From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Lorentzian or Galilean boosts & Noether theorem Date: 11 May 2000 22:52:26 GMT Newsgroups: sci.physics.research Summary: [missing] In article <8fc591$eu4$1@nnrp1.deja.com>, wrote: >The invariance of the Lagrangian under boosts (either Lorentzian or >Galilean) gives conservation of what? Heh. People keep asking this question, so by now I just pull out my handy little file of one of the previous discussions we've had about this... see below. Someone should put this in the FAQ. Someone like Nathan Urban or Chris Hillman! >At least I searched in many books and they didn't even mention that. Right. That's why everybody keeps asking this question.... >Is Noether's theorem is not valid in such cases or what? Yes, it's valid. >and if Noether's theorem still >work in these cases, why the heck books do not talk about the conserved >quantities corresponds to such invariances? Either 1) the textbook writers are too stupid to have thought about this issue, or 2) they have decided it's better to let everybody figure this out for themselves, or 3) they feel the answer is not sufficiently important to waste a precious paragraph on it. I don't know. When I become king of the universe, I will make all books on mechanics mention this issue. But for now, read these old articles: ......................................................................... Newsgroups: sci.physics,sci.math Subject: Re: Noethers Theorem and the Poincare Group From: John Baez In article <4o1019$r59@darkstar.UCSC.EDU> draconis@cats.ucsc.edu (Jacob Alexnder Mannix) writes: >what do those >funny lorentz "rotations" in the x-t, y-t, z-t planes (the boosts) >give rise to as conserved quantities? This question comes up repeatedly on sci.physics because for some stupid reason most textbooks are lazy to discuss these conserved quantities. Hopefully Matt McIrvin and others will repost the enlightening replies they gave to this question the last time it rolled around. And maybe I can get up the energy to work this into a FAQ for the next time it rolls around. Let me just say this: It is amusing to note that analogous observables also show up in nonrelativistic mechanics: not from the Lorentz boosts, of course, but from the "Galilei boosts". The Galilei group is the limit of the Poincare group as c -> infinity, and it's generated by spacetime translations, spatial rotations, and "boosts" of the form t -> t x -> x + vt y -> y z -> z and similarly for y and z. If we consider a single nonrelativistic free particle --- in 1-dimensional space, to keep life simple --- and describe its state by its position q and momentum p at t = 0, we see that the Galilei boost t -> t x -> x + vt has the following effect on its state: p -> p + mv q -> q In other words, a Galilei boost is just a translation in momentum space. In nonrelativistic quantum mechanics this should be familiar, though somewhat disguised. Here it is a commonplace that the momentum observable p generates translations in position space; in the Schroedinger representation it's just -i hbar d/dx. But by the same token the position observable q generates translations in momentum space. As we've seen, a translation in momentum space is just a Galilei boost. So basically, the generator of Galilei boosts is the observable q. Ugh, but there is a constant "m" up there in the formula for a Galilei boost. So I guess the observable that generates Galilei boosts is really mq. If we generalize this to a many-particle system we'd get the sum over all particles of their mass times their position, or in other words: the total mass times the center of mass. Now this seems weird at first because it's not a conserved quantity! Wasn't Noether's theorem supposed to give us a conserved quantity? Well, it does, but since our symmetry (the boost) was explicitly time-dependent --- it involved "t" --- our conserved quantity will also be explicitly time-dependent. What I was just doing now was working out its value at t = 0. If we work out its value for arbitrary t I guess we get: the total mass times the center of mass minus t times the total momentum. Using the fact that total mass is conserved we can turn this conserved quantity into something perhaps a bit simpler: the center of mass minus t times the velocity of the center of mass. Something similar applies to the relativistic case. .......................................................................... From: toby@ugcs.caltech.edu (Toby Bartels) Newsgroups: sci.physics Subject: Re: Noethers Theorem and the Poincare Group Date: 24 May 1996 17:49:49 GMT Organization: California Institute of Technology, Pasadena, CA USA Lines: 20 Message-ID: <4o4srt$or8@gap.cco.caltech.edu> References: <4o1019$r59@darkstar.ucsc.edu> Jacob Alexnder Mannix wrote in part: >I have the infinitesimal rotations in the x-y, y-z, and z-x planes, >giving me conservation of each component of angular momentum, and by >now, you probably see what I am leaving for last, but what do those >funny lorentz "rotations" in the x-t, y-t, z-t planes (the boosts) >give rise to as conserved quantities? Does it coorespond to some >sort of relationship between the momentum and energy, like >E^2 = (pc)^2 - (mc^2)^2 give or take a minus sign or power of 2, or >something completely strange? Symmetry under rotation in the xy plane corresponds to conservation of L_z = r_x p_y - r_y p_x (where r and p are linear position and momentum). In the same way, symmetry under hyperbolic rotation in the tx plane corresponds to conservation of r_t p_x + r_x p_t = t p_x - x E. In other words, the system's speed in the x direction is p_x / E. -- Toby toby@ugcs.caltech.edu .......................................................................... From: torre@cc.usu.edu (Charles Torre) Newsgroups: sci.physics Subject: Re: Noethers Theorem and the Poincare Group Message-ID: <1996May23.120128.80788@cc.usu.edu> Date: 23 May 96 12:01:28 MDT References: <4o1019$r59@darkstar.UCSC.EDU> Organization: Utah State University Lines: 50 In article <4o1019$r59@darkstar.UCSC.EDU>, draconis@cats.ucsc.edu (Jacob Alexnder Mannix) writes: > I have a question concerning a physical interpretation of some of > the conserved quantities associated with the Poincare group (lorentz > boosts, rotations in 3-space, and translations). ... > but what do those > funny lorentz "rotations" in the x-t, y-t, z-t planes (the boosts) > give rise to as conserved quantities? This is a good question which comes up with some regularity and is rarely given much treatment in the texts I am familiar with. I will make a brief attempt to explain it. From the point of view of relativity there is no invariant way to separate the boosts from the rotations because, in effect, different inertial observers will not agree on the definitions of the various x-t, x-y, etc. planes. So, ultimately, one should simply view the 6 conserved quantities associated with the boosts and rotations as the relativistic form of angular momentum. This is completely analogous to the fact that there is no invariant way to separate energy from momentum - different inertial observers will see different mixtures-and so one speaks only of the "4-momentum". Still, this does not quite answer the question. In a given inertial reference frame I can identify 3 of the six conservation laws, the ones coming from spatial rotational symmetry in the given frame, as corresponding to familiar notions of angular momentum. What of the other 3 coming from boosts? Actually this question can be asked already in Newtonian physics which employs "Galilean relativity". One can show that a Galilean invariant Lagrangian will be symmetric (up to a total time derivative) with respect to boosts (the non-relativistic boosts, not Einstein's). The corresponding conserved quantity is an example of an "explicitly time dependent constant of the motion" and has the physical interpretation of the initial position of the center of mass of the system. So, one can view the extra 3 conservation laws associated with boosts in a Poincare invariant theory as being a relativistic generalization of this. The Einstein-relativistic generalization of Galilean-relativistic angular momentum and initial center of mass is a bit complicated since the Galilean version of these quantities involves, in effect, a preferred notion of simultaneity, which is of course absent in Einstein relativity. You can usually find discussion of bits and pieces of these issues in a mechanics book which treats special relativity, such as Goldstein and/or which bothers to discuss Galilean relativity such as Landau and Lifshitz. Charles Torre Department of Physics Utah State University Logan, UT 84322-4415 USA torre@cc.usu.edu .......................................................................... From: toby@ugcs.caltech.edu (Toby Bartels) Newsgroups: sci.physics Subject: Re: Noethers Theorem and the Poincare Group Date: 24 May 1996 18:11:15 GMT Organization: California Institute of Technology, Pasadena, CA USA Lines: 43 Message-ID: <4o4u43$pou@gap.cco.caltech.edu> References: <4o1019$r59@darkstar.ucsc.edu> Jacob Alexnder Mannix wrote in part: >One of my professors put it as a conserved >current, but I don't think I want to get into what a current means >for system of classical particles in spacetime. It doesn't (unless you want to express it in terms of delta functions). If you have a system of particles, that a quantity is conserved means that the sum of this quantity for all particles is conserved. OTOH, if you have a system of fields, that a quantity is conserved means that the integral of its density over all space is conserved. This can be expressed locally and more elegantly in terms of a current. This isn't difficult to see; I'll use an example that may be familiar to you. Electric charge is a conserved quantity; d Q / d t = 0. But if your charge is spread out into a field, you'll measure it with rho (the charge density) and the 3vector j (the current density). (These can be put into a 4vector (rho, j), so density is the time component of current density.) You may have seen charge conservation expressed as @ rho / @ t + div j = 0 (where `@' indicates partial differentiation and `div' means divergence). Integrating this equation over 3D volume, you get d Q / d t + (integral over all space) (div j) = 0. Gauss's theorem tells you (integral over all space) (div j) is just the integral over space's boundary of j. Since space's boundary is at infinity, and j = 0 at infinity, integrating @ rho / @ t + div j = 0 gives you d Q / d t = 0. So @ rho / @ t + div j is a local equation for conservation of charge. Now, if you remember that rho is the time component j_t of the current density, you can say @ j_t / @ t + @ j_x / @ x + @ j_y / @ y + @ j_z / @ z = 0. This equation is essentially div j = 0, only now j and div are 4 dimensional. This makes the 4 dimensional j a conserved 4D current density. Just as j's time component measures the charge density and j's space components measure the flow of the charge, you can keep track of energy density and energy flow in a field with a 4 dimensional conserved energy current. The same thing goes for momentum, angular momentum, and even t p_x - x E. -- Toby toby@ugcs.caltech.edu ......................................................................... From: torre@cc.usu.edu (Charles Torre) Newsgroups: sci.physics Subject: Re: Noethers Theorem and the Poincare Group Message-ID: <1996May24.085145.80834@cc.usu.edu> Date: 24 May 96 08:51:45 MDT References: <4o290j$r8v@www.oracorp.com> Distribution: world Organization: Utah State University Lines: 60 In article <4o290j$r8v@www.oracorp.com>, daryl@racorp.com (Daryl McCullough) writes: > columbus@pleides.osf.org (Michael Weiss) writes: >>Time translation symmetry corresponds to conservation of energy. > Actually, I don't understood in what sense conservation of energy > is an instance of Noether's theorem. The auto-didactic way that I > understand Noether's theorem is this: > > Suppose that we have a parameterized family of paths x(l,t) > such that L(x,v) is invariant under changes of the parameter l. > In that case, we can write: > > dL/dl = @L/@x dx/dl + @L/@v dv/dl = 0 > > (where I am using @ to mean partial derivative) > > Using the Lagrangian equation of motion @L/@x = d/dt (@L/@v), > together with v = dx/dt and interchanging the order of differentiation > gives: > > (d/dt (@L/@v)) dx/dl + @L/@v (d/dt (dx/dl)) = 0 > > or > d/dt ( @L/@v dx/dl) = 0 > > Thus the quantity Q = @L/@v dx/dl is conserved (it's value is constant). > > Now, the problem with applying this to time symmetry is this: the > Lagrangian is *not* invariant under translations in time! In the > case where l = t, we don't get dQ/dt = 0, we get instead > > dQ/dt = dL/dt > > (where Q = @L/@v dx/dt) > So, we can still get an invariant quantity by letting Q' = Q - L, > but it seems to me that this is not an instance of Noether's theorem. Your point is well-taken, but Noether's theorem IS meant to handle this case. In one version of the theorem a "symmetry" of the Lagrangian is an infinitesimal transformation that changes the Lagrangian by a total derivative of a function. In this case, the conserved quantity includes that function in its definition as you surmised above. I think these symmetries are sometimes called divergence symmetries. I also seem to recall that if one formulates the variational problem in terms differential forms on "jet space" then the divergence symmetry is just a traditional symmetry of the Lagrangian n-form, where n is the dimension of the space of independent variables. By "traditional" symmetry I mean: view the infinitesimal symmetry as a sort of vector field in jet space (the space of independent variables, dependent variables, all their derivatives), the Lagrangian n-form has vanishing Lie derivative along this vector field. Charles Torre Department of Physics Utah State University Logan, UT 84322-4415 USA torre@cc.usu.edu