Mechanics Physics 151


 Beryl Warner
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1 Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2)
2 What We Dd Last Tme! Dscussed multpartcle systems! Internal and external forces! Laws of acton and reacton! Introduced constrants! Generalzed coordnates! Introduced Lagrange s Equatons!... and ddn t do the dervaton " Let s pck t up and start from there
3 Today s Goals! Derve Lagrange s Eqn from Newton s Eqn! Use D Alembert s prncple! There wll be a few assumptons! Wll make them clear as we go! Introduce Hamlton s Prncple! Equvalent to Lagrange s Equatons! Whch n turn s equvalent to Newton s Equatons! Does not depend on coordnates by constructon! Dervaton n the next lecture
4 Lagrange s Equatons Recpe d L L = dt q! q 0 Knetc energy Lqqt (,!, ) T V Potental energy Lagrangan! Express L = T V n terms of generalzed coordnates { q }, ther tmedervatves { q! }, and tme t! The potental V = V(q, t) must exst!.e. all forces must be conservatve
5 Vrtual Dsplacement! Consder a system wth constrants! Ordnary coordnates r ( = 1...N)! Generalzed coordnates q ( = 1...n)! Imagne movng all the partcles slghtly r r +δ r r1 = r1( q1, q2,..., qn, t) r2 = r2( q1, q2,..., qn, t) " rn = rn( q1, q2,..., qn, t) Vrtual dsplacement! Note that δr must satsfy the constrants 3N coordnates not ndependent δr q q +δ q = r q δq n coordnates ndependent
6 D Alembert s Prncple! From Newton s Equaton of Moton F = p! F p! = 0! Part of the force F must be due to constrants F = F + f ( )! Appled force s known! Constrant force f (usually) does no work! Movement s perpendcular to the force! Excepton: frcton appled force constrant force ( a)! Now multply F + f p! = 0 by δr and sum over F F ( r, r,..., r,..., r, ) ( a) = ( a) t 1 2 N fδ r = 0
7 D Alembert s Prncple F p! r ( a) ( ) δ = 0! Force of constrants dropped out because! Called D Alembert s Prncple (1743) constrant force s out of the game. You can forget (a) fδ r = 0! Now we swtch from r to q 1st term r = F q Q q q δ = δ Q F r q! Unt of Q not always [force]! Q q s always [work] Generalzed force
8 D Alembert s Prncple 2nd term δ r δ r = p! r = p! q = m!! r δ q q, q! A bt of work can show! D Alembert s Prncple becomes r!! d T T = dt q! q 2 2 r d v v q dt q! 2 q 2 δ q d T T Q δ q = 0 dt q! q mv T 2 2
9 Lagrange s Equatons d T T Q δ q = dt q! q! Generalzed coordnates q are ndependent d T T = Q dt q! q! Assume forces are conservatve Q r V = V r F = q q q 0 F These are free Almost there! = V Throw ths back n
10 Lagrange s Equatons ( T V) d T = 0 dt q! q V q! = 0 q! d L L Fnally = 0 L= T( q dt q, q!, t) V( q, t)! q! Assume that V does not depend on Done!
11 Assumptons We Made! Constrants are holonomc! We always assume ths! Constrant forces do no work! Forget frctons F = V! Lagrange s Eqn. tself s OK f V depends explctly on t V q! = 0 q!! Appled forces are conservatve! Potental V does not depend on Wll revew the last assumpton later r = r( q, q,..., q, t) 1 2 n fδ r = 0
12 Example: TmeDependent! Transformaton functons may depend on t! Generalzed coordnate system may move! E.g. coordnate system fxed to the Earth! An example r = r( q, t) sprng constant K natural length l mass m on a ral l + r angular velocty α
13 Example: TmeDependent x= ( l+ r)cosαt! Transformaton functons: y = ( l + r)snαt m 2 2 m 2 2 2! Knetc energy T = x! + y! = r! + ( l+ r) α 2 2! Potental energy V = K 2 { } { } r 2 m { ( ) } K L= r! + l+ r α r Lagrange s Equaton d L L mr m α 2 ( l r ) Kr 0 dt = + + = r!!! r
14 Example: TmeDependent d L L mr m α 2 ( l r ) Kr 0 dt = + + = r!!! r 2 2 mα l mr!! + ( K mα ) r 0 2 = K mα! If K > mα 2, a harmonc oscllator wth! Center of oscllaton s shfted by! If K < mα 2, moves away exponentally! If K = mα 2, velocty s constant ω =! Centrpetal force balances wth the sprng force K 2 mα m
15 Note on Arbtrarty! Lagrangan s not unque for a gven system! If a Lagrangan L descrbes a system df( q, t) L = L+ works as well for any functon F dt! One can prove d df df dt q dt! q dt = 0 usng df F F = q! + dt q t
16 Assumptons We Made! Constrants are holonomc! We always assume ths! Constrant forces do no work! Forget frctons F = V! Lagrange s Eqn. tself s OK f V depends explctly on t V q! = 0 q!! Appled forces are conservatve! Potental V does not depend on Let s revew the last assumpton r = r( q, q,..., q, t) 1 2 n fδ r = 0
17 VeloctyDependent Potental! We assumed Q and = 0 so that d T T = Q dt q! q! We could do the same f we had Q U d U = + q dt q! V = q L= T( q, q!, t) U( q, q!, t) V q! Ths had to be 0 d ( T V) ( T V) = 0 dt q! q U = U( q, q!, t) Generalzed, or veloctydependent potental
18 EM Force on Partcle! Lorentz force on a charged partcle F = q[ E+ ( v B)]! E and B felds are gven by A E = φ t B= A! Force s vdependent " Need a vdependent potental U = qφ qa v works check Veloctydependent. Can t fnd a usual potental V Physcs 15b! Lagrangan s 1 = + A v 2 2 L mv qφ q
19 Monogenc System! If all forces n a system are derved from a generalzed potental, U d U ts called a monogenc system Q! U s a functon of! Lorentz force s monogenc! A monogenc system s conservatve only f! Or U q! U = = t 0 qqt,!,! Lagrange s Equaton works on a monogenc system = + q dt q! U = U( q)
20 Hamlton s Prncple! We derved Lagrange s Eqn from Newton s Eqn usng a dfferental prncple! D Alembert s prncple uses nfntesmal dsplacements! It s possble to do t wth an ntegral prncple Hamlton s Prncple
21 Confguraton Space! Generalzed coordnates q 1,...,q n fully descrbe the system s confguraton at any moment! Imagne an ndmensonal space! Each pont n ths space (q 1,...,q n ) corresponds to one confguraton of the system! Tme evoluton of the system " A curve n the confguraton space real space confguraton space confguraton space
22 Acton Integral! A system s movng as! Lagrangan s q = q () t = 1... n Lqqt (,!,) = Lqt ((), qt!(),) t ntegrate I t 2 = Ldt Acton, or acton ntegral t1! Acton I depends on the entre path from t 1 to t 2! Choce of coordnates q does not matter! Acton s nvarant under coordnate transformaton
23 Hamlton s Prncple The acton ntegral of a physcal system s statonary for the actual path! Ths s equvalent to Lagrange s Equatons! We wll prove ths! Three equvalent formulatons! Newton s Eqn depends explctly on xyz coordnates! Lagrange s Eqn s same for any generalzed coordnates! Hamlton s Prncple refers to no coordnates! Everythng s n the acton ntegral We wll also defne statonary Hamlton s Prncple s more fundamental probably...
24 Statonary! Consder two paths that are close to each other! Dfference s nfntesmal! Statonary means that the dfference of the acton ntegrals s zero to the 1st order of δq(t)! Smlar to frst dervatve = 0 t δ ( δ,! δ!,) (,!,) 0 I 2 Lq qq qtdt 2 Lqqtdt t1 t1 = + + =! Almost same as sayng mnmum! It could as well be maxmum t confguraton space t 1 qt () t 2 qt () +δ qt () δqt ( ) = δqt ( ) = 0 1 2
25 Infntesmal Path Dfference! What s δq(t)?! It s arbtrary sort of! It has to be zero at t 1 and t 2! It s wellbehavng Contnuous, nonsngular, contnuous 1 st and 2 nd dervatves Don t worry too much confguraton space qt () t 2 qt () +δ qt ()! Have to shrnk t to zero! Trck: wrte t as δqt () = αη() t! α s a parameter, whch we ll make " 0! η(t) s an arbtrary wellbehavng functon t 1 η( t ) = η( t ) = 0 1 2
26 Hamlton " Lagrange! To derve Lagrange s Eqns from Hamlton s Prncple t δ ( δ,! δ!,) (,!,) 0 I 2 Lq qq qtdt 2 Lqqtdt t1 t1 = + + = t! Defne t I Lqt t qt!! t tdt ( 2 α ) ( ( ) + αη ( ), ( ) + αη ( ), ) t1! δi s then lm ( ) (0) α [ I α I ] 0 I α α = 0 dα I! We must show that = 0 leads to Lagrange s Eqns α α = 0 A bt of work. Wll do t on Thursday
27 Summary! Derved Lagrange s Eqn from Newton s Eqn! Usng D Alembert s Prncple Dfferental approach! Assumptons we made:! Constrants are holonomc " Generalzed coordnates! Forces of constrants do no work " No frctons! Other forces are monogenc " Generalzed potental! Introduced Hamlton s Prncple! Integral approach! Defned the acton ntegral and statonary! Dervaton n the next lecture Q U d U = + q dt q!
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