{"id":84,"date":"2018-11-02T06:26:47","date_gmt":"2018-11-02T06:26:47","guid":{"rendered":"http:\/\/phyp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=84"},"modified":"2019-04-29T05:51:28","modified_gmt":"2019-04-29T05:51:28","slug":"symmetry-transformations-and-conservation-laws","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/chapter\/symmetry-transformations-and-conservation-laws\/","title":{"rendered":"Symmetry Transformations and Conservation Laws"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n \r\n\r\n \r\n\r\n1. Introduction:\r\n\r\n \r\n

Principle of symmetry transformations plays a key role in the formulation of Quantum Field theories in many branches of physics like particle physics, condensed matter physics, general theory of relativity, string theories etc. Conservation laws in physics have a deep relationship with symmetry operations. The requirement of invariance of the Lagrangian under symmetry transformations so that the physical results remain unchanged leads to conservation laws. In what fallows we will consider some space time transformations leading to corresponding conservation laws. Suppose we have a Lagrangian of the system\u00a0 \u00a0where\u00a0 \u2019s stand for the set generalized coordinates. If we now make a transformation of the coordinate from\u00a0 , the Lagrangian in new coordinates is expressed as\u00a0 . The different solutions\u00a0 \u00a0obtained from\u00a0 \u00a0and\u00a0 \u00a0must have equivalent trajectories (path). The coordinates transformations can be discrete or continuous. An example of a discrete transformation is \u2018mirror reflection\u2019 under which<\/p>\r\n\"\"\r\n

A transformation is continuous if the transformed coordinates have a continuous dependence on a suitable parameter defining the transformation. An example is rotational transformation about the z-axis under which<\/p>\r\n \r\n\r\n\"\"\r\n

    \r\n \t
  1. Space time Translational symmetry and conservation of Energy and Linear Momentum.<\/strong><\/li>\r\n<\/ol>\r\n \r\n

    The space is assumed to be homogeneous and isotropic. A physical system at one place in space and time developes the same way as at any other point. The laws of physics are invariant under the space-time translation. Let us consider a Lagrangian of a dynamical system of n particles. The Lagrangian is a function of position coordinates and velocities. Assuming the constraints to be holonomic, the Lagrangian can be written as<\/p>\r\n \r\n\r\n\"\"\r\n

    Where the potential does not depend on the velocities.<\/p>\r\n \r\n

    1) Symmetry under space translation:<\/strong><\/p>\r\n \r\n

    Consider an infinitesimal translation of the coordinate<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n

    If the Lagrangian remains invariant so that the physical results remain unchanged, we have . Since \u00a0is an arbitrary displacement<\/p>\r\n\"\"\r\n

    and from Lagrange\u2019s equation\u00a0 of motion<\/p>\r\n \r\n\r\n\"\"\r\n\r\nNow\r\n\r\n\"\"\r\n\r\nFrom (3.8)\r\n\r\n\"\"\r\n\r\nTotal linear momentum of the system is conserved.\u00a0\u00a0 IF the motion of the system is described by a set of\r\n\r\ngeneralized coordinates \u00a0\u00a0we define the quantity\r\n\r\n\"\"\r\n\r\nas \u2018generalized momenta\u2019 or \u2018conjugate momenta. In general the conjugate momenta may not be identical to the mechanical momenta and may have dimensions different from MLT-1 . The dimension of mechanical momenta. In terms of generalized force \u00a0\"\"\u00a0the\u00a0\u00a0 Lagrangian equation\r\n\r\n \r\n\r\n\"\"\r\n\r\nIf in the expression of the Lagrangian, a particular coordinate\u00a0\"\" \u00a0or \u2018ignorable\u2019 and obviously for a cyclic coordinate\u00a0does not appear explicitly, it is called \u2018cyclic\u2019\r\n\r\n\"\"\r\n

    Hence the momentum conjugate to acyclic coordinate is conserved. ii) Symmetry under time translation. <\/strong>Under time translation<\/p>\r\n \r\n\r\n\"\"\r\n\r\nThe Lagrangian is transformed to\u00a0\"\"\r\n\r\n \r\n\r\nwhere\r\n\r\n \r\n\r\n\"\"\r\n

    If the Lagrangian remains invariant under time translation<\/p>\r\n \r\n\r\n\"\"\r\n

    Where we have used the Lagrange equation in the first term on the right to express\u00a0\u00a0\"\"<\/p>\r\n\"\"\r\n\r\n\"\"\r\n

    which implies that the total energy is conserved.<\/p>\r\n\r\n

      \r\n \t
    1. Rotational symmetry<\/strong><\/li>\r\n<\/ol>\r\n \r\n

      Let a vector \u00a0\u00a0rotates around the z-axis and rotates to \u00a0after rotating by an angle \u00a0\u00a0as shown.\u00a0 The tip of the vector \u00a0moves along a circle of radius .<\/p>\r\n \r\n\r\n\"\"\r\n\r\nLangrangi equation is given by\r\n\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n

      Total angular momentum is conserved.<\/p>\r\n

      Invariance of the Lagrangian under rotation is justified because of the isotropy of space.<\/p>\r\n\r\n

        \r\n \t
      1. Neother\u2019s Theorem<\/strong><\/li>\r\n<\/ol>\r\n \r\n

        Suppose we have a Lagrangian which depends on the generalized coordinates, generalized velocities and time\"\"\u00a0and a continuous symmetry operation denoted by \"\"\u00a0which depends on the parameter set \u00a0and \"\".For example, the symmetry operation of rotation about the Z-axis is parameterized by a continuous angle of rotation .\u00a0 The transformed Lagrangian \u00a0is a function of\u00a0\u00a0\"\"<\/p>\r\n \r\n\r\nIf\u00a0\"\" is a symmetry operation of the system, we must have\"\"\"\"\r\n\r\nSince eqn. (3.30) holds for all s.\r\n\r\n \r\n\r\nThis is Noether\u2019s Theorem<\/strong>.\r\n\r\n \r\n

        Noether\u2019s Theorem states that if a Lagrangian is invariant under a continuous symmetry operation parameterized by M<\/em> parameters , then there are M<\/em> conserved quantities associated with the symmetry transformation.<\/p>\r\n \r\n

        This theorem formed the corner stone in the development of Quantum Field Theory.<\/p>\r\n \r\n

        Example: Consider a system with the Lagrangian<\/p>\r\n \r\n\r\n\"\"\r\n

        \u00a0are the generalized coordinates and the potential depends only on T. The Lagrangian does not involve the coordinates \u00a0and , therefore they are cyclic coordinates. We then have the momenta conjugate to \u00a0and . Conserved.<\/p>\r\n \r\n\r\n\"\"\r\n

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

          Homogeneity and Isotropy of space leads to the conservation of energy linear momentum and angular momentum shown in the table.<\/p>\r\n \r\n\r\n\r\n\r\n\r\n\r\n\r\n
          Symmetry<\/span><\/td>\r\nLagrangian<\/span><\/td>\r\nConserved quantity<\/span><\/td>\r\n<\/tr>\r\n
          1. Space translation Homogeneity\r\n<\/span><\/td>\r\nInvariant under space transition\r\n<\/span><\/td>\r\nLinear momentum<\/span><\/td>\r\n<\/tr>\r\n
          2.Time translation homogeneity of time\r\n<\/span><\/td>\r\nInvariant under translation in time and no explicit dependence on time\r\n<\/span><\/td>\r\nTotal Energy<\/span><\/td>\r\n<\/tr>\r\n
          3. Rotation Isotropy of space\r\n<\/span><\/td>\r\nInvariant under rotation<\/span><\/td>\r\nAngular momentum<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n

          There are seven constraints (integrals of motion) for a closed system; total energy, three components of linear momentum and three components of angular momentum.<\/p>\r\n\r\n