hapter 1 Revew of Newtonan Mechancs 1.1 Why Study lasscal Mechancs? Quantum lmt Relatvstc lmt General relatvty Mathematcal technques Frst approxmaton Intuton 1.2 Revew of Newtonan Mechancs Basc defntons onservaton laws onservaton of lnear momentum onservaton of angular momentum onservaton of energy Wor-energy theorem. Exstence of a potental 1
hapter 2 Beyond the Second Law Introducton General coordnate nvarance Symmetres and the resultng conservaton laws onstrants Second law resstant problems onnecton wth quantum mechancs 2
hapter 3 The Acton Functonal 3.1 Functonals Informed dscusson of Lagrangan methods s helped by ntroducng the dea of a functonal. To understand t, thn of a functon, f x, as a mappng from the reals to the reals, f : R R that s, gven one real number, x, the functons hands us another real number, f x. Ths generalzes readly to functons of several varables, for example, f x s a map from R 3 to R whle the electrc feld E x, t maps E : R 4 R 3, snce each choce of four coordnates x, y, z, t gves us three unque components of the electrc feld at that pont. In ntegral expressons, we meet a dfferent sort of mappng. onsder F [x t] x t where we ntroduce square bracets, [ ], to ndcate that F s a functonal. Gven any functon x t, the ntegral wll gve us a defnte real number, but now we requre the entre functon, x t, to compute t. Defne F to be a functon space, n ths case the set of all ntegrable functons x t on the nterval [, t 2 ]. Then F s a mappng from ths functon space to the reals, F : F R Over the course of the twenteth century, functonals have played an ncreasngly mportant role. Introduced by P. J. Danell n 1919, functonals were used by N. Wener over the next two years to descrbe Brownan moton. Ther real mportance to physcs emerged wth R. Feynman s path ntegral formulaton of quantum mechancs n 1948 based on Drac s 1933 use of the Wener ntegral. We wll be nterested n one partcular functonal, called the acton or acton functonal, gven for the Newtonan mechancs of a sngle partcle by S [x t] L x, ẋ, t where the Lagrangan, L x, ẋ, t, s the dfference between the netc and potental energes, L x, ẋ, t T V 3
3.2 Some hstorcal observatons At the tme of the development of Lagrangan and Hamltonan mechancs, and even nto the 20 th century, the dea of a unquely determned classcal path was deeply entrenched n physcsts thnng about moton. The great determnstc power of the dea underlay the ndustral age and explaned the motons of planets. It s not surprsng that the probablstc preons of quantum mechancs were strongly ressted 1 but experment - the ultmate arbter - decrees n favor of quantum mechancs. Ths strong belef n determnsm made t dffcult to understand the varaton of the path of moton requred by the new approaches to classcal mechancs. The dea of varyng a path a lttle bt away from the classcal soluton smply seemed unphyscal. The noton of a vrtual dsplacement dodges the dlemma by nsstng that the change n path s vrtual, not real. The stuaton s vastly dfferent now. Mathematcally, the development of functonal calculus, ncludng ntegraton and dfferentaton of functonals, gves a language n whch varatons of a curve are an ntegral part. Physcally, the path ntegral formulaton of quantum mechancs tells us that one consstent way of understandng quantum mechancs s to thn of the quantum system as evolvng over all paths smultaneously, wth a certan weghtng appled to each and the classcal path emergng as the expected average. lasscal mechancs s then seen to emerge as ths dstrbuton of paths becomes sharply peaed around the classcal path, and therefore the overwhelmngly most probable result of measurement. Bearng these observatons n mnd, we wll tae the more modern route and gnore such notons as vrtual wor. Instead, we see the extremum of the acton functonal S [x t]. Just as the extrema of a functon f x are gven by the vanshng of ts frst dervatve, df 0, we as for the vanshng of the frst functonal dervatve, δs [x t] δx t Then, just as the most probable value of a functon s near where t changes most slowly,.e., near extrema, the most probable path s the one gvng the extremum of the acton. In the classcal lmt, ths s the only path the system can follow. 3.3 Varaton of the acton and the functonal dervatve For classcal mechancs, we do not need the formal defnton of the functonal dervatve, whch s gven n Not so lasscal Mechancs for anyone nterested n the rgorous detals. Instead, we mae use of the extremum conon above and use our ntuton about dervatves. The dervatve of a functon s gven by 0 dx df dx lm f x + ɛ f x ɛ 0 ɛ Notce that the lmt removes all but the part of the numerator lnear n ɛ, f x + ɛ f x f x f x + ɛ df dx vanshes, we do not need the dx part, but only f x + ɛ df dx + 1 2 ɛ2 d2 f dx + 2 f x. The functon at x cancels and we are left wth the dervatve. If the dervatve df f x + dx f x lnear order 0 where we have set ɛ dx. Applyng the same logc to the vanshng functonal dervatve, we requre δs [x t] S [x t + δx t] S [x t] lnear order 0 δs s called the varaton of the acton, and δx t s an arbtrary varaton of the path. Thus, f x t s one path n the xt-plane, x t + δx t s another path n the plane that dffers slghtly from the frst. The 1 Ensten s frustraton s captured n hs asserton to ornelus Lanczos,... dass [Herrgott] würfelt... ann ch enen Augenblc glauben. I cannot beleve for an nstant that God plays dce [wth the world]. He later abbrevated ths n conversatons wth Nels Bohr, Gott würfelt ncht..., God does not play dce. Bohr repled that t s not for us to say how God chooses to run the unverse. See http://de.wpeda.org/w/gott_würfelt_ncht 4
varaton δx s requred to vansh at the endponts, δx δx t 2 0 so that the two paths both start and fnsh n the same place at the same tme. In defnng the varaton n ths way, we avod certan subtletes arsng from places where the paths cross and δx t 0, and also the formal need to allow δx to be completely arbtrary rather than always small. The varaton s suffcent for our purpose. Now consder the actual form of the varaton when the acton s gven by S [x t] L x, ẋ, t wth L x, ẋ, t T V. For a sngle partcle n a poston-dependent potental V x, the acton s gven by S [x t] 1 2 mẋ2 V x and settng δx t h t, so we need to fnd S [x + h] S [x] lnear order. Snce we requre both paths, x t and x t + h t, to go between the same endponts at and t 2, we must have h h t 2 0. The vanshng varaton gves 0 δs [x t] S [x + h] S [x] lnear order 1 2 2 m ẋ + ḣ V x + h 1 2 mẋ2 V x lnear order 1 1 2 m ẋ 2 + 2ẋ ḣ + ḣ2 V x + h 2 mẋ2 V x lnear order Now we drop the small quadratc term, ḣ 2, cancel the netc energy 1 2 mẋ2 along the orgnal path x t, and expand the potental n a Taylor seres, 0 mẋ ḣ V x + h + V x lnear order x mẋ ḣ V x + h V x + O h 2 +... + V lnear order mẋ ḣ h V x Our next goal s to rearrange ths so that only the arbtrary vector h appears as a lnear factor, and not ts dervatve. We ntegrate by parts. Usng the product rule to wrte d mẋ h mẍ h + mẋ ḣ and solvng for the term, mẋ ḣ, that we actually have, mẋ ḣ d mẋ h mẍ h, the vanshng varaton of the acton mples 0 d mẋ h mẍ h h V x 5
mẋ t 2 h t 2 mẋ h mẍ + V x h mẍ + V x h snce h t 2 h 0. Fnally, suppose the ntegrand, except h t s nonvanshng at some pont x t. Then, snce the ntegral must vanshng for all h t, consder a choce of h parallel to the drecton of mẍ + V x and nonvanshng only n an nfntesmal regon about x t. Then the ntegral s approxmately mẍ t + V x t h t > 0 Ths s a contradcton, so mẍ t + V x t 0. Snce the pont x t was arbtrary, the expresson must vansh at every pont by the same argument, and we have mẍ V x Ths s Newton s second law where the force s derved from the potental V. 3.4 The Euler-Lagrange equaton For many partcle systems, we may wrte the acton as a sum over all of the partcles. However, there are vast smplfcatons that occur. For example, n a rgd body contanng many tmes Avogadro s number of partcles, the rgy constrant reduces the number of degrees of freedom to just sx - three to specfy the poston of the center of mass, and three more to specfy the drecton and magntude of rotaton about ths center. More generally, the use of general coordnates and constrants may gve expressons only vaguely remnscent of the sngle partcle netc and potental energes. Therefore, t s useful to tae a general approach, supposng the Lagrangan to depend on N generalzed coordnates q, ther veloctes, q, and tme. We tae the potental to depend only on the postons, not the veloctes or tme, so that L q, q, t T q, q, t V q Despte the generalty of ths form, we may fnd the extrema of the acton, whch are the equatons of moton n the coordnates q. arryng out the varaton as before, the th poston coordnate may change by an amount h t, whch vanshes at and t 2. Followng the same steps as for the sngle partcle, vanshng varaton gves, 0 δs [q 1, q 2,..., q N ] S [q + h ] S [q ] lnear order T q + h, q + ḣ, t V q + h T q, q, t V q 1 T q, q, t + T h + q ḣ 1 h T q + T V h q q 1 ḣ T q V q 1 h V q lnear order T q, q, t V q lnear order 6
where the Taylor seres to frst order of a functon of more than one varable contans the lnear term for each, f x + ɛ, y + δ f x, y + ɛ f x + δ f + hgher order terms y The center term contans the change n veloctes, so we ntegrate by parts, 1 T ḣ q The full expresson s now, 1 1 0 d T d T h h q q h t 2 T t 2 h T q q 1 1 d T h q 1 h h T d T V q q q q d q 1 d T h q where L T V and we use V q 0 to replace T wth L n the velocty dervatve term. Now, snce each h s ndependent of the rest and arbtrary, each term n the sum must vansh separately. The result s the Euler-Lagrange equaton, d 0 3.1 q q 3.5 General coordnate covarance of the Euler Lagrange equatons Here we show that the Euler-Lagrange equaton s covarant under general coordnate transformatons. By ths we mean that f the Euler-Lagrange equaton V x d ẋ x 0 s satsfed n one set of coordnates, x, then t wll hold n any other, y, V y d ẏ y 0 where x y j s the nvertble coordnate transformaton. For the two vectors to vansh together requres there to be a lnear map from one to other,.e., there exsts some J j such that V j J j V j, or d ẋ x J j d ẏ j y j It s clear what J j j1 must be f L s ndependent of velocty, we requre N x j1 7 J j y j
but the chan rule tells us that N x y j x y j j1 Therefore, J j s the Jacoban matrx of the coordnate transformaton, Lagrangan equaton hold n any coordnate system f and only f d ẋ N x y j d x j1 ẏ j y j y j x. In concluson, the Euler- 3.2 for any two, x y. We prove that ths s the case by dervng the relatonshp between the Euler-Lagrange equaton for x t and the Euler-Lagrange equaton for y t. onsder the varatonal equaton for y, computed n two ways. Snce the acton may be wrtten as ether S [ x ] or S [ y ], we have S [ y ] S [ x y ] Frst, we may mmedately wrte the Euler-Lagrange equaton by varyng S [ y t ]. Followng the steps that led us to Eq.3.1, that s, varyng and ntegratng by parts, we have δs δ L y, ẏ, t 1 1 y δy + ẏ δẏ y d ẏ δy as expected. Now compare ths to what we get by varyng S [ x y ] wth respect to y t: 0 δs δ,1 L x y, t, ẋ y, ẏ, t x x y δy + x ẏ δẏ + ẋ ẋ y δy + ẋ ẏ δẏ 3.3 Snce x s a functon of y and t only, x ẏ 0 and the second term n the frst parentheses vanshes. Now we need two denttes. Explctly expandng the velocty, ẋ, the chan rule gves: ẋ dx d x y t, t x y ẏ + x t 3.4 so dfferentatng, we have one dentty, ẋ ẏ x y 8
For the second dentty, we dfferentate eq.3.4 for the velocty wth respect to y : ẋ y 2 x y y j ẏj + 2 x y t x y j y ẏ j + x t y d x y Now return and substtute nto the varaton 0 δs,1,1,1,1 Fnally, ntegrate the fnal term by parts,,1 d x ẋ y δy x x y δy + x ẏ δẏ + ẋ ẋ y δy + ẋ ẏ δẏ x x y δy + d x ẋ y δy + x y δẏ x x y δy + d x ẋ y δy + x d y δy x x y δy + d x ẋ y δy,1,1,1 d x ẋ y δy d x ẋ y δy x x ẋ y δy fnal ẋ y δy ntal x d ẋ y δy where δy vanshes at the endponts. The vanshng varaton now becomes 0,1 x d x ẋ y δy The ntal equalty of the two forms of the acton, S [ y ] S [ x y ] mples δs [ y ] δs [ x y ] and therefore y d ẏ δy x d x ẋ y δy 0 1 1 [ y d ẏ,1 x d ẋ x and the ndependence and arbtrarness of the varaton, δy mples covarance: y d ẏ x d x ẋ y 9 y,1 ] δy 0 d x ẋ y δy
The concluson we reach s that no matter what coordnates q we choose for a problem, we may always wrte the equaton of moton as q d q 0 The same s true of the acton. Rather than wrtng the Euler-Lagrange equaton, we may wrte the acton as the ntegral of the Lagrangan and wrte the Lagrangan n terms of whatever coordnates we choose, S [ q ] L q, q, t Varyng ths wth respect to the q wll gve the correct form of the equatons. 3.6 Noether s Theorem There are mportant general propertes of Euler-Lagrange systems based on the symmetry of the Lagrangan. The most mportant symmetry result s Noether s Theorem, whch we prove below. We then apply the theorem n several mportant specal cases to fnd conservaton of momentum, energy and angular momentum. 3.7 Noether s theorem for the Euler-Lagrange equaton Symmetres may be ether dscrete or contnuous. Dscrete symmetres le party, tme reversal, or the four rotatons of a square, have only a fnte number of possble transformatons. By a contnuous symmetry, we mean a symmetry dependent upon a real, contnuous parameter such as a rotaton through an angle θ, where θ may be any number between 0 and 2π. In essence, Noether s theorem states that when an acton has a contnuous symmetry, we can derve a conserved quantty. To prove the theorem, we need clear defntons of a conserved quantty and of what we mean by a symmetry. Def: onserved quanttes We have shown that the acton S [x t] L x, ẋ, t s extremal when x t satsfes the Euler-Lagrange equaton, x d ẋ 0 3.5 Ths conon guarantees that δs vanshes for all varatons, x t x t + δx t whch vansh at the endponts of the moton. Let x t be a soluton to the Euler-Lagrange equaton, eq.3.5 of moton. Then a functon of x t and ts tme dervatves, f x t, ẋ t..., s conserved f t s constant along the paths of moton, df 0 x t 10
Defnton: Symmetry of the acton Sometmes t s the case that δs vanshes for certan lmted varatons of the path wthout mposng any conon at all. When ths happens, we say that S has a symmetry: A symmetry of an acton functonal S [x] s a transformaton of the path, x t λ x j t, t that leaves the acton nvarant, S [ x t ] S [ λ x j t, t ] regardless of the path of moton x t. In partcular, when λ x represents a contnuous transformaton of x 2, we may expand the transformaton nfntesmally, so that x x x + ε x δx x x ε x Snce the nfntesmal transformaton must leave S [x] nvarant, we have δ ε S S [ x + ε x ] S [ x ] 0 whether x t satsfes the feld equatons or not. If an nfntesmal transformaton s a symmetry, we may apply arbtrarly many nfntesmal transformatons to recover the nvarance of S under fnte transformatons. Here λx s a partcular functon of the coordnates. Ths s qute dfferent from performng a general varaton we are not placng any new demand on the acton, just notcng that partcular transformatons do not change t. Notce that nether λ nor ε s requred to vansh at the endponts of the moton. We are now n a poston to prove Noether s theorem. Note that we carefully dstngush between the symmetry varaton δ ε and a general varaton δ. Theorem Noether: Suppose an acton dependent on N ndependent functons x t, 1, 2,..., N has a contnuous symmetry so that t s nvarant under δ ε x x x ε x where ε x are fxed functons of x t. Then the quantty s conserved. Proof: I The exstence of a symmetry means that x λ ẋ ε x 0 δ ε S [x t] t2 x t x ε x + 1 x t ẋ n dε x Notce that δ ε S vanshes dentcally because the acton of δ ε s a symmetry. No equaton of moton has been used. Integratng the second term by parts we have 0 x ε x + d ẋ ε x d ẋ ε x t 2 ẋ ε x x d ẋ ε x + 2 Techncally, what we mean here s a Le group of transformatons, but the defnton of a group lnes up well wth our ntuton of symmetry. Groups are sets closed under an operaton whch has an dentty, nverses and s assocatve. For symmetres, each transformaton leaves the acton nvarant, so the combnaton of any two does as well, showng closure. The dentty s just no transformaton at all, nverses are just undong the transformaton we ve just done, and assocatvty s natural f you can pcture t compoundng three transformatons AB t doesn t matter whether we fnd the effect of D AB and then fnd D, or f we fnd E B frst then loo at the effect of AE. It just means the symmetry transformaton AB s well-defned no matter whch way we compute t, as long as we eep the orgnal order. 11
Ths expresson vanshes for every path. Now suppose x t s a soluton to the Euler-Lagrange equaton of moton, x d ẋ 0 Then along any such classcal path x t, the ntegrand vanshes and t follows that 0 δs [x] t 2 ẋ ε x t I t 2 I for any two end tmes,, t 2. Therefore, and s a constant of the moton. I di 0 x, ẋ ẋ ε x 3.8 Examples of conserved quanttes n Euler-Lagrange systems 3.8.1 yclc coordnates and conjugate momentum We begn ths secton wth some defntons. Def: yclc coordnate For example, n the sphercally symmetrc acton all three veloctes cyclc. A coordnate, q, s cyclc f t does not occur n the Lagrangan,.e., S [r, θ, ϕ] Def: onjugate momentum q 0 [ 1 2 m ṙ 2 + r 2 θ2 + r 2 sn 2 θ ϕ 2 ] V r ṙ, θ, ϕ are present and the coordnates r, θ are present, but ϕ 0. Therefore, ϕ s The conjugate momentum, p, to any coordnate q s defned to be p q For a sngle partcle n any coordnate-dependent potental, V x, the acton may be wrtten as S [x] so the momenta conjugate to the three coordnates x are [ ] 1 2 mẋ2 V x p ẋ mẋ 12
reproducng the famlar expresson for the momentum of a partcle. The conjugate momentum for a partcle s not always smply mv. If the partcle moves n a velocty dependent potental, the form changes. The prncpal example of ths s the Lorentz force law, whch follows from the velocty-dependent potental F q E + ẋ B V x, ẋ qφ x qẋ A x where B A. hec ths. The acton, s S [x] t 2 [ 1 2 mẋ2 qφ + qẋ A ], so we see mmedately dfferentate to fnd the conjugate momentum so that the momentum conjugate to x s To chec the equaton of moton, we vary: 0 δs [x] [ [ π ẋ mẋ + qa π mẋ + qa mẋ δẋ q φ δx + qδẋ A + qẋ mẍ δx q φ δx qδx Ȧ + qẋ ] A x δx ] A x δx where we have dscarded the surface term from ntegratng the velocty varaton by parts. Note that the fnal term contans a double sum, so we need the explct summaton rather than a second dot product. Then, extractng the varaton δx, we expand the total tme dervatve of the vector potental. Snce A A x, t wth no velocty dependence, we have Now regroup, 0 0 [ mẍ q φ x q da ] A + qẋ x δx mẍ q φ x q A t + j mẍ q The term n bracets must vansh, and we recognze A φ x + A + q t j fnal sum as the cross product of the curl wth ẋ. Then Aj x x j ẋj mẍ q E + ẋ B + q j ẋ j Aj x ẋ j A j x δx A x j δx A x as the components of the curl and the j Exercse: Expand ẋ B n terms of components of the velocty, where B A to show that the th component s [ẋ B] ẋ j Aj x A x j j 13
3.8.2 yclc coordnates and conserved momentum We have the followng consequences of a cyclc coordnate. Theorem: yclc coordnates If a coordnate q s cyclc then the momentum conjugate to q s conserved. Proof: Ths follows mmedately from Noether s theorem, snce, f L s ndependent of q t s unchanged by replacng q wth a translaton q q + a for any constant a. The acton s therefore nvarant under δq a and Noether s theorem gves the conserved quantty I x, ẋ, q ε x q x, ẋ, q a q But p q x,ẋ, q q s the momentum conjugate to q and a s constant, so p q s conserved. 3.8.3 Translatonal nvarance and conservaton of momentum Now consder full translatonal nvarance. We loo frst at a sngle partcle, then at many partcles. Suppose the acton for a 1-partcle system s nvarant under arbtrary fnte translatons, or nfntesmally, lettng a ε, x x + a δx x x ε We may express the nvarance of S under δx ε explctly, 0 δ ε S t2 x δx + ẋ δẋ t2 x δx + d ẋ δx d t 2 t2 ẋ ε x d + ẋ δx ẋ ε For a partcle whch satsfes the Euler-Lagrange equaton, the fnal ntegral vanshes. Then, snce and t 2 are arbtrary we must have ẋ ε p ε conserved for all constants ε. Snce ε s arbtrary, the momentum p ẋ conjugate to x s conserved as a result of translatonal nvarance. Now consder an solated system,.e., a bounded system wth potentals dependng only on the relavte postons, x a x b of the N partcles a, b 1,..., N. We may wrte the acton for ths system as S [x] 1 2 mẋ2 a V x a x b b a a1 Then shftng the entre system by the same vector a, x a x a a 14
leaves S nvarant snce x a x b x a a x b a x a x b x a ẋ a Accordng to Noether s theorem, the conserved quantty s I x, ẋ ẋ ε x 1 ẋ 2 mẋ2 a V x a x b a b a a1 mẋ aa a1 Fnally, snce a may be any constant vector we must have P mẋ a so that the total momentum s conserved for an solated system. 3.8.4 Rotatonal symmetry and conservaton of angular momentum 2 dm onsder a 2-dmensonal system wth free-partcle Lagrangan L x, y 1 2 m ẋ 2 + ẏ 2 V ρ a1 where ρ x 2 + y 2 s the radal dstance from the orgn. Then rotaton x x x cos θ y sn θ y y x sn θ + y cos θ for any fxed value of θ leaves the acton unchanged, S [x] L nvarant. hec ths! For an nfntesmal change, θ 1, the changes n x, y are ε 1 δx x x x cos θ y sn θ x x 1 1 2! θ2 +... y θ 1 3! θ3 +... x yθ ε 2 δy y y xθ 15
Therefore, from Noether s theorem, we have the conserved quantty, ẋ ε x mẋε 1 + mẏε 2 mẋ yθ + mẏ xθ θm xẏ yẋ as long as x and y satsfy the equatons of moton. Snce θ s just an arbtrary constant to begn wth, and we can dentfy the angular momentum, J m ẏx ẋy xp y yp x J x p as the conserved quantty. It s worth notng that J s conjugate to a cyclc coordnate. If we rewrte the acton n terms of polar coordnates, r, ϕ, t becomes S [r, ϕ] so that ϕ s cyclc. The momentum conjugate to ϕ s 1 2 m ṙ 2 + r 2 ϕ 2 p ϕ ϕ mr 2 ϕ Dfferentatng tan ϕ y x, 1 cos 2 ϕ ϕ ẏ x yẋ x 2 gvng the same result, ϕ xẏ yẋ x 2 cos 2 ϕ xẏ yẋ x 2 x 2 r 2 p ϕ mr 2 ϕ m xẏ yẋ J We wll generalze ths result to 3-dmensons after a complete dscusson of rotatons. 3.9 onservaton of energy onservaton of energy s related to tme translaton nvarance. However, ths nvarance s more subtle than smply replacng t t + τ, whch s smply a reparameterzaton of the acton ntegral. Instead, the conservaton law holds whenever the Lagrangan does not depend explctly on tme so that t 0 The total tme dervatve of L then reduces to dl x ẋ + ẋ ẍ + t x ẋ + ẋ ẍ 16
Usng the Lagrange equatons to replace n the frst term, we get dl d d x d ẋ ẋ ẋ + ẋ ẋ Brngng both terms to the same sde, we have d ẋ ẋ L 0 ẋ ẍ so that the quantty E ẋ ẋ L s conserved. The quantty E s called the energy. For a sngle partcle n a potental V x, the conserved energy s E ẋ ẋ L ẋ 1 ẋ 2 mẋ2 V x 1 mẋ ẋ 2 mẋ2 V x 1 2 mẋ2 V x 1 2 mẋ2 + V x For the velocty-dependent potental of the Lorentz force law, S [x] [ ] 1 2 mẋ2 qφ + qẋ A so that E ẋ 1 ẋ 2 mẋ2 qφ + qẋ A ẋ mẋ + qa 1 1 2 mẋ2 qφ + qẋ A 2 mẋ2 qφ + qẋ A 1 2 mẋ2 + qẋ A qφ + qẋ A 1 2 mẋ2 + qφ s conserved. 17
3.10 Scale Invarance Physcal measurements are always relatve to our choce of unt. The resultng dlatatonal symmetry wll be examned n detal when we study Hamltonan dynamcs. However, there are other forms of rescalng a problem that lead to physcal results. These results typcally depend on the fact that the Euler-Lagrange equaton s unchanged by an overall constant, so that the actons S L S λ L have the same extremal curves. Now suppose we have a Lagrangan whch depends on some constant parameters a 1,..., a n n adon to the arbtrary coordnates, L L x, ẋ, a 1,..., a n, t These parameters mght nclude masses, lengths, sprng constants and so on. Further, suppose that each of these varables may be rescaled by some factor n such a way that S changes by only an overall factor. That s, when we mae the replacements x αx t βt ẋ α β ẋ a γ a for certan constants α, β, γ 1,..., γ n we fnd that L αx, α β ẋ, γ 1 a 1,..., γ n a n, βt λl x, ẋ, a 1,..., a n, t for some constant λ whch depends on the scalng constants. Then the Euler-Lagrange equatons for the system descrbed by L αx, α β ẋ, γ 1 a 1,..., γ n a n, βt are the same as for the orgnal Lagrangan, and we may mae the replacements n the soluton. onsder the smple harmonc oscllator. The usual Lagrangan s If we rescale, then the rescaled Lagrangan s L 1 2 mẋ2 1 2 x2 x αx m βm γ t δt L 1 βα 2 2 δ 2 mẋ2 1 2 γα2 x 2 and as long as β δ γ, we have S γα 2 S as a scalng symmetry. Scalng x doesn t depend on the other 2 scales, so there s no nformaton there. 18
Now consder a system wth unt mass and unt sprng constant, m 0 1 0 1 and suppose ths system s perodc, wth perod T 0. Then rescalng, the mass, sprng constant and perod become m βm 0 β γ 0 γ T δt 0 and scale nvarance tells us that a perodc soluton also holds for the scaled m, and T as long as δ β γ m. Therefore m T T0 and the frequency s proportonal to m. 19