Lecture 5 8 251 Spring 2007 Lecture 5 Topics Nonrelativistic strings Lagrangian mechanics Reading Zwiebach Chapter 4 Non Relativistic Strings Study nonrelativistic strings rst to develop intuition and math notation before moving to the relativistic strings that we actually care about Non relativistic string Characterized by Tension T0 T0 Force Energy Length Mass Length 0 T0 0 v 2 Natural velocity v T0 0 M L v 2 Transverse Oscillation Mark point P on string and see it moving up and down y P t x P t x P x not dependent on t Small Oscillation y t x 1 x Consider small section of string 1 Lecture 5 8 251 Spring 2007 Approximate tensions on endpoints as equal good for transverse waves terrible for longitudinal y y t x dx T0 t x x x 2y T0 2 t x dx x 2y 0 dx 2 t dF T0 2y 1 2y 0 2 x T0 0 t2 The Wave Equation t x are parameters Motion described by y t x If had motion in more than 1 dimension y t x Stretching of string dx2 dy 2 dx dx 1 dy dx 2 1 1 dx dy dx 2 small 2 l General form of wave equation 2f 1 2f 0 x2 v 2 t2 v velocity of wave v T0 0 General Solution y x t h x v0 t h x v0 t Note the h s are function of 1 variable x v0 t not 2 variables x and t inde pendently Boundary Conditions Behavior of endpoints at all times special points at all times Open string y t x 0 0 Dirichlet condition for xed end point y t x 0 0 x Free BD Neumann condition 2 Lecture 5 8 251 Spring 2007 For free endpoint hoop on string means string must be perp here Initial Conditions All points on string at some t0 all points at special time y t 0 y x t 0 x Example Fixed Endpoints y t 0 h v0 t h v0 t 0 Let u v0 t h u h u h u h u y t x a 0 h a v0 t h a v0 t h a v0 t h a v0 t h a v0 t Let u a v0 t h u 2a h u Variational Principle Consider point mass m doing 1D motion x t Assume x ti xi x tf xf Under the in uence of potential V x Know 3 Lecture 5 8 251 Spring 2007 Possible motions Not possible Given a path 4 Lecture 5 8 251 Spring 2007 Functional S x t not a function of time Hamilton s Principle Principal path makes S stationary Call true path x t Consider new path x t x t S x t x t S x x 2 Assume x ti 0 x tf 0 Lagrangian L t Kinetic Energy Potential Energy t2 t2 1 2 S L t dt m x t V x t dt 2 t1 t1 tf 1 2 S x x m x x V x x dt 2 ti tf tf V 1 1 2 S x mx x x t x t dt m x t V x 2 2 x 2 ti t i x2 Need to eliminate second term tf mx x be true ti V x x t x t dt must go away for S x x S x x 2 to Call this the variation S tf d S dt mx x m x x V x t x t dt ti Integrate by parts tf dS mx tf x tf mx ti x ti dt x t mx V x t ti x tf ti 0 from before The integral tf ti dt x t mx V x t must be 0 too so mx V x t 5 Lecture 5 8 251 Spring 2007 String Lagrangian T Kinetic energy Potential Energy 1 2 0 dx L 0 a 2 lT0 string y t a 1 dx 0 2 y x 2 T0 1 1 2 2 dx 0 y t T0 y t 2 2 tf S L t dt ti Call L Lagrangian Density L 1 y 1 y 0 2 2 t 2 t So tf S a dt ti 0 y y dxL t x y ti x 0 y tf x 0 Don t know y x 0 t or y x a t tf a L L S dt dx y y y y ti 0 6 Lecture 5 8 251 Spring 2007 Let P t L y P x L y tf S ti tf S a dt ti 0 a t y x y P P t x 0 t a tf P P x t P x y x a dx y x t dxP t y tfi x 0 t x 0 ti y ti y tf 0 Must have P t P x 2y 2y 0 0 2 T0 2 t x t x Some kind of conservation law like J 0 tf dtP x y x a x 0 ti tf dt P x t x a y t x a P x t x 0 y t x 0 ti For 0 a P x t x y t x Dirichlet condition y t x xed y t x 0 Free boundary condition P x t x 0 y x 0 Neumann condition 7