{"id":185,"date":"2018-11-05T04:53:34","date_gmt":"2018-11-05T04:53:34","guid":{"rendered":"http:\/\/phyp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=185"},"modified":"2019-04-29T05:56:00","modified_gmt":"2019-04-29T05:56:00","slug":"hamiltonian-principle-and-lagrange-equations-ii","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/chapter\/hamiltonian-principle-and-lagrange-equations-ii\/","title":{"rendered":"Hamiltonian Principle and Lagrange equations II"},"content":{"raw":"
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  1. Introduction<\/strong><\/li>\r\n<\/ol>\r\n \r\n

    A particle in a central\u2013force field may execute bounded or open noncircular motion depending on the nature of the force. If the particle in executing one complete revolution does not return to its original position it signals a deviation from the inverse \u2013 square law force however slight. An irregularity in the motion of the planet mercury was observed. It was observed that the perihelion of mercury that the semi-major axis advances at the rate of roughly 574 sec. of arc per century. Calculation of the influence of other planets predicted an advance of approximately 531 sec of arc per century leaving a deficit of about 40 sec of arc per century. Einstein\u2019s General Theory of Relativity in post \u2013 Newtonian approximation was able to account for this difference of 43 sec of arc per century and became one of the greatest triumphs of General Relativity. In this unit we will discuss the conditions for closed orbits, stability of the orbits and an estimate of the advance of the perihelion of mercury in the presence of deviations from the inverse-square law.<\/p>\r\n\r\n

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    1. Open and Closed Orbits:<\/strong><\/li>\r\n<\/ol>\r\n \r\n\r\nWe saw earlier that the radial velocity of a particle in a central field is given by\r\n\r\n \r\n\r\n\"\"\r\n\r\nThe velocity will vanish at the roots of \u00a0and\u00a0\"\"\r\n\r\n\"\"\r\n\r\nThe motion is therefore confined in the region \u00a0\"\"and these points are the turning points\r\n\r\n \r\n

      of the motion. For certain value of the potential (Ur) , there may be only one root in which case the motion is circular. If the motion of the particle is such that the orbit is closed after finite number of round travel between \u00a0and , the central potential satisfies certain conditions. If the orbit after finite number of oscillations does not come back to the initial orbit, the orbit is said to be open as shown below:<\/p>\r\n\"\"\r\n\r\n\"\"\r\n

      Example: <\/strong>Show that if the potential \u00a0varies as some integer power of radial distance a closed non-circular path will result only if \u00a0or +1.<\/p>\r\n \r\n

      In terms of the variable , equation (11.4) can be written as<\/p>\r\n \r\n\r\n\"\"\r\n

      The integral (11.5) can be expressed in terms of circular functions provided the radical in the integral (11.5) can be expressed in the form<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n

      For the stability of the orbit, it is required that \u00a0is imaginary \u00a0will increase exponentially making the \u00a0orbit to be unstable. Therefore the stability of the orbit is ensured provided<\/p>\r\n\"\"\r\n

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      1. Perihelion of Mercury<\/strong><\/li>\r\n<\/ol>\r\n \r\n

        We mentioned in the introduction that the perihelion of Mercury that is semi-major axis was found to advance at the rate of roughly 574 sec. of arc per century. This would happen if the law of force deviated to the inverse square law. We will assume the presence of a small \u00a0force in addition to the inverse square law to understand the advance in the perihelion. This small \u00a0force could for example arise due to the pressure of other gravitating bodies i.e. other planets, comets, etc. apart from the sun which provides the major inverse square force. We take the potential \u00a0to be<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n

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