W = Fdz = mgdz = mg dz = mgz. dv F = ma = m dt = =

Transcription

1 Theoretil Physis Prof. Ruiz, UNC Asheille, dotorphys on YouTube Chpter W Notes. The Priniple of Lest Ation W. Grity, Time, nd Lgrngins. Referene: Elish Huggins, "Grity, Time, nd Lgrngins," The Physis Teher 48, pp (Noember 00. The Pound-Rebk experiment (publition in 960 mesured the frequeny shift in light s light treled down seen-story shft. If you drop bll through height z, tht bll gins kineti energy by wy of the work done by the fore of grity. z z z W = Fdz = mgdz = mg dz = m This work is trnslted into kineti energy. Remember when we disussed work erlier in our ourse? Now we use d F = m = m. z d zd dz z d W = m dz = m dz m dz m d m 0 = 0dz = = 0 dz 0 Equting these we he W = m = m Ner the Erth the grittionl field n be tken to be onstnt. The potentil energy is defined s U = m, where z = 0 is t the Erth's surfe. Then, if you fll drop stone from building nd it flls to the ground, you get this energy trnslted into kineti energy when fritionl fores re negleted. For ny gien height, the totl energy is the sum of the kineti energy nd potentil energy. E = m + m Mihel J. Ruiz, Cretie Commons Attribution-NonCommeril-ShreAlike 3.0 Unported Liense

2 Compring two different heights during fll, you he If you drop stone from rest t height z ground m + m = m + m ( z = 0 where the finl eloity is = z, then = 0. If you let it fll to the 0+ m = m + 0 =, you get the erlier result We re going to reple the kineti energy with the energy of photon. For photon γ, the energy is gien by E hf γ =. The "flling" photon of ourse nnot speed up. Insted, the energy gined results in higher frequeny. So strt by stting the onsertion of energy for the photon tht "flls" to the ground. hf + m = hf + 0 top bottom Now use E top top = to substitute for the mss top m m = E hf =. hftop hftop + = hf bottom hf hf = hf + top Let's sole for the bottom: bottom top We bring the hf top out to the left next. ( hf bottom top ( bottom top = hf + f = f + Mihel J. Ruiz, Cretie Commons Attribution-NonCommeril-ShreAlike 3.0 Unported Liense

3 We now ompre the periods. Remember Compring the periods, ( f bottom = ftop + T = f ( T bottom Ttop = +... Also, note tht Remember your Tylor expnsion for dded to smll number from our first lss n months go? It is ( + ε + nε ε =. Here we he nd n =. Then ( T bottom = Ttop nd top bottom ( T = T +. A nie quik rule to remember is this one: + ε ε for ε. Our top lok runs fster. ( T top = Tbottom + During one seond of bottom time, T bottom =, the top lok dnes n extr T =. Let's remember this rule s follows: The gin per seond by lok t height z T = is. mount: extr We re shortly going to onsider time diltion in speil reltiity. So s not to onfuse tht time effet with this one, we will lbel the lok-height effet. We hoose GR for generl reltiity sine generl reltiity dels with elertion nd grittion. We restte our rule with the GR lbel. T =. The GR Rule: The gin per seond by lok t height z is GR Mihel J. Ruiz, Cretie Commons Attribution-NonCommeril-ShreAlike 3.0 Unported Liense

4 Now if tht lok strts moing round we he the speil reltiisti time diltion we studied erlier. Engineers must keep trk of both effets for Globl Positioning Stellite (GPS. For the speil reltiisti time diltion effet we he T = T / 0 The T 0 is the proper time, i.e., the time kept by the lok in the moing frme being obsered. Remember lwys tht your proper time is the time kept by wth in your poket. Time slows down for you in the moing frme. A Tylor expnsion gies T T = T( 0 = T 0 T0 = ( T This brings us to our seond rule whih we lbel SR for speil reltiity. The SR Rule: The loss per seond by lok moing t T is SR =. Now it is time for Feynmn's Gme. There re two loks on tble in room. You tke one nd I tke the other. We eh trel with our loks doing whteer we wnt but we must bring our loks bk in one hour ording to the room lok. The winner is the one whose lok gins the most time. T = GR TSR = Strtegy : You should moe your lok up s high s you n. Strtegy : It is wste to moe sidewys Tht loses time for you. Strtegy 3: Don't speed up too muh moing ertilly s tht loses time. Your sore is gien by dding up ll the effets for eh T = s tik of the room lok. For eh tik of the room lok, plyer's gin or loss is gien by the sum of the two reltiisti effets. Mihel J. Ruiz, Cretie Commons Attribution-NonCommeril-ShreAlike 3.0 Unported Liense

5 Tplyer = TGR + TSR TGR = T SR = 3600 n n Tsore = n= Now, we go oer to n integrl using our three rules delt to d, disrete rible to ontinuous, 3summtion sign to "snke." 3600 n n TSore = n n= h n n TSore = 0 ( ( dn We n reple n with t for time. h t t TSore = 0 m ( ( We wnt to mximize this. If me multiply by you he the Lgrngin in the integrnd nd S is lled the tion. We wnt to minimize the tion to win the gme. h m TSore S = m ( t m( t 0 The tion in more generl terms is the integrl of the differene between the kineti energy nd the potentil energy. We lso reert to the more ommon x rible. b S = m ( t V ( x The integrnd is the Lgrngin. L = m t V x ( ( Mihel J. Ruiz, Cretie Commons Attribution-NonCommeril-ShreAlike 3.0 Unported Liense

6 W. Lest Ation. Referene: Feynmn R, Leighton R, nd Snds M. The Feynmn Letures on Physis (3 olumes 964, 966. Refer to Volume, Chpter 9. The Priniple of Lest Ation. We would like to minimize the following from our gme in the preious setion. h S = m ( t mgx( t 0 We see here the differene between the kineti nd potentil energies. Let the integrtion go from point "" to point "b" nd mke the substitutions eloity nd V ( x dx = for the = mgx for the potentil energy. This gies us more generl expression for the potentil energy. In this prtiulr se, x represents the height. dx ( ( b S = m V x We re in serh for the idel pth tht minimizes this integrl. Cll this idel pth xt (, i.e., the x with br oer it. Then, n rbitrry pth xt ( n be expressed s sum of the idel pth plus some deition from the idel. xt ( = xt ( + η( x For the speed we he dxt ( dxt ( dη( x = + nd for the potentil V( x = V( x+ η. Mihel J. Ruiz, Cretie Commons Attribution-NonCommeril-ShreAlike 3.0 Unported Liense

7 For the eloity we get dx ( ( b S = m V x beomes b dx dη ( S = m + V ( x + η dx dη dx dx dη dη ( + = ( + + ( dx dη dx dx dη + = + + higher order terms ( ( For the potentil energy, we do Tylor series expnsion. + η = + η+ η + V ( x V ( x V '( x V "( x higher order Note tht the deitions ( x η( x quntity. η re smll nd ery lose to the ide pth, i.e.,. So we pln to neglet higher order terms, i.e., higher powers of smll b dx ( dx dη S = m + m V ( x V '( x η b dx ( dx dη S = m V ( x + m V '( x η Mihel J. Ruiz, Cretie Commons Attribution-NonCommeril-ShreAlike 3.0 Unported Liense

8 The rition in the tion due to the pth tht wnders wy from the idel is gien by b dx dη δs = m V '( x η For the ide pth, this is zero. So we set S 0 tht will desribe our idel pth. δ = in serh for differentil eqution b dx dη δs = m V '( x η = 0 The η rible is rbitrry. We wnt tht outside the brkets but we do not he η by itself for the other term. But we he deritie o η. So we use integrtion by prts to lift the deritie off it. = + d dx d x dx d η η η = η dx dη d dx d x η b d dx d x δs = m η m η V '( x η = 0 Mihel J. Ruiz, Cretie Commons Attribution-NonCommeril-ShreAlike 3.0 Unported Liense

9 b δ dx b d x S m m V '( x 0 η η η = + = b dx m η = 0 The integrted term sine you must strt t the beginning point "" nd go to point "b" for ny of your pths. This mens there is no digression for the points t the beginning nd the end. This lees us with δs = m η+ V x η = b d x '( 0, where now we n ftor out the rbitrry η. b d x δs = m + V '( x η = 0 Now we use the rbitrry trik. Sine the η re rbitrry deitions s we hoose the different wrong pths, for the best pth, eerything else must be zero. d x m + V '( x = 0 Do you reognize this? It is Newton's Seond Lw. Let's drop the br now tht we found our idel pth. d x m + V '( x = 0 F = m, where F dv ( x = nd dx d x =. Mihel J. Ruiz, Cretie Commons Attribution-NonCommeril-ShreAlike 3.0 Unported Liense

10 W3. The Lgrngin. In our preious setion we minimized the tion dx ( ( b S = m V x nd rried t Newton's Seond Lw F = m, where F dv ( x = nd dx d x =. The integrnd in the tion integrl is lled the Lgrngin L, where dx L= m V( x L = m V x L = KE PE ( The Lgrngin is found by writing down the kineti energy nd subtrting from it the potentil energy. Now we rrie t n elegnt nd sophistited wy to write Newton's Seond Lw s minimiztion priniple. Strt with the usul F = m, where L = m V x dv ( x F = nd dx ( d x =. F dv ( x L = = dx nd x d d L m = m = Mihel J. Ruiz, Cretie Commons Attribution-NonCommeril-ShreAlike 3.0 Unported Liense

11 F = m beomes L d L = x. Finlly, we reert to Newton's nottion of time deritie nd get the following. or The boe is lso lled n Euler-Lgrnge Eqution. When there re more thn one dimension, you get one of these for eh of the dimensions. Then they re lled the Euler-Lgrnge equtions. Mihel J. Ruiz, Cretie Commons Attribution-NonCommeril-ShreAlike 3.0 Unported Liense