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## PHYS608

by: Citlalli Sauer

20

0

27

# PHYS608 PHYS608

Citlalli Sauer
SDSU
GPA 3.75

M.Bromley

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COURSE
PROF.
M.Bromley
TYPE
Class Notes
PAGES
27
WORDS
KARMA
25 ?

## Popular in Physics 2

This 27 page Class Notes was uploaded by Citlalli Sauer 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 20 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 4 Outline Show us ya Lagrange a new D Alembert s principle Section 14 o Lagrange s Equation Section 14 o and some examples Section 16 Reversed effective force I 0 last time D Aleinbert s principle and generalised Force a d d o expanding the 2nd reversed effective Force term d z s dZ s dZ 377 i i j i 0 again distributive b b 61 a i 5 73 mi 1 1dt239aqjdt Zdt39aqj zdzt39dt aqj 0 chain rule the 2nd right hand term of Eq1 1aa8 2 Z ni 4 dt dqj an k 3an at 593015 861739 Reversed effective force II 0 now for the rst of the right hand terms of Eq1 aaacf am 8z7 0 thus we simplify the reversed effective force term d cm 37 57 i 5 39 2 dt r m dig 8 qj Z i mg 5 ji dt 1 zaqj z raqj Q7 D Alembert s Principle Take II 0 So now reversing such that 0 d 677 o and simplifying reversed effective and generalised forces d 827 827 fin z 5 0 gt maqj Q1 d a use 2amp6 EL EL i52 0n the rst two terms 13 d3 13 d a 1 2 a 1 i2 1 7 z z 5 0 dt 327 aqj 2mlvl C17 1 M j 0 but T m z7 2 gives new D Alei nbert s Principle d 8T 8T 5 0 Z idt adj 3613 M Q j D Alembert s Independence 7 7 39 39 d z39 original D Alembert s Prlnmple ft ua0 and the new form of D Alembert s Principle d 8T 8T 5 0 7 dt adj 3617 Q7 Q If the qj are linearly independent ie holonomic ie Zj chj 0 iff all cj 0 then 1 61 8TQ dt j with n equations for each generalised coordinate in Cartesian co ords 3 0 since no curvature 7 Easy Examples free particle contd free particle in polar coords I free particle in polar coords I Lagrange s Equations 39 i 7 M 39 39 i lt9T lt9T o holononnc D Alernbeit s PrinCiple dt adj aqj Q7 0 when the generalised Forces are Fi V V then a a a 37 av Q 1 ViV 1 j Z Z L o The last step can be seen by simple example Thus age Wa m LW 0 assuming that V is independent of time conservative 1 a a 0 a note that L T V is not the only choice of Lagrangian Simple vector calculus example Prescription a Can we de ne a Lagrangian ie Holonomic system a with applied forces derivable from ordinary or a generalised potential 0 and workless constraints 1 Write T and V in generalised coordinates 2 Form 3 from T V 3 Get equations of motion in each coordinate j from 1 a a 0 a transforming T from eg Cartesian into generalised coordinates is generally the hardest bit potential problem Cartesian Lecture 4 Outline Show us ya Lagrange a new D Alembert s principle Section 14 o Lagrange s Equation Section 14 o and some examples Section 16 Reversed effective force I 0 last time D Aleinbert s principle and generalised Force a d d o expanding the 2nd reversed effective Force term d z s dZ s dZ 377 i i j i 0 again distributive b b 61 a i 5 73 mi 1 1dt239aqjdt Zdt39aqj zdzt39dt aqj 0 chain rule the 2nd right hand term of Eq1 1aa8 2 Z ni 4 dt dqj an k 3an at 593015 861739 Reversed effective force II 0 now for the rst of the right hand terms of Eq1 aaacf am 8z7 0 thus we simplify the reversed effective force term d cm 37 57 i 5 39 2 dt r m dig 8 qj Z i mg 5 ji dt 1 zaqj z raqj Q7 D Alembert s Principle Take II 0 So now reversing such that 0 d 677 o and simplifying reversed effective and generalised forces d 827 827 fin z 5 0 gt maqj Q1 d a use 2amp6 EL EL i52 0n the rst two terms 13 d3 13 d a 1 2 a 1 i2 1 7 z z 5 0 dt 327 aqj 2mlvl C17 1 M j 0 but T m z7 2 gives new D Alei nbert s Principle d 8T 8T 5 0 Z idt adj 3613 M Q j D Alembert s Independence 7 7 39 39 d z39 original D Alembert s Prlnmple ft ua0 and the new form of D Alembert s Principle d 8T 8T 5 0 7 dt adj 3617 Q7 Q If the qj are linearly independent ie holonomic ie Zj chj 0 iff all cj 0 then 1 61 8TQ dt j with n equations for each generalised coordinate in Cartesian co ords 3 0 since no curvature 7 Easy Examples free particle free particle in polar coords I free particle in polar coords II Lagrange s Equations 39 i 7 M 39 39 i lt9T lt9T o holononnc D Alernbeit s PrinCiple dt adj aqj Q7 0 when the generalised Forces are Fi V V then a a a 37 av Q 1 ViV 1 j Z Z L o The last step can be seen by simple example Thus age Wa m LW 0 assuming that V is independent of time conservative 1 a a 0 a note that L T V is not the only choice of Lagrangian Simple vector calculus example Prescription a Can we de ne a Lagrangian ie Holonomic system a with applied forces derivable from ordinary or a generalised potential 0 and workless constraints 1 Write T and V in generalised coordinates 2 Form 3 from T V 3 Get equations of motion in each coordinate j from 1 a a 0 a transforming T from eg Cartesian into generalised coordinates is generally the hardest bit Space potential problem Cartesian potential problem Polar

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