R is rolling down an inclined planed without slipping write down the Lagrangian of the system and obtain the equation of motion.<\/p>\r\n \r\n\r\n\"\"\r\n

        The kinetic energy of the disc is the sum of translational and rotational energy<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n

        (Note that if instead of disc, we had a cylinder, a sphere, a ring or a spherical shell, the acceleration can be obtained by putting the corresponding expression for the moment of inertia).<\/p>\r\n\r\n

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

          For a double pendulum consisting of the masses \u00a0and \u00a0tied by string of length \u00a0and \u00a0supported at a point in the horizontal plane, obtain the Lagrangian and the equations of motion. Solve them for small amplitude.<\/p>\r\n \r\n

          Ans.: Set up the coordinate system as shown<\/p>\r\n\"\"\r\n\r\nThe coordinate of the pendulum can be expressed in terms of angles \u00a0and \u00a0as\r\n\r\n\"\"\r\n\r\n\"\"\r\n

          The caxpled set of differential equations can be solved by assuming the solution to be<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nThus we get the two frequencies.\r\n

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          1. Particle on a cycloid :<\/strong><\/li>\r\n<\/ol>\r\n \r\n

            A particle is moving on a cycloid , \u00a0under the action of gravity set up the Lagrange\u2019s equation of motion and find the frequency for Solution: An infinitesimal distance element on the surface of the cycloid is<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n

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            1. Summary<\/strong>\u00a0<\/strong><\/li>\r\n<\/ol>\r\n