Physics 6010, Fall 2010 Symmetries and Conservation Laws: Energy, Momentum and Angular Momentum Relevant Sections in Text: 2.6, 2.


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1 Physics 6010, Fll 2010 Symmetries nd Conservtion Lws: Energy, Momentum nd Angulr Momentum Relevnt Sections in Text: 2.6, 2.7 Symmetries nd Conservtion Lws By conservtion lw we men quntity constructed from the coordintes, velocities, ccelertions, etc. of the system tht does not chnge s the system evolves in time. When the equtions of motion re second order, conservtion lws typiclly rise s functions on the velocity phse (possibly with explicit time dependence). Conservtion of energy, momentum nd ngulr momentum re fmilir exmples. Conservtion lws re lso known s constnts of the motion. They re llso clled first integrls of the equtions of motion. We will see why little lter. For exmple, consider hrmonic oscilltor in one dimension. coordinte be x. The Lgrngin is L(x, ẋ) = 1 2 mẋ2 1 2 kx2, where m, k re constnts. The eqution of motion is of the form mẍ + kx = 0. Let the generlized The energy is given by E(x, ẋ) = 1 2 mẋ kx2. The energy is conserved becuse we hve the identity de = ẋ(mẍ + kx), so tht, when E is evluted on solution x = x(t) to the equtions of motion, the resulting function E(t) E(x(t), dx(t) ) stisfies de(t) = 0. Indeed, the generl solution to the eqution of motion is x(t) = A cos(ωt + α), where A nd α re constnts (determined, e.g., by initil conditions) nd ω = km. We then hve E(x(t), dx(t) ) = 1 2 ka2, which is indeed time independent. I do not think I need to impress upon you the importnce of conservtion lws in physics. On the prcticl side, one cn use conservtion of energy, momentum, etc. to 1
2 unrvel mny spects of the motion of system without hving to explicitly integrte the equtions of motion. Indeed for systems with one degree of freedoms, conservtion lw usully determines everything! More generlly, if there re enough conservtion lws it is possible to completely solve for the motion. On the other hnd, I cnnot emphsize enough the fct tht lmost ll dynmicl systems re not simple enough for us to study their motion by integrting the eqution of motion, i.e., there re usully not enough conservtion lws to completely determine the motion. Still, even in these cses, the conservtion lws provide some of the principl clues we hve s to the dynmicl behvior of such systems. At deeper level, we use conservtion lws to guide us in our quest to find wht re the physicl lws governing the universe. Throughout the history of physics we hve repetedly revised our formultion of the lws of nture. The current stte of the rt involves the stndrd model of strong nd electrowek forces long with Einstein s generl theory of reltivity for the grvittionl force. In building these theories, the myrid of conservtion lws observed in nture* form the foundtion for the work of the theoreticl physicist. It is resonble to suppose tht future genertions of physicists will further revise our theory of mtter nd interctions of mtter, but it much less likely tht these theories will not incorporte conservtion lws. One of the principl dvntges of the Lgrngin formultion of mechnics (nd its field theoretic generliztions) is the ese with which one cn understnd the existence of conservtion lws. The fundmentl result we wnt to explore now is tht the existence of conservtion lws stems from the existence of symmetries of the Lgrngin. Trnsltion Symmetry nd Conservtion of Momentum. Let us begin by noting very esy result: when (generlized) coordinte does not pper in the Lgrngin, then conserved quntity results. When coordinte, q 1 sy, is bsent in the Lgrngin we sy tht q 1 is cyclic or ignorble. In this cse we hve q 1 = 0. The EL eqution for the q 1 degree of freedom is then simply (exercise) d q 1 = 0, which sys tht the quntity is conserved. For Newtonin prticle for which the q 1 q i re the Crtesin coordintes of the prticle, the conserved quntity resulting from * Of course there is energy, momentum nd ngulr momentum, but recll we lso hve conservtion of chrge, bryon number, lepton number, wek isopsin, strngeness, chrm, etc. 2
3 cyclic coordinte is just the corresponding component of the usul Newtonin momentum (exercise). This motivtes the following definition. For given degree of freedom, q i, the quntity p i (q, q, t) = q i is clled the cnonicl momentum conjugte to q i. We see tht if q i is cyclic, its conjugte momentum is conserved. Note tht if we define E i s the EL expressions, E i (q, q, q, t) := q i d q i, then the conservtion lw ssocited with cyclic q 1 follows from the identity d p 1 = E 1. If coordinte, sy, q 1, is cyclic, this obviously implies tht the Lgrngin is invrint under trnsltions of this coordinte (becuse the Lgrngin doesn t depend upon tht coordinte!): L(q 1 + constnt, q 2,..., q 1, q 2,..., t) = L(q 1, q 2,..., q 1, q 2,..., t). Thus trnsltion in the q 1 coordinte does not chnge the form of the Lgrngin. We sy tht trnsltion in q 1 is symmetry of the Lgrngin, or tht the Lgrngin is invrint under trnsltions in q 1. So, invrince of the Lgrngin with respect to trnsltions in the q i coordinte leds to conservtion of the cnonicl momentum p i. Note tht the EL equtions will not depend upon q 1 either, so tht they re the exct sme equtions no mtter wht vlues q 1 tkes the equtions re invrint under trnsltions in q 1. We sy tht trnsltions in q 1 re symmetry of the EL equtions. But it is the symmetry of the Lgrngin which leds to the conservtion lw.* As mentioned bove, the choice of the term momentum to describe p i stems from the fct tht for Lgrngin for system of Newtonin prticles: L = 1 2 m (1) r 2 (1) m (2) r 2 (2) +... V ( r (1), r (2),..., t), we hve (exercise) p (i) = m (i) r(i). * In generl, every symmetry of the Lgrngin is symmetry of the equtions of motion. But the converse is not true. 3
4 We will cll m (i) r(i) the mechnicl momentum of the i th prticle. For system of prticles moving under the influence of potentil vi Newton s second lw, the Crtesin components of the cnonicl moment re the sme s tht of the mechnicl moment. If the Lgrngin is invrint under trnsltion of one of the coordintes, this mens the potentil energy is likewise invrint, nd conversely (exercise). It is cler why this leds to conservtion lw from the Newtonin point of view: if the potentil is trnsltionlly invrint in certin direction, then its derivtive in tht direction which gives the force in tht direction vnishes, leving the corresponding momentum component unchnged in time. When we use generlized coordintes to define the configurtion of the system it is possible tht the cnonicl momentum is not wht you would usully cll the mechnicl momentum. Moreover, one my hve trnsltionl symmetries which do not mnifest themselves vi cyclic coordintes. For exmple, consider prticle moving in 3d under the influence of centrl force derivble from the potentil energy V = V ( r ). Becuse of the sphericl symmetry of the problem, it is nturl to use sphericl polr coordintes; the Lgrngin is (exercise) L = 1 2 m(ṙ2 + r 2 θ2 + r 2 sin 2 θ φ 2 ) V (r). Clerly the coordinte φ is cyclic. The resulting conservtion lw is d p φ = 0, where p φ = φ = mr2 sin 2 θ φ. In this cse the conserved quntity is the zcomponent of ngulr momentum. More on this shortly. It is instructive to consider the specil cse V = 0 in sphericl polr coordintes. Of course, in Crtesin coordintes the Lgrngin is trnsltionlly invrint in ech of (x, y, z). Tht trnsltionl symmetry still exists in sphericl polr coordintes, it is just not pprent. We will see how to think bout this issue shortly. Another importnt cse where the mechnicl nd cnonicl moment differ even for Crtesin coordintes occurs for prticle motion in prescribed electromgnetic field. Recll tht the Lgrngin is L = 1 2 m r 2 qφ + q c A r. The cnonicl momentum is, in vector nottion, (exercise) p( r, r, t) = m r + q c A( r, t). 4
5 The cnonicl momentum in this cse need not hve immedite physicl significnce since it depends upon the choice of vector potentil, which is not uniquely determined by given electromgnetic field. Still, it is the cnonicl momentum which will be conserved when coordinte is cyclic.* To summrize, if coordinte does not pper in the Lgrngin, i.e., the Lgrngin dmits trnlsltionl symmetry in coordinte, its conjugte momentum will be conserved. Time trnsltion Symmetry nd Energy Conservtion We consider system with generlized coordintes q i nd Lgrngin L = L(q, q, t). We ssume tht L is unchnged by time trnsltion: L = L(q, q) t = 0. This lck of explicit t dependence mens the Lgrngin is the sme function on the velocity phse spce for ll time. This lso implies the equtions of motion re the sme differentil equtions for ll time t (exercise).* Thus we sy tht such systems dmit time trnsltion symmetry. Now consider the restriction of the Lgrngin to curve q i (t) stisfying the EL equtions. Denote this restriction by L(t): L(t) := L(q(t), dq(t) ). The dependence of L on time comes solely through its dependence on q i (t) nd q i (t). Thus, on such curve, (exercise) dl(t) = q i qi + q i qi, where it is understood tht ll quntities on the right hnd side of the eqution re evluted on the curve. On curve stisfying the EL equtions we hve q i = d q i. * Also, it turns out tht it is the cnonicl momentum which is to be represented by derivtive opertor in the position representtion of quntum mechnics. * Such equtions, with no explicit dependence on the independent vrible, re clled utonomous. 5
6 So, ssuming the curve q i (t) stisfies the EL equtions, we hve ( ) dl d = q i q i + q i qi = d ( ) q i qi. We conclude tht, when evluted on solutions q i (t) of the EL equtions, we hve the result: ( ) d q i qi L = 0. This leds us to define the cnonicl energy E(q, q, t) of system described by the Lgrngin L(q, q, t) s E(q, q, t) = q i qi L. Our result is tht, when t = 0, nd when evluted on curve stisfying the equtions of motion, the cnonicl energy is conserved: d E(q(t), q(t)) = 0. We cn express energy conservtion s n identity which holds when the Lgrngin does not depend upon time. For generl Lgrngin L = L(q, q, t) we hve (exercise) ( d E = qi q i d ) q i t. From this identity you cn see tht E, when viewed s function on velocity phse spce is unchnged s you move long curve stisfying the EL equtions provided the Lgrngin hs no explicit time dependence. The use of the fmilir term energy to lbel this conservtion lw stems from the fct tht for the usul type of Newtonin system the quntity E(q, q, t) corresponds to our fmilir definition of energy. For exmple, prticle moving in given potentil hs the Lgrngin L = 1 2 m r 2 V ( r, t). We hve using Crtesin coordintes (exercise) so tht ẋ i ẋi = mẋ i ẋ i = 2T, E = 2T (T V ) = T + V. 6
7 Thus we recover the usul definition of energy s the sum of kinetic nd potentil energies. We lern, then, tht this quntity is conserved for prticle moving in given potentil provided the potentil is time independent. More generlly, whenever L = T V nd the kinetic energy T (q, q, t) is homogeneous function of degree two in the velocities, we hve tht (exercise) q i qi = 2T, nd hence, E = T + V. It should be noted tht, unless the bove requirement is stisfied, the conserved quntity ssocited with time trnsltion invrince of Lgrngin need not be the mechnicl energy T + V. In prticulr, it is possible tht the cnonicl energy is conserved, but is not the mechnicl energy. Conversely, the mechnicl energy might be conserved, but need not in generl be the sme s the cnonicl energy. Rottionl Symmetry nd Conservtion of Angulr Momentum Here we demonstrte tht rottionl symmetry of Lgrngin leds to conservtion of ngulr momentum. You hve lredy seen n exmple of this: the prticle moving in centrl force hs the sphericl coordinte φ s cyclic vrible. The conjugte momentum tht is conserved is the z component of ngulr momentum. The kinetic energy is invrint under rottions bout ny xis; for centrl force the potentil energy V = V (r) nd hence the Lgrngin L = T V is invrint under rottions bout ny xis. This implies tht we cn choose the zxis long ny direction nd the corresponding component of ngulr momentum will be conserved. Thus ll components of ngulr momentum will be conserved for prticle moving in centrl force. More generlly Lgrngin which is rottionlly invrint bout some xis will hve the totl ngulr momentum long tht xis conserved. In wht follows we show this directly. Let us begin with the Lgrngin for single prticle, L( r, r, t). We wnt to impose the condition tht the Lgrngin is rottionlly invrint in order to see the consequences. Thus we need to get mthemticl hndle on how the position nd velocity of prticle chnge under rottions. Recll tht to specify rottion one needs to pick n xis of rottion nd n ngle. Given these dt, one cn write down formuls for how vectors trnsform. For simplicity, we use very importnt strtegy: focus on infinitesiml rottions. The ide is tht finite rottions cn be built up by mny infinitesiml trnsformtions. In prticulr, Lgrngin is invrint under rottions bout some xis if nd only if it is invrint bout infinitesiml rottions bout tht xis. 7
8 We consider the chnge δl in the Lgrngin produced by n infinitesiml rottion round n xis defined by the unit vector n by n ngle ɛ << 1. First, we point out tht under such n infinitesiml rottion we hve δ r = ɛ n r, nd δ r = ɛ n r. This is esily verified with judicious choice of coordintes, which you should verify s n exercise. (Choose your zxis long n, compute the effect of rottion round by ɛ on r or r nd expnd everything to first order in ɛ. You cn lso convince yourself of the vlidity of the bove formuls by drwing some pictures.) Anywy, under rottion bout n by ɛ << 1 we hve the chnge in the Lgrngin to first order in ɛ given by (exercise) Here we hve set x i = (x, y, z). δl = x i δxi + ẋ i δẋi = x i ɛ( n r)i + ẋ i ɛ( n r) i. Suppose the Lgrngin is rottionlly invrint (e.g., the prticle moves in centrl force field) bout n. Then we know priori the following identity holds x i ( n r)i + ẋ i ( n r) i = 0. In terms of the cnonicl momentum p conjugte to r, the EL equtions re E i x i ṗ i = 0. The rottionl symmetry identity, written bove, cn be expressed in terms of the EL equtions s ( p + E ) ( n r) + p ( n r) = 0. This identity cn be rewritten s (exercise) d [ n ( r p)] = ( n r) E. This is the sttement tht the component long n of the ngulr momentum, M = r p, is conserved when evluted on solutions to the EL equtions. If the rottionl invrince is vlid for ny choice of n then ll components of M will be conserved. Note however tht 8
9 it is the cnonicl momentum tht fetures in this conservtion lw, not the mechnicl momentum. We thus cll M the cnonicl ngulr momentum nd we cll r m v the mechnicl ngulr momentum. Here re some elementry exmples of rottionl symmetry nd conservtion of ngulr momentum. A prticle moving in centrl force field hs Lgrngin: L = 1 2 m r 2 U( r ). This Lgrngin is clerly invrint under rottions. As relly good exercise you should verify the infinitesiml rottionl invrince identity: x i ( n r)i + ẋ i ( n r) i = 0. Thus ll components of ngulr momentum re conserved. This cn, of course, be checked directly by using the equtions of motion. Using the fct tht the prticle obeys Newton s second lw with force directed long the position vector r, you cn esily prove tht r p is conserved (exercise). As nother exmple, consider prticle moving in uniform force field F. This mens tht the Crtesin components of F re constnts. A Lgrngin for this system is (exercise) L = 1 2 m r 2 + F r. This Lgrngin would, e.g., describe the motion of prticle ner the erth s surfce. While the kinetic energy is rottionlly invrint bout ny xis, the potentil energy in this exmple is not invrint under ll rottions. This is becuse the vector F is fixed once nd for ll; F is not llowed to rotte long with r nd r. Another wy to see this is to suppose tht we choose our zxis long F. Then the potentil energy is of the form U = F z. Under rottion bout nything but the zxis this function will chnge! The potentil energy is invrint under rottions bout n xis prllel to F (exercise). Another good exercise: verify x i ( F r) i + ẋ i ( F r) i = 0. for this Lgrngin. Consequently the component of ngulr momentum long F will be conserved. Agin, using the equtions of motion (EL equtions), m r = F, you cn verify the conservtion lw explicitly (exercise). 9
10 Spcetime symmetry nd Closed Systems nd the Glileo Group The usul model of spce nd time, which is tcitly prt of Newtonin mechnics, ssumes tht spce nd time re homogeneous nd tht spce is isotropic. Thus closed system (isolted from the externl world) should behve the sme obey the sme dynmicl lws no mtter where in the universe it is locted, when the system is studied, nd no mtter how the system is oriented in spce. This suggests tht the Lgrngin for such system should be unchnged under spce nd time trnsltions of the whole system s well s rottions of the whole system. Given the connection between symmetries nd conservtion lws we then expect corresponding conservtion lws. We explore this briefly here. Let us consider closed isolted system of intercting prticles lbeled by positions, r (1), r (2), etc. Think of the prticles s living inside blck box which does not interct with the outside world. The ssumption of homogeneity of time implies tht the Lgrngin for the system does not hve ny explicit time dependence nd the totl cnonicl energy is conserved, s discussed previously. By the ssumption of the homogeneity of spce the physicl system behves the sme no mtter where it is locted in the universe. The Lgrngin will be unchnged under sptil trnsltion of the entire system. This mens tht the Lgrngin is invrint under the trnsformtion r () r () + b, i where b is ny constnt vector. Lbeling the Crtesin coordintes s x i, = 1, 2,..., () i.e., x i (1) = (x 1, y 1, z 1 ), etc. this implies (exercise) b i x,i i () = 0. The EL equtions for the motion of the prticles imply (exercise) where d b i p i() d b p() = 0,,i p i() = ẋ i, () nd we hve denoted the vector cnonicl momentum for ech prticle by p (). quntity P = p () The represents the totl liner momentum for the system. We see tht the invrince of physicl lws under spce trnsltion by b implies tht the component of the totl momentum 10
11 long b is conserved. Of course, since b is rbitrry, ll 3 components of P re conserved for closed system becuse of the homogeneity of spce. For Newtonin system, the Lgrngin is thus of the form L = 1 2 m () r () V ( r (1), r (2),...). The conserved totl energy is of the form E =,i ẋ i () ẋ i () L = 1 2 m () r () + V ( r (1), r (2),...). The conserved totl momentum is P = r () = m () r(). Although I won t prove it here, it is worth noting tht the requirement of symmetry of the Lgrngin under sptil trnsltions of the system implies tht the potentil energy function cn only depend upon the position vectors through their pirwise vector differences, i.e., V = V ( r (1) r (2), r (1) r (3),...). As n exmple, let us consider the ErthSun system ignoring ll other externl interctions. The Lgrngin is of the form L = 1 2 m E r 2 E m S r 2 S V ( r E r S ). Here we usully tke the potentil energy to be of the form V = Gm Em S r E r S. But more sophisticted choices for V re possible. In ny cse, the homogeneity of spce demnds tht V = V ( r E r S ), so the Lgrngin is invrint under ny trnsltion of the form r E r E + b, r S r S + b. Neither the momentum of the Erth nor tht of the Sun is conserved. But, becuse the Lgrngin is trnsltionlly invrint, the totl momentum: P = m E re + m S rs is conserved. 11
12 Let us now consider the implictions of sptil isotropy. The isotropy of spce implies tht closed system will hve Lgrngin which is invrint under rigid rottions of ll the prticles in the system. We hve prticles lbeled by positions r () nd velocities r (). Suppose the system dmits rottionlly invrint Lgrngin. More precisely, suppose the Lgrngin is ssumed invrint under simultneous rottion of ll the positions nd velocities bout n xis n (through the origin). The infinitesiml invrince condition generlizes in strightforwrd mnner (exercise): ( r () ( n r () ) + p () ( n r () ) As before, we cn restrict this to curve stisfying the EL equtions to find (exercise) ( p() ( n r () ) + p () ( n r () )) = 0. ) = 0. As before we cn rerrnge this s (exercise) ( d [ n ( r () p () )] = d n ) M = 0. Thus invrince of the Lgrngin under simultneous rottion bout the xis n of ll degrees of freedom leds to conservtion of the component long n of the totl ngulr momentum, M totl = M. As n exmple, let us return yet gin to the ErthSun system (2body centrl force problem). The Lgrngin is of the form L = 1 2 m E r 2 E m S r 2 S V ( r E r S ). Becuse the sclrs v E, v S, nd r E r S re invrint under rottions, the Lgrngin is invrint under ny simultneous rottion of the Erth nd Sun vribles (exercise). The conserved ngulr momentum is (exercise) P = m E r E r E + m S r S r S. To summrize, closed systems cn lwys be expected to hve conservtion lws for the totl energy, totl momentum, nd totl ngulr momentum by virtue of the homogeneity nd isotropy of spce nd time. I should emphsize tht it is possible to hve conservtion lws for open systems s well. We hve lredy seen exmples, but let me revisit this ide. For exmple, the Erth Sun system we hve discussed is closed system dmitting the usul conservtion lws. 12
13 But we often pproximte the motion of the system by ssuming the sun is fixed in spce, letting the dynmicl system consist of the motion of the Erth in the fixed potentil of Sun. So, if we simplify our model by ssuming we cn choose reference frme in which the Sun is fixed, e.g., r Sun = 0, then the resulting system (Erth in centrl force field due to the environment the Sun) is not closed system nd need not hve ll the spce nd time symmetries. In this cse the system lcks spce trnsltion invrince since it does mtter to the erth where it is in spce (reltive to the fixed sun). Consequently, the Erth s liner momentum is not conserved. But the Lgrngin still hs no explicit time dependence nd is still rottionlly invrint (bout xes going through the sun) nd so the cnonicl energy, E = 1 2 m E r 2 E Gm Em S, r E is conserved, s is the ngulr momentum, M = m E ṙ E. Of course, both of these conservtion lws re used to gret effect in studying the motion of the Erth. 13
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