{"id":682,"date":"2018-11-12T08:39:47","date_gmt":"2018-11-12T08:39:47","guid":{"rendered":"http:\/\/phyp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=682"},"modified":"2018-11-12T08:52:34","modified_gmt":"2018-11-12T08:52:34","slug":"lagrangian-and-hamiltonian-of-a-relativistic-particle","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/chapter\/lagrangian-and-hamiltonian-of-a-relativistic-particle\/","title":{"rendered":"Lagrangian and Hamiltonian of a Relativistic Particle."},"content":{"raw":"
    \r\n \t
  1. Introduction<\/strong><\/li>\r\n<\/ol>\r\n \r\n

    Defining the Lagrangian and Hamiltonian functions in special theory of relativity as we have done in Newtonian mechanics, is not possible. We cannot define a potential energy function because the potential energy function is defined in a particular frame of reference. So, it would be difficult to establish a Lorentz covariant formulation of the Lagrangian. We can of course do that for free particles or particles moving in electro\u2013magnetic field because of the fast that electrodynamics is naturally covariant.<\/p>\r\n\r\n

      \r\n \t
    1. Lagrangeian and Hamiltonian of a Relativistic Particle<\/strong><\/li>\r\n<\/ol>\r\n \r\n

      A non relativistic particle of mass m in a force field given by the potential U has the\u00a0Lagrangian\u00a0\"\"and its momentum is related to the Lagrangian through<\/p>\r\n \r\n\r\n\"\"\r\n

      We now consider a dynamical system characterized by generated coordinates \u00a0such that the conjugate momenta are still given by<\/p>\r\n \r\n\r\n\"\"\r\n

      where the integral is over the word line between two events.<\/p>\r\n \r\n\r\n\"\"\r\n

      We can thus write the Lagrangian of a free relativistic particle in analogy with<\/p>\r\n\"\"\r\n\r\nCharged Particle in Electromagnetic Field.<\/strong>\r\n\r\n \r\n

      As mentioned earlier electrodynamics is invariant under Lorentz transformation. The interaction of a particle with electromagnetic field depends as a single parameter called charge of the particle. The properties of electro \u2013 magnetic fields are characterized by a four potential \u00a0whose time and spatial components are the scalar \u00a0and vector \u00a0potential respectively.<\/p>\r\n \r\n\r\n\"\"\r\n

      In terms of the scalar and vector potential, the electric and magnetic fields are given by the well known relations:<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n

      We get the familiar expression of the Lorentz force.From the Lagrangian we can find the Hamiltonian function for a particle in the field from the general expression.<\/p>\r\n\"\"\r\n\r\n\"\"\r\n

        \r\n \t
      1. Covariant Formulation<\/strong><\/li>\r\n<\/ol>\r\n \r\n

        In the Lagrangian and Hamiltonian formulation given above, the time coordinate is treated as distinct from the spatial coordinates. The formulation is thus not manifestely covariant. We should ideally have a description in terms of four vectors. Thus in the Minkowski space we should be able to write the Lagrangian as a function of four position vector \u00a0and the four velocity defined as<\/p>\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\nExample 1:<\/strong>\r\n\r\n \r\n

        Consider the motion of a charged particle in an electromagnetic field given by the four potential.We know that the vector potential is arbitrary and the electric and magnetic fields are not changed if we demand \u00a0to satisfy the Lorentz condition:<\/p>\r\n \r\n\r\n\"\"\r\n

        A covariant expression of the Lagrangian can be written as<\/p>\r\n \r\n\r\n\"\"\r\n

        where the sum over repeated indices here and else where is implied.The Lagrangian so constructed is a Lorentz scalar<\/p>\r\n\"\"\r\n\r\n\"\"\r\n

        a Lorentz scalar and is not equal to the total energy.<\/p>\r\n \r\n\r\nSummary<\/strong>\r\n

          \r\n \t
        • Relativistic expression of the Lagrangian of a free particle is given by.<\/li>\r\n \t
        • Relativistic expression of the Lagrangian of a charged particle in the electromagnetic field.<\/span><\/li>\r\n \t
        • Covariant expression of the Lagrangian in the electromagnetic field.<\/span><\/li>\r\n<\/ul>","rendered":"
            \n
          1. Introduction<\/strong><\/li>\n<\/ol>\n

             <\/p>\n

            Defining the Lagrangian and Hamiltonian functions in special theory of relativity as we have done in Newtonian mechanics, is not possible. We cannot define a potential energy function because the potential energy function is defined in a particular frame of reference. So, it would be difficult to establish a Lorentz covariant formulation of the Lagrangian. We can of course do that for free particles or particles moving in electro\u2013magnetic field because of the fast that electrodynamics is naturally covariant.<\/p>\n

              \n
            1. Lagrangeian and Hamiltonian of a Relativistic Particle<\/strong><\/li>\n<\/ol>\n

               <\/p>\n

              A non relativistic particle of mass m in a force field given by the potential U has the\u00a0Lagrangian\u00a0\"\"and its momentum is related to the Lagrangian through<\/p>\n

               <\/p>\n

              \"\"<\/p>\n

              We now consider a dynamical system characterized by generated coordinates \u00a0such that the conjugate momenta are still given by<\/p>\n

               <\/p>\n

              \"\"<\/p>\n

              where the integral is over the word line between two events.<\/p>\n

               <\/p>\n

              \"\"<\/p>\n

              We can thus write the Lagrangian of a free relativistic particle in analogy with<\/p>\n

              \"\"<\/p>\n

              Charged Particle in Electromagnetic Field.<\/strong><\/p>\n

               <\/p>\n

              As mentioned earlier electrodynamics is invariant under Lorentz transformation. The interaction of a particle with electromagnetic field depends as a single parameter called charge of the particle. The properties of electro \u2013 magnetic fields are characterized by a four potential \u00a0whose time and spatial components are the scalar \u00a0and vector \u00a0potential respectively.<\/p>\n

               <\/p>\n

              \"\"<\/p>\n

              In terms of the scalar and vector potential, the electric and magnetic fields are given by the well known relations:<\/p>\n

              \"\"<\/p>\n

              \"\"<\/p>\n

              \"\"<\/p>\n

              \"\"<\/p>\n

              We get the familiar expression of the Lorentz force.From the Lagrangian we can find the Hamiltonian function for a particle in the field from the general expression.<\/p>\n

              \"\"<\/p>\n

              \"\"<\/p>\n

                \n
              1. Covariant Formulation<\/strong><\/li>\n<\/ol>\n

                 <\/p>\n

                In the Lagrangian and Hamiltonian formulation given above, the time coordinate is treated as distinct from the spatial coordinates. The formulation is thus not manifestely covariant. We should ideally have a description in terms of four vectors. Thus in the Minkowski space we should be able to write the Lagrangian as a function of four position vector \u00a0and the four velocity defined as<\/p>\n

                 <\/p>\n

                \"\"<\/p>\n

                \"\"<\/p>\n

                \"\"<\/p>\n

                Example 1:<\/strong><\/p>\n

                 <\/p>\n

                Consider the motion of a charged particle in an electromagnetic field given by the four potential.We know that the vector potential is arbitrary and the electric and magnetic fields are not changed if we demand \u00a0to satisfy the Lorentz condition:<\/p>\n

                 <\/p>\n

                \"\"<\/p>\n

                A covariant expression of the Lagrangian can be written as<\/p>\n

                 <\/p>\n

                \"\"<\/p>\n

                where the sum over repeated indices here and else where is implied.The Lagrangian so constructed is a Lorentz scalar<\/p>\n

                \"\"<\/p>\n

                \"\"<\/p>\n

                a Lorentz scalar and is not equal to the total energy.<\/p>\n

                 <\/p>\n

                Summary<\/strong><\/p>\n

                  \n
                • Relativistic expression of the Lagrangian of a free particle is given by.<\/li>\n
                • Relativistic expression of the Lagrangian of a charged particle in the electromagnetic field.<\/span><\/li>\n
                • Covariant expression of the Lagrangian in the electromagnetic field.<\/span><\/li>\n<\/ul>\n","protected":false},"author":3,"menu_order":27,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["ashok-goyal"],"pb_section_license":""},"chapter-type":[],"contributor":[58],"license":[],"class_list":["post-682","chapter","type-chapter","status-publish","hentry","contributor-ashok-goyal"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/wp-json\/pressbooks\/v2\/chapters\/682","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":5,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/wp-json\/pressbooks\/v2\/chapters\/682\/revisions"}],"predecessor-version":[{"id":706,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/wp-json\/pressbooks\/v2\/chapters\/682\/revisions\/706"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/wp-json\/pressbooks\/v2\/chapters\/682\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/wp-json\/wp\/v2\/media?parent=682"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/wp-json\/pressbooks\/v2\/chapter-type?post=682"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/wp-json\/wp\/v2\/contributor?post=682"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp01\/wp-json\/wp\/v2\/license?post=682"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}