Popular in Course
Popular in Physics 2
This 10 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 Staff in Fall. Since its upload, it has received 5 views. For similar materials see /class/225322/phys608-san-diego-state-university in Physics 2 at San Diego State University.
Reviews for PHYS608
Report this Material
What is Karma?
Karma is the currency of StudySoup.
Date Created: 10/20/15
Lecture 5 Outline Variational Principle 0 one nal Lagrange example Section 16 o Hamilton s Variational Principle Section 21 o Calculus of Variations Section 22 Another Lagrange Example 0 Time dependant rotating straight Wire Variational Principles Appeared in different forms over time eg Fermat s principle of least time of light rays through media leads to the Snell law of refraction Otherwise Hero gt Fermat gt Newton gt Leibniz gt Bernoulli Brothers gt Maupertuis gt Lagrange gt Gauss gt Hertz gt Hamilton 193435 Out of all the possible paths the system will minimise the time integral of kinetic and potential energy differences Basic postulate of classical physics Later will show that Lagrange s principle follows space Hamilton s Variational Principle For monogenic systems in which all forces are derivable from scalar potentials except constraints sz dt 7 t which may depend on Ii The motion of the system from time 151 to 152 is such that t2 t2 1 L dt T Vdt t1 751 has a stationary value for the actual path of motion t2 d h dQ2 616 qq tdt0 t1 1 2 dt dt Sometimes I is called the action units of energytime Variational principle s appeared in different forms Calculus of variations intro 0 Consider a 2 D path in 9 y space 0 De ne the variation in y as d y dy2 dyl dy Wad Em WSW 5 0 Similarly variation and integration operations commute b b 2 6 ydx6ydx and 6J 6fyxdx0 a 901 ll93 y293 y193 and 0 ie we want stationary J to second order 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 yx O with set of paths WU 06 3567 0 067793 0 by defn 771 1 771 2 0 nice functions yx O77r M mm a j Zm a as dx 0 want stationary point when Ell ia0 O 0 since d is independent of 9 differentiate under the sign UL 962 df x2 df dy df d3 df dx d d dd A dd 9 A1 dy dd d3 dd dx dd 9 x1 dy dd d3 dzrdd 0 now second part we integrate by parts 1 Calculus of variations technique 8y o integrating by parts f vdu uv f udv With u af dd 8f d 8f candv g gsO 8 g thusdv 8 gdx 9623 323 3f3y x2 962 d 3f 33 x 33 393304 33 304 dx 33 304 1 331 1 0 but 3931 71931 g ng nx2 O dJ 5fi if 0556 da x1 33 dx 3 304 0 thus at the stationary value of a 0U 8fi a a 0520 x1 33 dx 3 304 0 da 0 but g a 77x is an arbitrary function a0 Calculus of variations Lemma Fundamental C of V lemma if for all arbitrary nice 77x 2 Manxdx O then 0 between 931 S a S 932 thus J is only stationary with 04 when afdafgt 8 2 E 8 2 which resembles the Lagrange equations Simple c of V example 0 Shortest distance between two points in a plane