Physics 105 Lecture Notes: An Introduction to Symmetries and Conservation Laws

Transcription

1 c Reserve for eucational use only. Physics 105 Lecture Notes: An Introuction to Symmetries an Conservation Laws Guest Lecturer: Any Charman 445 Birge Hall Department of Physics, University of California, Berkeley (Date: September 19 21, 2011) Here we elaborate an expan on the material of Taylor, Classical Mechanics, section 7.8. We explore the connections between continuous symmetries of Lagrangian ynamical systems an ynamical invariants, formalize in the celebrate Noether s Theorem. Also, congratulations to Hal an family, an best wishes to all! will not make Hal too haggar. Hopefully the new Haggar IMPORTANT ANNOUNCEMENT : in-class miterm on Friay, September 23, Please bring a blank bluebook. The mathmetical sciences particularly exhibit orer, symmetry, limit, an proportion; an these are the greatest forms of the beautiful. Aristotle Symmetry, as wie or as narrow as you may efine its meaning, is one iea by which through the ages we have trie to comprehen an create orer, beauty, an perfection. Hermann Weyl I. MOTIVATION AND OVERVIEW A. Why Do We Care About Symmetry? The concept of symmetry is central to contemporary an classical physics. Symmetry is eeply tie to both our aesthetic ugements of the beauty of physical theories as well as out practical ability to solve problems. Symmetries may reveal important facts about nature, an simplify our escriptions of physical systems or their behavior. Themes of symmetry, transformation, an conservation will recur often in this class an throughout your stuies of physics. Some go as far to say that every funamental law of nature is an expression of symmetry in one form or another. Certainly the various forms of Noether s celebrate theorem, connecting symmetries of an action to conserve quantities uner the evolution generate by it, as well as generalizations an thematically-relate work by Sophus Lie, Élie Cartan, Albert Bäcklun, an others, provie founation stones for builing up both classical an quantum physics as now unerstoo. Although they might isagree on other issues, most working physicists woul agree on this funamental point. Steven Weinberg has written that it is increasingly clear that the symmetry group of nature is the eepest thing we unerstan about the worl toay. Lee Smolin sai that The connection between symmetries an conservation laws is one of the great iscoveries of twentieth century physics. But I think very few non-experts will have hear either of it or its maker Emmy Noether, a great German mathematician. But it is as essential to twentieth century physics as famous ieas like the impossibility of exceeing the spee of light. Tsung-Dao lee sai that since the beginning of physics, symmetry consierations have provie us with an extremely powerful an useful tool in our effort to unerstan nature. Graually they have become the backbone of our theoretical formulation of physical laws.

2 2 B. What Is Symmetry? Derive from the ancient Greek for agreement in proportion or arrangement, the wor symmetry involves some notion that certain things remain unchange even as others aspects o change. Imagine, as in class, that I hol up a perfect cue ball. You close your eyes momentarily, an when you open them, are unable to say whether I have rotate the cue ball or not. A symmetry is sai to arise when something looks the same in some sense after transforming it in some way. Of course, we must flesh out what the something is, in what sense it looks the same, an uner what sorts of changes or transformations. These will epen on the context at han, an on what we hope the symmetry may accomplish for us. In classical physics, the transformations are thought of as applying to a mechanical system for example, physically shifting it, or rotating it, or imagining reflecting it in a mirror or else to generalize coorinates or other observables an/or parameters escribing the configuration an/or behavior of the system. The symmetry manifests as the invariance of certain properties of the system or its ynamical evolution, an/or of certain observable quantities associate with the system. 1. Active versus Passive Perspectives Opportunities for confusion arise because at various stages we can often think of a transformation in an active sense, transforming the actual configuration or evolution of a physical system (for example, actually rotating the positions an velocities of a system of particles) or in a passive sense, transforming our escription of a system or its behavior (for example, rotating the coorinate system in which we escribe the positions of the particles). Unfortunately, our answers will very often come out backwars, inverte, or otherwise incorrect if we misinterpret active transformations as passive or vice versa. Whenever it matters, when tackling a given problem or examining a certain topic, the best thing to o is to try state explicitly which perspective is being assume where. 2. Functional Forms versus Function Values Also note that in physics, by invariance sometimes we mean that the value of some function or observable quantity is unchange, while in other cases we mean that the functional form of some functions or ifferential equations are unchange, although specific values or solutions, respectively, may change. The latter is sometimes escribe as a covariance rather than invariance, although beware that these various term have several meanings. 3. Continuous versus Discrete Symmetries Another very important istinction is that between continuous an iscrete transformations. For example, translations an rotations are continuous because we can translate by arbitrarily small shifts or rotate by arbitrarily small angles, an because these transformations can be thought of as being close to close others or eforme continuously into them. But mirror reflections are iscrete because we either perform the reflection in a given plane or not there is no way to perform a fraction of the reflection, or a meaningful way to speak of a very small reflection, or continuously vary the reflection. Both types of transformations an the associate symmetries are very important in physics, but iscrete symmetries ten to play a less important role in classical mechanics than in quantum mechanics, so except for a few wors we will focus here on continuous symmetries.

3 3 4. Symmetry Groups Whether continuous or iscrete, transformations associate with symmetries always come together in groups that mathematicians conveniently call groups. Such groups of symmetry transformation, leaving the system invariant in some specifie sense, will have the following properties: 1. oing nothing leaves the system invariant in any sense, so is always a trivial symmetry transformation; 2. if one transformation leaves the system invariant, as oes a secon transformation, then the combine transformation consisting of first one then the other will obviously also o so, so is also a symmetry; 3. obviously, enacting one transformation followe by a secon, an then followe by a thir, is the same as enacting the first, an then the secon followe by the thir; 4. if a transformation leaves the system invariant, then the transformation consisting of un-oing or inverting the first will leave the system invariant; 5. first performing then un-oing a symmetry transformation is of course equivalent to oing nothing. These properties efine a group of symmetry transformations. Any subset of transformations which itself satisfies all of these conitions, an for which combinations of any of these transformations oes not lea to any others outsie the subset, is known as a subgroup. Group theory is in part, the mathematical escription of symmetry. We say a symmetry group is generate by some subset if any symmetry transformation can be thought of as the concatenation of (possibly repeate) transformations from among the set of generators. In physics, the most important iscrete symmetry group is generate by ust three types of generalize flips or inversions: parity, or spatial reflection of configurations an traectories through the origin; charge conugation, which replaces the occurrence of any particle with its anti-particle; an time-reversal, which reverses the irection of all particle traectories. For continuous groups, the generators will epen on continuously-variable real parameters specifying exactly how much translation, or rotation, etc. in some given irection is to be performe. For many groups of interest to physics, these parameters are typically chosen such that as their values get closer to zero, the transformations o less an less, that is, an resemble more closely the ientity transformation. Full symmetry groups may involve several inepenent parameters, although we shall that normally we can restrict attention to sub-groups each epening on only one parameter. Some continuous symmetry group may inclue sub-groups which are not continuously connecte to the ientity, but rather some other iscrete transformation, like parity inversion or time reversal. C. Physical Symmetries an Conserve Quantities Issues of symmetry in our attempts to escribe an unerstan the physical worl, an what happens or seems to happen in it, go back, like most of Western thought, at least to the ancient Greek civilization, when various philosophers ispute an ebate whether change was funamental an inevitable or superficial an contingent, or whether it was the existence of change or lack of change that neee explanation, or to what extent the particular, concrete, transient, visible material worl coul be explaine by un-changing, abstract, absolute, an necessary universal principles. Many of these questions have persiste. Exact opinion towar notions of symmetry epene in part on where one fell in the Newton/Leibniz ebate as to whether space an time are funamentally relational or absolute. Some physicists still ten to regar symmetries as something that in a sense, nature must work har to preserve. Other physicists ten towar a contrary perspective, in which symmetries are precisely those things about which nature is inifferent. Either way, uncovering an exploiting those symmetries will be useful.

4 4 1. Symmetries in Classical Mechanics In classical mechanics, sometimes the concept of symmetry applies usefully at the level of a single obect for example, etermining the gravitational fiel from a fixe, spherically-symmetric mass istribution. But in ynamics we are primarily concerne with the evolution of systems over time, an rarely if ever o iniviual configurations of systems, or else iniviual traectories of systems, exhibit fully the kins of rotational, translational, or other symmetry in which we are intereste. This is usually because initial conitions pick out iniviual traectories which break the full symmetry. So to assess symmetry in ynamical settings, we really nee to look not at iniviual traectories but rather at: 1. invariance in the form of the equations-of-motion iniviual traectories satisfy; 2. invariance of the entire set of traectories satisfie by the equations-of-motion, starting from all possible initial conitions; or 3. invariance of the action which generates the equations-of-motion, if such a variational formulation exists. Each of these perspectives has its uses, but one reason that the Lagrangian/action escription is so useful is that we can assess symmetry at the level of ust one function, rather that in terms an infinite set of (usually several) functions over all time, or of an entire system of couple ifferential equations. 2. Infinitesimal, Near-Ientity Symmetry Transformations In physics, continuous symmetries of interest are not only, well, continuous, but ifferentiable in a wellefine sense. In fact, we can in effect imagine Taylor expaning such transformations with respect to a group parameter, say ɛ. In such an expansion, the O(1) term typically represents the ientity transformation, reflecting the fact that for small ɛ the transformation only shifts things a little bit. The term linear in ɛ represents the leaing non-trivial action of the transformation, whereas for sufficiently small ɛ we can neglect the terms proportional to ɛ 2 or higher-orer terms. For ɛ regare as arbitrarily small, the result is calle an infinitesimal, near-ientity transformation. (A better terminology woul be ientity-plus-infinitesimal transformation, but no matter). Such expansion yiel infinitesimal transformations from finite ones. For continuous groups epening on one parameter, we can move in the opposite irection, an think generating continuous but finite transformations by concatenating an infinite number of infinitesimal ones. One of the very nice things about Noether s Theorem is that we can restrict attention to symmetries uner ust such infinitesimal, near-ientity transformations. Usually this simplifies our work significantly, since we can in effect ignore all but two terms in an infinite series. Near-ientity transformations must remain invertible for sufficiently small ɛ. In aition symmetry transformations o not commute generically; that is, if we have two transformations, applying the first then the secon is not in general the same thing as applying the secon then the first. But this lack of commutativity oes not arise for infinitesimal transformations, an it is for this reason that we will be able to look separately at each one-parameter symmetry group rather than groups involving several continuous parameters for example, rotations aroun a single axis in the limit of small rotation angles, instea of the larger group consisting of all possible spatial rotations. For Lagrangian systems, we shall consier infinitesimal symmetries which funamentally act on the generalize coorinates, an possibly on the evolution time parameter. The behavior uner the transformation of the generalize velocities cannot be chosen inepenently, but is inferre by the chain rule, after insisting that transforme generalize velocities remain total transforme-time erivatives of the corresponing transforme generalize coorinates.

5 5 3. An Just What Is a Constant of the Motion? When the numerical value of a certain observable remains constant in time as a physical system evolves, regarless of the initial conitions, then we say in various contexts that observable is conserve uner the evolution, or is a conserve quantity, or is a ynamical invariant, or is a constant of the motion, or is a first integral of the motion. For our purposes we can use any of these terms interchangeably. Note that although the value assume by the ynamical invariant oes not change in time along any one traectory, it will generally epen on the initial conitions, that is, on which traectory the system happens to fin itself. (At least, if it oes not, the invariant is too trivial to be particularly useful). Also note the function whose value is conserve uner the time evolution may have explicit time epenence (which is then balance by implicit time variation of other terms), but a ynamical invariant is often more useful if it is a function only of the system variables an not explicitly of the evolution time. On the other han, even a time-epenent ynamical invariant can be quite useful if it can be evaluate without knowing the full solution. Of course, given an invertible classical system, formally we can always efine time-epenent invariants as follows: given the time t an all particle s positions an momenta, output the unique positions an momentum at some initial time. While possible in principle, this is rarely helpful in practice unless we alreay know the solutions to the equations-of-motion. 4. An Example: Kepler s Laws As a simple example familiar to you from introuctory physics, let us consier the case of a planet orbiting a much more massive star, as escribe by Kepler s Laws erive in turn from Newton s Laws of Motion an Universal Gravitation, assuming for simplicity that both boies are point masses an that the center-of-mass coincies with the position of the massive star to a goo approximation. Note that both kinetic energy an potential energy are invariant uner simultaneous translations of the star an planet by any amount in any irection, an are invariant uner rotations of the planet/star system by any angle aroun any axis. Also note that the energy epens only on the instantaneous positions an velocities of the planet an star, an not explicitly on some absolute time. Aitional, somewhat less obvious symmetries also exist, which we will mention below an which Hal may iscuss later in the course. Essentially none of these symmetries are immeiately apparent in any one allowe orbit, in which relative to the star the planet travels with variable spee along an ellipse, with the star fixe at one foci, while the center-of-mass of the combine system remains either fixe or moves at constant velocity in a given inertial frame. The plane of the ellipse, its eccentricity, the location of it foci an orientation of its axes all remain fixe. That is, the ellipse trace out by the planet oes not rotate, expan, contract, istort, or precess. But if we shift the ellipse (an star an planet) in space by any fixe amount, or rotate by any amount aroun any axis, or else shift the planet to a new position an corresponing velocity along the orbit, we obtain other traectories, equally vali apart from the initial conitions which picke our our specific orbit. Yet this is not to imply that the symmetries o not play an important role in the behavior of iniviual traectories. It is a eeply beautiful, or ironic, fact (epening on your point-of-view) that any one orbit manifestly breaks the symmetries, an consists of a fixe ellipse of fixe shape in a fixe plane, precisely because of the symmetries in the equations-of-motion, or equivalently in the Lagrangian or action.

6 6 D. The Bottom Line Both symmetries an conservation laws are important in part because they simplify our escription an moeling of physical systems. In a sense both reuce the complexity of the problem, an constrain the kin of behaviors we can observe or not. Both involve invariance in the former case, invariance uner symmetry transformations applie to the system or our escription of it; in the latter, invariance in time of the value of certain observables as the system evolves. What is perhaps not immeiately obvious is that these two seemingly istinct forms of invariance are in fact interrelate. The previous example of planetary motion is typical of a general feature of Lagrangian ynamical systems symmetry uner a certain continuous group of transformations at the level of the Lagrangian or the equations-of-motion inevitably leas to traectories which themselves may not be iniviually symmetric, but along which which certain quantities are necessarily fixe in time. We explore these connections in more etail next. II. DYNAMICAL INVARIANTS FROM DYNAMICAL SYMMETRIES Here we will examine these connections between symmetries an conservation laws from the point of view of forces, equations-of-motion, Lagrangians, an actions, through both examples an some general results. A. Some Mathematical Notation an Assumptions Bol-face Latin or Greek letters will enote multi-component quantities collecte into column vectors. In particular, we shall enote a collection of N so-calle generalize positions or generalize coorinates by q = ( q 1,..., q N ) T, an refer either to a particular point on a traectory at time t, or the function of time specifying such a traectory, by q(t) (Which is implie will hopefully be clear epening on the context). Often to work with more compact expressions we will rop various arguments of functions if the meaning is clear. Over-ots will enote total time erivatives, so that the generalize velocities are enote by q(t) = t q(t) = ( q 1 (t),..., q N (t) ) T. Partial erivatives are enote in the usual shorthan as well, for example ( ) f q = f T, q 1,..., f q N (1) Either we will explicitly inclue the arguments of which f is a function, or it will hopefully be clear form context what other variables are being hel fixe in the partial ifferentiation. SImilarly, 2 f q q is a matrix of secon partial erivatives, etc. When confusion might otherwise arise, we will use expressions like f to enote a total erivatives with respect to a transforme time variable. In expressions like L t q = L q, recall that in taking the partial erivatives, we regar the generalize coorinates an the generalize velocities all as inepenent variables of which the Lagrangian is a function. A partial erivative with respect to one of these variables is performe while holing fixe all the others. Then we formally evaluate (or imagine evaluating) the expression for a traectory q(t), substituting q(t) for q an tq(t) for q, an then we take the remaining total time erivative. All this is a bit glosse over in our usual shorthan notation, an can cause some confusion. Even if the q o not enote Cartesian components of some position vector, we will still use the usual shorthan to enote a Eucliean inner prouct between such N-imensional vectors, for example, q q = N q i q i. (2) i=1

7 This notation will allow us to avoi writing a lot if sums explicitly, but can lea to misunerstaning. Very often we will eal with systems consisting of n particles or other sub-systems. A useful compact notation to partition the components is then the irect-sum representation, for example, q = n x, (3) =1 7 where q q = x x = x l x l, (4) l an where x enotes the generalize coorinates (often ust the actual spatial position) of the th subsystem. Whether a given instance of a ot prouct is over the whole space of generalize coorinates or over a single-particle s coorinates shoul be clear from context, but only in the case of Cartesian components is it the usual rotationally-invariant inner prouct for three-imensional vectors. We consier only Lagrangians over a finite number of egrees-of-freeom which epen on the generalize coorinates, possibly explicitly on time, an at most on the first erivative in time of the generalize coorinates, i.e., functions of the form L ( q, q, t ). We will be a little sloppy an often use the same notation to enote the function where all its arguments are regare as inepenent variables, or the its value along some traectory, as a function of time. We shall everywhere assume that any functions we introuce may be ifferentiate as many times as neee. B. Ignorable Coorinates The Euler-Lagrange equations, state that the time rate-of-change of the generalize momenta p L q coorinates q are equal the generalize forces L q. L t q = L q (5) conugate to the generalize A particular generalize coorinate q i is usually terme cyclic, kinosthenic, or ignorable if it oes not actually appear in the Lagrangian. (At one time these terms ha slightly ifferent meanings accoring to some authors, but now are use more or less interchangeably. The rather unilluminating term cyclic arose in the context of so-calle action-angle representations, where the ignorable coorinate coul inee be thought of, at least abstractly, as an angle associate with a uniform rotation in a suitable space.) Anyway, in the case of an ignorable coorinate the corresponing generalize force obviously vanishes ientically, so the conugate (corresponing) momentum vanishes: t p i = L q i = L q i = 0. (6) But an equivalent way of saying that L q i = 0 ientically (that is, whatever the values of the generalize coorinates an velocities) is that the Lagrangian L is invariant uner infinitesimal shifts in the generalize coorinate q i, starting from any generalize position, so we immeiately we see our first irect connection between an infinitesimal symmetry an a conserve quantity. Obviously, everything else being equal, it can be useful to choose generalize coorinates so that as many as possible are ignorable. For a single conserve quantity associate with a one-parameter family of infinitesimal symmetries (e.g., translations in one fixe irection, or rotation about one fixe axis), it is in fact always possible in principle, if not necessarily simple in practice, to choose generalize coorinates where this one ynamical invariant emerges as a momentum conugate to an ignorable coorinate (that oes not appear in the transforme Lagrangian). It may not be possible to o this simultaneously for multiple invariants.

8 Although the Lagrangian formulation gives us great freeom to choose generalize coorinates, it is also not necessarily convenient to use up this freeom in fining ignorable coorinates. Our goal in the remainer of these notes is in effect to generalize an expan on this connection between symmetries an conservation in cases where the symmetry is not so obvious as an ignorable coorinate explicitly missing from a Lagrangian. Before we construct the full apparatus of Noether s Theorem, we will look at some familiar cases. 8 C. Canonical Linear Momentum from Translation Invariance of the Lagrangian We can generalize what we mean by invariance uner infinitesimal transformations to cases where it is not necessarily ust a matter of of a shift in one generalize coorinate. As a simple but commonly-encountere example, consier an isolate system of n classical particles moving in three imensional space with a potential energy of the form U(x 1,..., x n, t) that is invariant uner simultaneous translations of all Cartesian positions x, = 1,..., n by arbitrary shifts proportional to in some fixe isplacement vector u: U(x 1 + ɛu,..., x n + ɛu, t) = U(x 1,..., x N, t) (7) for any sufficiently small choice of real parameter ɛ, an where again u R 3 is a fixe, time-inepenent vector specifying the common irection of spatial isplacement in three-imensional space for all particle positions. Note that uner the translations x x + ɛ u, the corresponing velocities are unchange: ẋ ẋ + t ɛ u = ẋ + 0 = ẋ. Hence the usual kinetic energy, K = K(ẋ 1,..., ẋ n ) = 1 2 m ẋ 2 (8) is invariant uner such translations, an hence the Lagrangian L = K U is also invariant: L(x 1 + ɛu,..., x n + ɛu, ẋ 1..., ẋ n, t) = L(x 1..., x n, ẋ 1..., ẋ N, t). (9) Now, ifferentiating both sies of this last relation with respect to ɛ, an then evaluating at ɛ = 0 an for some traectories x (t) satisfying the equations-of-motion, we fin ɛ L ɛ=0 = u L x = 0, (10) = L x, an the fact that u = u û is assume time- an using the Euler-Lagrange equations t inepenent, we infer L ẋ t û L ẋ = 0, (11) or in other wors: the component of the total linear momentum P = in the irection of û is conserve uner the time evolution. If this hols for three linearly-inepenent choices of u, then it must hol for all components of the momentum, an we can conclue the total linear momentum P is conserve. Note in the particular case where the x represent Cartesian coorinates of positions, an the kinetic energy is given by the stanar form (8), the total momentum is ust P = p, where p = L ẋ = mẋ, as woul be expecte. L ẋ

9 If, in particular, the potential energy is of the form of aitive pairwise interactions between particles that are a function only of istance, i.e., if U(x 1,..., x n, t) = <k Φ k ( x x k, t ), (12) 9 then (all components of) the total linear momentum P must be conserve, even though momentum may be exchange between particles ue to their interactions. Notice that the potential energy is not invariant if we shift ust one particle s coorinates, so iniviual particle momenta p nee not be conserve. However, if in aition, external potential energy terms that break translation invariance are present, for example if U(x 1,..., x n, t) = Ψ (x, t) + <k Φ k ( x x k, t ), (13) where at least some Ψ (x, t) are non-constant, then the total linear momentum of the system is no longer conserve. In effect, this is because the aitional terms can be seen to represent interactions with an external worl which can exchange momentum with the system of interest. D. Energy Conservation from Time Inepenence of the Lagrangian Instea of looking at infinitesimal shifts in the generalize coorinates, we can also consier the consequences of infinitesimal shifts in any explicit time epenence in the Lagrangian. Suppose the Lagrangian oes not epen explicitly on time at any fixe values of the generalize coorinates an generalize velocities. This means or equivalently that tl(q, q, t) = 0 for all t, (14) L(q, q, t + ɛτ) = L(q, q, t) for all t an ɛ, (15) given some choice of the fixe time offset τ. These are to hol at any prescribe values of q an q. But if instea we consier the value of the Lagrangian along a traectory, ifferentiate with respect to time, an invoke the Euler-Lagrange equations, we fin t L( q(t), q(t), t ) = q(t) L q + q(t) q + L t L = q(t) t Using the prouct rule, this becomes or after rearrangement, along the traectory. q + q(t) L q + L t. (16) t L = t q L q + L t, (17) t q L q L = L t (18) 1. Conservation of the Hamiltonian So if in fact L t = 0 (everywhere an always), then t q L q L = 0, (19)

10 10 an we obtain a conservation law associate with this invariance in the Lagrangian. The quantity in the brackets, H q L q L, is sufficiently important to earn its own name: the Hamiltonian, an abbreviation, usually H. (The Hamiltonian is more naturally regare as a function of the generalize coorinates an their conugate momenta, rather than the generalize coorinates an velocities, but that is a story for later). The Hamiltonian is relate to the Lagrangian an its erivatives in a way known as a Legenre Transformation, which you will learn more about later in this course as well as in other physics courses like Statistical Mechanics. 2. When Can the Hamiltonian Be Ientifie with the Total Energy? At least in most cases where the Lagrangian L has no explicit time epenence (an in other cases as well) this conserve Hamiltonian H can be ientifie with the system s total energy. In particular, when the Lagrangian is given by L = K( q) U(q), while the energy E = K(q, q) + U(q), we see that we can ientify H with E provie or H = q K q (K U) = E = K + U, (20) q K q = 2 K (21) Remembering Euler s Homogeneous Function Theorem, we realize that as long as the kinetic energy K( q) is smooth, this conition can hol if an only if the kinetic energy K is a homogenous function of egree 2 of the generalize velocities q: K(q, λ q) = λ 2 K(q, q) for all real, positive λ. (22) Note this is true for the usual expression for kinetic energy of a system of particles (expresse in Cartesian coorinates), K = m ẋ 2, as well as for the rotational kinetic energy of a rigi boy, K = I k ω ω k, where the I k are the Cartesian components of the moment-of-inertia tensor, an ω the components of angular velocity. More generally, this homogeneity requirement hols in any case that the kinetic energy can be expresse as a quaratic form in the generalize velocities, possibly with coefficients that epen on the coorinates. This particular, but commonly-encountere, form for the kinetic energy is preserve uner any invertible, time-inepenent an velocity-inepenent coorinate transformations. Suppose the generalize coorinates q are relate to another set of coorinates r by time-inepenent, invertible transformations: q = q(r), r = r(q), (23a) (23b) where q ( r(q ) ) = q an r ( q(r ) ) = r, such that the kinetic energy is a quaratic form with respect to the generalize velocities ṙ: K ( r, ṙ ) = 1 2ṙT M ( r ) ṙ, (24) for some symmetric matrix M = M(r) = M T which may be a function of the generalize coorinates r, but not the generalize velocities ṙ. In the two coorinate systems, the generalize velocities are relate by q = q r ṙ, ṙ = r q where r q = ( q r ) 1, an an thus the kinetic energy can be re-expresse as (25a) q, (25b) K(q, q) = 1 2 qt r q TM ( r(q) ) r q q, (26)

11 which is a quaratic form with respect to the transforme generalize velocities q. Notice that the matrix r q TM ( r(q) ) r q is symmetric, an will be positive efinite (an hence invertible) if M is positive efinite A Caveat Ambiguous statement s like the laws of nature o not care when we start or clocks can lea to some misunerstaning as to exactly what form of invariance is neee to establish energy conservation. While we have use the erivative t L( q(t), q(t), t ) in our proof of energy conservation, we are not assuming a symmetry of the form L ( q(t + ɛτ), q(t + ɛτ), t + ɛτ ) = L ( q(t), q(t), t ). While in a few systems (notably, free particles) the value of Lagrangian happens to be conserve uner time evolution in this way, the symmetry associate with energy involves shifts in the explicit time epenence of the Lagrangian, at fixe values of the generalize coorinates an velocities. The shift of the time parameter everywhere, that is both the explicit an implicit time epenence of the Lagrangian evaluate on a traectory, represents in effect merely a change in our book-keeping an is a symmetry of sorts in any ynamical system, albeit a rather trivial one which woul not in generalize to any interesting conservation laws. E. Angular Momentum Conservation from Rotational Invariance of the Lagrangian Now that you have the iea, I will actually leave it to the reaer to emonstrate conservation of total angular momentum J = x p irectly form the rotational invariance of the Lagrangian. The proof is similar to the case of linear momentum. If you get stuck, o not worry; we will erive this conservation law irectly from Noether s Theorem below. F. Galilean Invariance an Center-of-Mass Motion In Newtonian (that is, pre-relativistic) theories, recall that the laws of nature are expecte to be the same in all inertial frames relate by any one of ten types of transformations: all spatial translations, all spatial rotations, time translations, or Galilean boosts by any velocity. The latter means that one inertial frame is moving at a fixe velocity V relative to the other. Uner a Galilean boost, the corresponing coorinates are relate by t = t, x = x V t, (27a) (27b) in a passive sense if the prime frame is moving at V relative to the unprime frame but otherwise has its coorinate axes parallel to the corresponing unprime axes. The corresponing velocities then transform as ẋ = ẋ V. (28) Consier a Lagrangian for n classical particles, of the form L = K U, where once again the kinetic energy is of form (8) an the potential energy is of the form U(x 1,..., x n, t) = <k U k ( x x k, t ). (29)

12 Invariance uner Galilean boost is easiest to see at the level of the equations-of-motion. Because V is constant an the potential energy terms epen only on relative positions, uner a boost it is obvious that both the accelerations ẍ an forces ( U k x x k, t ) are left unchange. k However, let us look more closely at the behavior of the Lagrangian. Uner an active, infinitesimal Galilean boost by ɛv, the potential energy U remains invariant, while the kinetic energy becomes K(ẋ 1 + ɛv,..., ẋ n + ɛv ) = K(ẋ 1,..., ẋ n ) + ɛ MV t X cm + ɛ M V 2, (30) 12 where M = = m is the total mass, X cm = 1 M m x (31) is the center-of-mass, an t X cm = V cm = 1 M m ẋ = P M (32) is the center-of-mass velocity. So on the one han ɛ L(, x 1 + ɛv t,..., x n + ɛv t, ẋ 1 + ɛv,..., ẋ n + ɛv, t) ɛ=0 = MV t X cm. (33) On the other han, a formal use of the chain rule an Euler-Lagrange equations yiels ɛ L(x 1 + ɛv t,...,x n + ɛv t, ẋ 1 + ɛv,..., ẋ n + ɛv, t) ɛ=0 = = V t L x + V L ẋ V t t p + V p = V t t P + V P. (34) Equating these two expressions an rearranging, we have MV t X cm = V t tp + V P, (35) an from the prouct rule an the fact that V is a constant vector, we fin t V X cm = t V P M t, or equivalently: t ˆV X cm P M t = 0. (36) This is the time-epenent constant-of-the-motion associate with invariance uner a boost by V. If this hols for arbitrary boost irections, then we may conclue an in fact t Xcm P M t = 0, (37) X cm (t) = X cm (0) + P M t. (38) Notice that we i not actually use the fact that the total momentum P is conserve, or the fact that P = MV cm, although both are true in this case by virtue of the presume form for k an U. But in general it is important to note that spatial translations by a fixe isplacement, an Galilean boosts by a fixe velocity, are actually istinct symmetries.

13 13 III. NOETHER S THEOREM Some common strategies an patterns are beginning to emerge, but in orer to evelop the connection between symmetries an conservations laws for Lagrangian systems more systematically, we shall work up to a proof of Noether s (First) Theorem, an offer some iscussion an examples. There are various versions of Noether s Theorem, in varying egrees of generality. We will proof here the following: given a system with a finite number of egrees-of-freeom, an escribe by a seconorer Lagrangian, if the action is suitably invariant uner a one-parameter group of infinitesimal, nearientity transformations of the generalize coorinates an/or the evolution time, then there must exist a corresponing ynamical invariant whose value is conserve along all allowe traectories. We will not go into etail here, but it turns out the proof works backwars, an so the converse is also true: any conservation law of a certain form will be associate with a ynamical symmetry. A. Biographical Asie: Emmy Noether Amalie Emmy Noether ( ) has been consiere by Einstein an others, an continues to be consiere, to be the greatest woman mathematician in history. (Incientally, opinions as to the greatest male mathematician of all time are ivie. Popular caniates are Archimees, Newton, Euler, an Gauss). She ha to suffer many obstacles to prouce great mathematics. As a female, she was enie entrance into the gymnasium, or college preparatory school, in her native Germany. Despite support from Davi Hilbert, Felix Klein an other influential mathematicians of her ay, she went years without official acaemic positions ue to entrenche sexism in the German universities. Of Jewish heritage, she then ha to flee Germany with the rise of the Nazi party, an then ie young in America from complications from a tumor. While most of her work was in abstract algebra, she is most famous in the mathematical physics for one of the central results of Lagrangian ynamics, now known as Noether s Theorem. B. Symmetry Transformations an Action Invariance In its most basic version stuie here, Noether s theorem relates conservation laws for a finite-egreeof-freeom Lagrangian system to action invariance uner a one-parameter group of infinitesimal, nearientity, smooth symmetry transformations of the generalize coorinates an/or the time parameter. The inuce transformation of the generalize velocities are then fixe by assuming the generalize velocities remain interprete as the total time erivatives of the generalize coorinates. Every such one-parameter group of symmetries will be associate with its own conservation law. Again, working with the infinitesimal versions is generally much simpler. Although occasionally invariance uner finite transformations is obvious for example, we can confirm at a glance the rotational symmetry of an expression like 1 2 m ẋ 2, with infinitesimal transformations we en up being able to o more ifferentiation (mechanical, an relatively simple) rather than integration (more ifficult), or avoi systems of nonlinear algebraic equations (also har), an along the way nee only worry about the leaing terms in Taylor expansions. Because the transformations are infinitesimally close to the o-nothing ientity transformations, again note that such transformations will necessarily remain invertible when ɛ is regare as sufficiently small. Therefore such transformations can be regare either in a passive sense, as well-behave if ever-so-slight changes to the coorinates escribing the traectories, or in an active sense, as infinitesimal perturbations to the physical traectories. Both interpretations will have their uses, as long as we keep them straight.

14 14 1. How o Lagrangians Transform uner Coorinate Changes? First we shoul review how the Lagrangian itself transforms uner a passive change of coorinates of the form q = q (q, t), implying q = t q + q q q. In Newtonian (pre-relativistic) physics, it is often sai that the Lagrangian transforms as a scalar, meaning L (q, q, t) = L(q, q, t) (39) which is to say, uner a (passive) change of coorinates, the functional form of the Lagrangian changes so that its value at any instant in a given ynamical state oes not change, regarless of which mathematical coorinates are use to escribe it. In this way the Euler-Lagrange equations erive from the Lagrangians will generate the same physical traectories in either coorinate system. Notice that it is the value of the Lagrangian, not its actual functional form, which is kept fixe. In general, it woul not even make sense to try to keep the form fixe for example, if we transforme from Cartesian coorinates x, y, z to spherical coorinates, ρ, θ, φ, we cannot ust stick the secon transforme coorinate (i.e., the polar angle θ) into the argument slot for the secon original coorinate (i.e., Cartesian component y) in the original Lagrangian, for they o not serve the same role an o not even have the same units. However, we can aopt a still more expansive view of what counts as a legitimate transformation in our escription of traectories, an of what corresponing change to the Lagrangian leas to equivalent ynamical equations. So far, we have not messe with time. However, we may for example wish to introuce an overall shift in time (i.e., change in origin from which we measure elapse time) or an overall scaling (i.e., a change in the units in which we measure time). Actually, we can go further an accommoate virtually any smooth, invertible mappings from the original generalize coorinates q an original time t to transforme coorinates q an transforme time t : t = t (q, t), q = q (q, t), (40a) (40b) from which we can infer the corresponing transformation rule for the generalize velocities: q t q = q t t t = t q + q q q t t + q. (41) q t Note that we are assuming that the coorinate an temporal mappings o not epen on the generalize velocities, ensuring that the Euler-Lagrange equations remain secon-orer in either coorinate system. To etermine how the Lagrangian must in turn transform, we first consier the action functional, since integrals behave rather simply uner a change of integration variable. Hamilton s Principle tells us that the allowe traectories are those for which the variation δs in the action is at least of secon-orer when we imagine making first-orer infinitesimal variations in the traectories, keeping fixe the generalize positions at the fixe initial an final times. In orer to ensure that the action remains stationary in the new coorinates, the value of the action shoul be invariant, except possibly for terms epening on the generalize coorinates an time at the enpoints of the traectory, which are then unaffecte by the variation consiere in applying Hamilton s Principle. That is, we must require: S = S + F ( q(t 2 ), t 2 ) F ( q(t1 ), ) (42) for some function F (q, t) whose values at the initial an final points along the traectory provie any extra bounary terms that may emerge uner the coorinate transformations, an which we stress again is a function only of quantities that are fixe at the enpoints uner the variations allowe by Hamilton s Principle. Clearly δs = δs uner such infinitesimal variations,an so Hamilton s Principle will pick out the same physical traectories in either coorinate system, ust escribe ifferently.

15 15 Writing the various terms in (42) as efinite integrals, we fin t 2 t 2 t L ( q (t ), q (t), t ) = ( t L q(t), q(t ), t ) + t 2 t t F ( ) q(t), t, (43) t 1 where we have introuce the natural shorthan t 2 = t ( q(t 2 ), t 2 ) an t 1 = t ( q( ), ). Transforming the integration variable on the left-han sie, rearranging using linearity on the right, an suppressing some complicate neste functional epenence to avoi notational clutter, we have t 2 t t t L ( q, q, t ) = t 2 t L ( q, q, t ) + t F ( q, t ), (44) where both integrals are now performe with respect to the original time t. essentially arbitrary, it must be the case that the integrans are in fact equal: Because an t 2 are L (q, q, t ) t t = L(q, q, t) + tf (q, t), (45) which establishes the esire transformation rule. The aition of the tf term is known for historical reasons as a Lagrangian gauge transformation, an F (q, t) itself as a gauge generator or gauge function. 2. When Are Two Lagrangians Equivalent? The possibility of a gauge transformations arises because although a well-efine Lagrangian uniquely generates a set of Euler-Lagrange equations, the equations-of-motion o not uniquely etermine a Lagrangian. Notice that if we use the ientity transformation t = t an q = q, the transformation rule (45) becomes simply L (q, q, t) = L(q, q, t) + tf (q, t). (46) Clearly this relation hols true if F (q, t) is a constant function, but it is in fact true more generally. Inee it is straightforwar to work backwars, an use the Funamental Theorem of Calculus to verify that if two Lagrangians (which are functions of the same set of generalize coorinates an velocities, an time) iffer by a total time erivative, then they generate the same equations-of-motion. The Lagrangians are then sai to be gauge-equivalent or sometimes ust equivalent. To ensure that L epens at most only on the first erivatives of q(t), note that F can epen on q but not q. In terms of the corresponing action functionals, we have S = t 2 t L = t 2 t = S + F (q 2, t 2 ) F (q 1, ). t 2 ( ) t 2 L + F t = t L(q, q, t) + t tf (q, t) Thus we conclue that δs = δs uner the sort of variations consiere in Hamilton s Principle with the enpoints fixe in time an generalize position, an therefore these two actions are necessarily stationary for exactly the same functions q(t) satisfying these bounary conitions. The reaer may also fin it comforting if a bit teious to verify irectly from the Euler-Lagrange equations that gauge-equivalent Lagrangians inee o lea to the same equations-of-motion. A proof is presente in the Appenix A, which if nothing else confirms that a proof using the action functional is simpler. Interestingly, gauge-equivalence in this sense is a sufficient but not necessary conition for two Lagrangians to generate the same ynamical equations. There o exist inequivalent Lagrangians in the sense that they o not iffer by a total time erivative, yet o generate the same equations-of-motion. We offer it as a Take-Home Challenge Problem for the reaer fin an example of such Lagrangians. (HINT: think about what you have ust stuie, namely simple harmonic oscillators.). A solution is escribe in the Appenix. However, these sorts of exceptional cases o not seem to play a irect role in the unerstaning of conservation laws. (47)

16 16 3. Lagrangian Transformations Uner General Symmetry Transformations So far our transformation rule (45) tells us how the Lagrangian itself must transform uner any smooth, invertible, but otherwise arbitrary coorinate/time transformations. The recipe tells us to change the form of the Lagrangian function so as to preserve the value of the action, up to bounary terms. We cannot in general consier preserving a functional form of the Lagrangian for arbitrary transformations. But if the transformations can also be meaningfully interprete in an active sense, specifically as a shift of one physical traectory into another, escribe before an after by the same general type of coorinates (in particular, with the same units), then we coul legally insert the transforme variables into the ol Lagrangian, or the ol variables into the new Lagrangian. In fact, if the transformation is aitionally a symmetry, then the transforme Lagrangian shoul retain the same functional form before an after the transformation; that is, L (q, q, t ) = L(q, q, t ) as functions, an we obtain a precise mathematical statement of this ynamical symmetry: for some gauge function F (q, t). L(q, q, t ) t t = L(q, q, t) + tf (q, t), (48) The simplest case arises if t = t an F (q, t) is simply constant, in which case the value of the Lagrangian is unchange, namely L(q, q, t) = L(q, q, t), but the notion of ynamical symmetries associate with conservation laws is somewhat more general. 4. Lagrangian Transformations uner Infinitesimal Symmetry Transformations To treat infinitesimal, near-ientity symmetry transformations, we will parameterize the transformations by a real, imensionless parameter ɛ, an essentially restrict attention to arbitrarily small values of ɛ, taking the limits as ɛ 0 at the en of the calculation if neee. Along the way we can generally ignore terms that are O(ɛ 2 ) or of higher orer. Specifically, we can write the transformations for time an the generalize coorinates in the form: T = T (q, t; ɛ) = t + ɛ τ(q, t) + O(ɛ 2 ), Q = Q(q, t; ɛ)= q + ɛ ξ(q, t) + O(ɛ 2 ), (49a) (49b) where the corresponing rule for transforming generalize velocities becomes Q Q = T Q = t T t = q + ɛ ξ 1 + ɛ τ = q + ɛ ( ξ τ q) + O(ɛ 2 ), (50) an the total time erivatives of τ(q, t) an ξ(q, t) are to be calculate by the chain rule in the familiar way. For ɛ = 0 the transformations clearly reuce to the ientity transformation. Since the coorinate transformations are parameterize by ɛ, so too will be the transforme Lagrangian an the gauge generator F (q, t). Setting F (q, t) f 0 (q, t) + ɛ f(q, t) + ɛ 2 f 2 (q, t) +..., an expaning the symmetry conition (48) in powers of ɛ using the chain rule, we fin L + ɛ τl + ɛ τ L t + ɛ ξ L q + ɛ ( ξ τ q ) L q + O(ɛ2 ) = L + ɛ f + O(ɛ 2 ), (51) where everything is now evaluate using the original coorinates an time. Equating powers of ɛ, we arrive at our expression for symmetry uner the infinitesimal transformation (49): τ L + τ L t + ξ L q + ( ξ τ q ) L q f = 0 (52) for some function f(q, t). The symmetry conition inclues a certain irectional erivative of the Lagrangian, but also terms ue to the integration measure an the possible gauge transformation.

17 17 C. A Derivative-Base Proof Now we can finally offer a proof of Noether s Theorem, in more generality than is offere in your textbook. Again, various versions of the theorem (an the converse) are iscusse in the literature. Here we aim to emonstrate that if the action is invariant uner a group of infinitesimal, near-ientity, invertible transformations epening on time an the generalize positions (but no erivatives), then there exists a ynamical invariant, associate with the symmetry in question, conserve along any traectories satisfying the original Euler-Lagrange equations. First, we offer a irect assault by manipulating various erivatives. The proof is straightforwar but perhaps not entirely illuminating. Our strategy is to start from (??), a an subtract canceling quantities, rearrange, an simplify. These aitional terms will be esigne to allow us either to (a) combine various terms into total erivatives using the prouct rule, or (b) cancel combinations of terms using the Euler-Lagrange equations. So, starting with, we can write or τ L + τ L t τ L + τ L t τ q L q + ξ L q + ( ξ τ q ) L q f = 0 (53) L + q q + ξ L q + ( ξ τ q L ) q f = 0, (54) τ L + τ L f + ( L ξ τ q ) q + ( ξ τ q τ q L ) q = 0. (55) Aing an subtracting L t q, we obtain τ L + τ L f + ( ξ τ q ) L q ( L t q + ξ τ q L ) t q + ( ξ τ q τ q L ) q = 0. (56) Combining the first two terms together, an also the last two terms together, using the Prouct Rule, we fin t τ L t f + ( ξ τ q ) L q ( L t q + t ξ τ q L ) q = 0, (57) an finally combining various total time erivatives together using linearity, we have t ( τ L + ξ τ q L ) q f + ( ξ τ q ) L q L t q = 0, (58) so if the symmetry conition (49) hols, then along a traectory satisfying the Euler-Lagrange equations, the observable J (q, q, t) = τ L + ( L ξ τ q ) q f is conserve. D. An Action-Oriente Proof We can also prove Noether s Theorem using our action functional, which to my taste better reveals the connection between the conservation law an our original variational principle. We efine the ifference in the transforme versus original actions: T 2 t 2 ( ) ( ) S(ɛ) T L Q(T ; ɛ), Q(T ; ɛ), T t L q(t), q(t), t, (59) T 1 along some prescribe an fixe traectory q(t). Here T 2 T (q 2, t 2 ; ɛ) an T 1 T (q 1, ; ɛ), where q 2 = q(t 2 ) an q 1 = q( ), an the transforme traectory Q(T ; ɛ) is efine by Q(T ; ɛ) q ( t(t ; ɛ) ) ( + ɛ ξ q ( t(t ; ɛ) ) ), t(t ; ɛ), (60)