Lecture 6 8 251 Spring 2007 Lecture 6 Topics The relativistic point particle Action reparametrizations and equations of motion Reading Zwiebach Chapter 5 Continued from last time P t P x 0 t x P t 0 y t P x T0 y x Similar to J 0 t J 0 Q dx Free BC Neumann BC P x t x 0 a Py 0 dx y t 0 a t a dx P x x P x t x a P x t x 0 dx P t Py t 0 dxP t 0a 0 Conservation of momentum Free Relativistic Particle Non relativistic Action S 1 2 dt mv 2 Calculation dv dt 0 Relativistic Particles 1 Lecture 6 8 251 Spring 2007 Everyone should agree on action It s a Lorentz invar ds2 dx dx ds cdt 1 v 2 c2 cd s mc2 P So s mc2 tf ti ds mc c ds P dt 1 v 2 c2 Check Lagrangian v2 c2 1 v2 mc2 1 2 c2 1 mc2 mv 2 2 L mc 2 1 Taylor Expansion rest energy kinetic energy Momentum L P v 1 2 v 2 mc2 2 c 1 v2 c2 mv 2 2 1 vc2 Hamiltonian mc2 H p v L 2 1 vc2 Parameterization Have parameterization x the x s are functions of ds2 dx dx 2 Lecture 6 8 251 Spring 2007 ds dx d tf s mc ti dx d d dx dx d d d dx dx d d d d tf s mc dx dx d d dx dx d d d ti tf mc ti d d d So using a di erent parameter instead of gets same action s s is reparameterization invariant 2 Quick calculation to nd equation of motion from s mc 1 vc2 dt Should get derivative of rel momentum with respect to time 0 S mc dS S mc dS dS 2 dx dx dx dx d 2 d d dx dx d 2 2 dS dS 2 d d dS 2 3 Lecture 6 8 251 Spring 2007 dS d dx x d d ds Must vary with dx d and dx d but since is symmetric su cient to vary just dx d and multiply by 2 dS d dx x d d dS f d x dx S mc d d ds i f d dP x P x d d d i x i x f 0 f dS i dP x d d Equation of Motion dP 0 d This means that P constant on world line Constant as a function of any pa rameter dP dP d dt dt d 0 Therefore d d dx 0 ds d2 ds2 dx 0 0 0 if s Okay because is arbitrary But can t assign s d2 x d 2 0 d dx 0 ds d 4 Lecture 6 8 251 Spring 2007 Coupling to Electromagnetism Lorentz Force Equation dP q dx F dS c ds dP q dx F d c d q dx S mc dS A x d c P d P A Nevitz Schwartz Tensor 5
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