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by: Jamaal McGlynn


Jamaal McGlynn
GPA 3.78


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This 16 page Class Notes was uploaded by Jamaal McGlynn on Tuesday October 20, 2015. The Class Notes belongs to PHYS608 at San Diego State University taught by M.Bromley in Fall. Since its upload, it has received 22 views. For similar materials see /class/225318/phys608-san-diego-state-university in Physics 2 at San Diego State University.


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Date Created: 10/20/15
Lecture 6 Outline Lagrange s Eqns II o contd Calculus of Variations Section 22 o Calculus of Variations examples a Derivation of Lagrange s from Hamilton s Section 23 o if time intro to Lagrange Multipliers Section 24 Calculus of variations intro 0 Consider a 2 D path in as y space 0 De ne the variation in y as n 2 a a 2 any da dx x dx x da 0 Similarly variation and integration operations commute b b 5102 6 yda 6yda and 6 6fy d a da 0 a a L 1 ll93 31297 31197 and 0 ie we want stationary J to second orderl The varied path is in nitesimally different to the actual path we want consistent With external and constraint forces Calculus of variations setup 0 Assume correct path ya 0 with set of paths 24937 06 y937 0 06775 0 by defn 77331 77332 0 nice functions ya 077a 2 M f yltxagt ltxagtx dx x1 da 0 want stationary point when 3 la20 0 0 since 04 is independent of a differentiate under the sign dJ x2 df x2 8f8y 8f8y 81 851 dd A dd x A 8y8a8yda8xda x 2 2 Z Pia i g dx 1 8y 80 83 8513804 0 now second part we integrate by parts Calculus of variations technique a integrating by parts f vdu m f udv With u 3 o and v 2 so 3 2 5 thus d2 2 5 da m f 82y 8f8y x2 x2 d 8f 8y da da x 8y 8513804 8y 804 x1 x dx 8y 804 0 1 1 0 at startend x1 77331 x2 77332 dJ f afi if dx da x1 8y da 8 804 0 thus at the stationary value of 04 cu f2 1111 a a 0520 1 8y da 83 804 0 da 0 but 775 is an arbitrary function so 0 oz0 Calculus of variations Lemma Fundamental c of V lemma if for all arbitrary nice 775 2 Ma77ada 0 then 0 between 5131 g a g 5132 thus J is only stationary with 04 when a gang 8y da 83 known the Euler Lagrange equations Simple c of V example a Shortest distance between two points in a plane contd Classic c of V example Catenary o What is the length of a hanging curve contd Geodesic a Show that the Shortest distance are great Circles Euler s 2nd Formula 2 for the geodesic problem 5 7 12 sin2 6 dgp f6 V 92 sin2 6 easiest to work with Euler s 211d formula since 8 8y da 83 81 da 383 if 8 81 8f das da y x 8y8x 831813 8513 3183 383 813 which we use to rework i g afi 8f iaf d 33 yay ydac y39 d9 81 y y ydxay 3 Lagrange s Equations 2nd derivation o Deriving Lagrange s Eqns Via G of V 0 consider variation of many variables along a path 2 d d Mas1 fltq1ltxgtq2ltxgt m mnmdax0 o introduce the family of curves With parameter oz C11377 05 C11377 a C12377 05 C12377 a 0 ie we want extremum functions q a 0 with notation 2 3f 3 3f 36 azodoz 1 dozda 3 0 6J 8804 8 q39idoz 0 Again integrating the 2nd term by parts to move the a 2nd derivation contd so integrating by parts to move the a 2 2 8f 82 8f 8 2 d 8f 8 1 egg dacdozdx 811398051 f1 8 dadx 2 8 f d 8 f M Z 1 8 d9 8 1 where in netesemal path departures 6 99 dd oz0 and the qs are independent so also are fundamental lemma gt the coefficients of 6 must vanish 8 f d 8 f 0 8 d9 8C known as the Euler Lagrange differential eqns Hamilton s Variational Principle 0 For monogenic systems in which all forces are derivable from scalar potentials except constraints 8 a a0d04 with independent Virtual displacements 6 The motion of the system from time 251 to 252 is such that t2 t2 1 ltqrqmtgtdtj T vmt t1 151 has a stationary value for the actual path of motion t2 813 d 83 616 didt0 A 8 da 8 q so far coord constraints built into generalised coords System with constraints How does Hamilton s principle treat constraint forces Consider a number of semi holonomic constraints dQl dQn faq17 E7 7 dt Apply the variational principle to obtain eqnls of motion agna t0 Via method of Lagrange undetermined multipliers First introduce functions Aaq1 gm 2 such that iAafaO gt 6t2lt afagtdt0 a1 h a1 and combining with the action 6 fit dt 0 5 lt 3Aafagt dtOz tt2 dt a1 Lagrange undetermined multipliers o from the effective Lagrangian L L 221 Aafa a consider variations in 6 along with choice of Aa 0 since we may be coupled eg fq1 qg Sql 6q2 0 t2 d an an m a f 151 dt 8 3 zz an Qk we will show next time where this appears from but it means that we now have to solve n m eqns solving for each of the n coordinates ql qn 39 39 7w dq dqn 1nclud1ng m eqn s faq1 qn 1 5 w f 0 get the forces of constraint Qk through the solution


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