{"id":278,"date":"2018-11-05T08:08:47","date_gmt":"2018-11-05T08:08:47","guid":{"rendered":"http:\/\/phyp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=278"},"modified":"2019-04-29T06:02:58","modified_gmt":"2019-04-29T06:02:58","slug":"hamiltonian-formalismand-poison-bracket-11","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/chapter\/hamiltonian-formalismand-poison-bracket-11\/","title":{"rendered":"Central-Field Motion I"},"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

      We saw earlier that the radial velocity of a particle in a central field is given by<\/p>\r\n \r\n\r\n\"\"\r\n

      The motion is therefore confined in the region \u00a0and these points are the turning points of the motion. For certain value of the potential , 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\u00a0round 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

      To find the condition for closed orbit, we need to find the change in angle \u00a0after the particle has made one complete oscillation from \u00a0and \u00a0and back; thus<\/p>\r\n

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

      Where and \u00a0are integers. Thus after \u00a0oscillations \u00a0will change by \u00a0times an integer and would return to its original position i.e. after \u00a0periods the particle would have made \u00a0complete oscillations.<\/p>\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

      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

      \u00a0implies constant force and is not of interest, \u00a0corresponds to the inverse square for law and \u00a0corresponds to inverse-cube force law. The case of \u00a0is the familiar case of planetary motion in gravitational field which we considered in detail in the previous unit. The equation of the orbit was found to be \u00a0and in this obviously the orbit is closed because as, the radial is\u00a0 vector returns to its original position.<\/p>\r\n \r\n

      The case of inverse-cubic force law \u00a0is \u00a0will be treated when we evaluate the advance in the perihelion of mercury. We will find that in this case the orbit does not close on itself. The case of harmonic force \u00a0corresponds to \u00a0and can be solved in terms of circular functions and has a closed orbit.<\/p>\r\n\r\n

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      1. Orbit in a Linear Force Field<\/strong><\/li>\r\n<\/ol>\r\n \r\n

        We have the force \u00a0which can be derived from the potential function . The equation of the orbit expressed in the variable \u00a0is<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nSubstituting back \u00a0and \u00a0.\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\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.<\/p>\r\n \r\n

          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

          Now for , the eccentricity \u00a0, the orbit is an ellipse, but it is not a closed ellipse because as \u00a0the particle does not come back to its original position. In the presence of a small respective inverse-cubic force.<\/p>\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