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1. LAGRANGIAN Beauty, at least in theoretical , is perceived in the simplicity and compactness of the that describe the phenomena we observe about us. Dirac has emphasized this point and said “It is more important to have beauty in one’s equations than to have them fit experiment…. It seems that if one is working from the point of view of getting beauty in one’s equations, and if one has really a sound insight, one is on a sure of progress.” In this sense the beauty of lies in the fact that it can all be derived from the postulates of relativity together with just one hypothesis, which we call Hamilton’s principle. This includes all of and all of electricity and magnetism. In fact, if we postulate other interactions, such as the Yukawa potential, the mathematical form of these interactions is very restricted. The flexibility in the choice of natural laws is very limited. In the , as so-called “grand unified theories” are developed, it is expected that even this limited flexibility will be removed. One of the remarkable developments of modern physics has been the growing perception that the laws of physics are inevitable. Hawking may have gone beyond the realm of pure physics when he asked the question “Did God have any choice?” in the way She wrote the laws of physics. However, it seems that if the consists of three spatial and , and we require , then there is little choice in the laws of physics.

1.1. Hamilton’s principle and the postulates of relativity

Figure 1 Reference frames stated as his first law of that unless acted upon by an outside , a body at rest will remain at rest, and a body in motion will remain in uniform motion. A K in which this is true is called an inertial frame. The postulates of state that the laws of physics observed in an inertial reference frame K are identical to those observed in another inertial reference frame K ' that moves with respect to K as shown in Figure 1. Clearly, if the reference frame K' is also an inertial frame, it moves relative to K with at most a constant , and vice versa. Newton also said that “…absolute, true, and mathematical time, of itself, and from its own nature, flows equably and without relation to anything external.” We now know that this is true only in the nonrelativistic limit, but we assume here that it is true. Although the discussion really deserves to be fully relativistic, we restrict our attention to the nonrelativistic case. If time is absolute, then the coordinates r and r ' and the t and t ' in the two inertial reference frames are related by rrV' = − t , (1) tt' = , (2) where V is the velocity of K ' in K . These are known as a . Hamilton’s principle says that as a system moves from state a to state b , it does so along the that makes the

b Sdt= ∫ L (3) a an extremum, generally a minimum, subject to the constraint that the endpoints a and b (including both the coordinates and the times) are fixed. That is, in the notation of the ,

b δδSdt= ∫ L = 0 (4) a for variations δr of the trajectory that vanish at the endpoints, as shown in Figure 2. The quantity L is called the Lagrangian for the system, and its form depends on the nature of the system under consideration. The task in classical mechanics and classical theory therefore consists of two parts. First we must determine the Lagrangian L for the system, and we must find the that minimize the action S . As we shall see, the form of the Lagrangian follows from the postulates of relativity. Only the few parameters that appear in the equations must be determined from experiment.

Figure 2 Variation of a trajectory.

1.2. Lagrangian for a Free Up to this point we have not said anything about the physical system we are trying to describe, which may consist of , or fields, or both. We begin with a simple, structureless, , described by the coordinates r and time t . We hypothesize that the Lagrangian depends only on the coordinates, the time, and the velocity, but no higher of the , so that LL= (rv,,t) (5) where vr= ddt/ is the velocity of the particle. In fact, since and time are homogeneous, the Lagrangian of a cannot depend explicitly on the coordinates or the time, but only on the velocity. Otherwise, the behavior of the particle would be different at different places and different times. Thus, the Lagrangian must be simply LL= (v) (6) But the Lagrangian cannot depend on the direction of v , since space is isotropic, so it can depend only on the magnitude v2 = vv⋅ and have the form LT= (vv⋅ ) (7) for some T that we must determine. Hamilton’s principle for a free particle may now be stated in the form

b δδSvdt= ∫T ()2 = 0 (8) a Using the methods of the calculus of variations, we compute

bb b dddTTδr δδSdtdtdt=vv ⋅ =⋅22 v δ v =⋅ v = 0 (9) ∫∫T () 22 ∫ aadv a dt dv and integrating once by parts, we get

b b dddTT⎛⎞ δδSdt=⋅22rv − δ r ⋅ v = 0 (10) 22∫ ⎜⎟ dva a dt⎝⎠ dv The first term vanishes because δr = 0 at the endpoints. Since the second term vanishes for all variations δr , the rest of the integrand must vanish identically: dd⎛⎞T ⎜⎟v 2 = 0 (11) dt⎝⎠ dv Thus, v is a constant. The trajectory of a free particle is a straight line. The form of the function T (vv⋅ ) is determined by the requirements of Galilean relativity, which state that Hamilton’s principle must be equally valid in both the reference frames K ' and K . That is,

bb δδSdtdt''''0'''=⋅==+⋅+∫∫T'()vv δ T ⎣⎡ ()() v V v V⎦⎤ (12) aa

For this to be true, it is necessary that T ⎣⎡(vV''+⋅+) ( vV)⎦⎤ and T (vv''⋅ ) differ by at most the time of a function of the coordinates and the time, dtΛ(r ', ') TT⎡⎤()()()vV''-''+⋅+ vV vv ⋅= (13) ⎣⎦dt ' for in this case

bb dtΛ (r ', ') b δδδTT⎡⎤vV''-'''+⋅+ vV vv ⋅dt = dt '','0 =Λ r t = (14) ∫∫{}⎣⎦()()() ()a aadt ' since the variation of the coordinates vanishes at the endpoints. But dtΛ()r ', ' ∂Λ =∇'' Λ⋅v + (15) dt''∂ t so ∂Λ TT⎡⎤()()()vV''-''''+⋅+ vV vv ⋅=∇Λ⋅+ v (16) ⎣⎦ ∂t ' But T is independent of the coordinates and time, so ∇ 'Λ and ∂Λ∂/'t , which depend only on the coordinates and time, must be constants, and we get

TT[vv''2⋅+ Vv ⋅+⋅ ' VV] -( vv '' ⋅) = K12 ⋅+ v 'K (17) It is easily shown (by expanding in a series, for example), that this can be true only if the Lagrangian is 1 LT()vmvK=⋅=+ (vv ) 2 (18) 2 for some constants m and K . Since it disappears from the equations of motion when the variation is taken, we set K = 0 . We must determine the constant m by comparison with experiment.

1.3. Lagrangian for a Particle Interacting with a Field To describe the interaction of a particle with a field, we postulate a Lagrangian of the form 1 LU=−mv2 ()r, t . (19) 2 where the first term is just the Lagrangian of a free particle. The variation of the action is therefore bbdδr δδS=⋅ m∫ v dt −∫ U dt , (20) aadt But δ U =∇⋅δr , so upon integrating once by parts we get b b ⎛⎞dv δδSm=⋅vr − m +∇⋅ δ r dt =0 (21) a ∫⎜⎟U a ⎝⎠dt According to Hamilton’s principle the first term vanishes because δr = 0 at the endpoints. Then, since the integral must vanish for arbitrary variations δr , the rest of the integrand must vanish identically. We therefore obtain the equations of motion dv m = −∇U (22) dt Comparing this with experiment, we identify the constant m as the of the particle and U as the potential .

1.4. Invariance and

Figure 3 of a trajectory.

We return to (21) for a , which we may now write in the form

b b ⎛⎞dv δδSm=⋅vr − m +∇⋅ δ r dt =0 (23) a ∫⎜⎟U a ⎝⎠dt and consider the case of a translation of the entire trajectory by the constant amount δr = ε = constant (24) as illustrated in Figure 3. Provided that the potential U is under the translation δr = ε , the Lagrangian is unchanged. Therefore, the action is unchanged by the translation and δ S = 0 for this variation of the trajectory. But the result of the translation remains a valid trajectory, so the integral in (23) still vanishes identically. However, the variation δr is no longer zero at the endpoints. Therefore, we see from the first term in (23) that mmbmavvb =−= v0 (25) a () () That is, the quantity mv is conserved along the trajectory. We call those quantities that are conserved in a translationally invariant system the momenta, so the momentum must be p = mv (26) For a system of that attract and repel one another through central potentials, the Lagrangian has the form 112 LU=−∑∑mvii ij()rr i − j (27) iij22, where the factor of ½ appears in the second term because we have counted the interaction between each pair of particles twice. If the positions of all the particles are translated or rotated together, the Lagrangian is unchanged. Therefore, the total momentum and of a system of particles interacting according to (27) are conserved. We can also see this by looking at the of motion that we derive from (27), which is

ddpp∂L ∂Ukj kk−= +∑ =0 (28) dt∂∂rrki dt j The factor of one half has disappeared because in the sum over i two terms survive for ki= , these being Ukj and U jk= U kj . Summing over all particles we obtain

dp ∂U ∑∑k + kj = 0 (29) kkjdt ∂rk But ∂U ∂U kj=− kj (30) ∂rrkj∂ That is, the of interaction on the two particles are equal and opposite. Therefore, the sum vanishes by cancellation and the total canonical momentum of the interacting particles is conserved: dp ∑ k = 0 (31) k dt Note that this result depends on the fact that the interaction can be written in the form

(27), in which only the instantaneous positions appear in Uij . That is, the interaction is felt instantaneously by both particles, so every action has an equal and opposite , as represented by (30) and stated by Newton. This is valid only in nonrelativistic theory. In relativistic theory, the interaction propagates at the of light, and is not felt instantaneously by another particle. Therefore, the total momentum of the particles is not conserved. Instead, the total momentum of the particles and fields is conserved. Just as translational invariance is associated with linear momentum, is associated with angular momentum. For an of the position of a particle about the origin, the increment in the coordinates is linear in the of rotation and in the coordinates of the particle. The variation of the trajectory is therefore δrr= δω× (32) where δω is the rotation vector.

Figure 4 Rotation of a trajectory. When the trajectory is rotated as shown in Figure 4, the variation of the action is given by (23), as before. Substituting (32) for δr , we now get

b b ⎛⎞dv δδSm=⋅×vrω − m +∇⋅U δ r dt (33) ()a ∫⎜⎟ a ⎝⎠dt For a system which is rotationally invariant, the action is unaffected by this transformation, so δ S = 0 . But the trajectory remains valid, so the integral on the right still vanishes. Therefore, upon rearranging the triple product, we find that

bb m vr⋅×==⋅×δδωω0 rp (34) ()aa() That is, lr=×p is a constant of the motion. When the Lagrangian is invariant under a rotation in space, the angular momentum l is conserved. For a set of particles that interact through central potentials, as described by (27), the Lagrangian is invariant under a rotation of the positions of all the particles. Therefore, we can repeat the arguments used for linear momentum to show that the total angular momentum of all the particles is conserved. 2.

2.1. Canonical Momentum In the Lagrangian formulation of mechanics, the variables in the Lagrangian are the coordinates r and t , and the velocity vr= ddt/ , 1 LUL=−mv2 ()rrv,,, t = ( t ) (35) 2 The definition of the velocity vr= ddt/ is used explicitly in the variation of the trajectory to obtain the Euler-Lagrange equations of motion. But another approach is possible, called Hamiltonian mechanics. In the Hamiltonian formulation of mechanics we introduce the canonical momentum P and change from the variables r , v , and t to the variables r , P , and t , and we give the coordinates and momenta equal standing. The space spanned by the coordinates r and the momenta P is called the of a system, and the motion of a system is described by a trajectory in phase space with t acting as a parameter along the trajectory. For example, the nonrelativistic motion of a one-dimensional , for which Pp= , is an ellipse in the x - px phase , as shown in Figure 5, with time acting as a parameter. To change variables from ()rv,,t to ()r,P,t we use what is called a , and to make the new variables independent of one another we carry out the variation of the trajectory separately for r and P , ignoring the connection between position and momentum. The resulting equations of motion, called the canonical equations of motion, are simpler and more symmetric than the Lagrangian equations. However, there are now twice as many equations of motion since each is a first-order instead of second- order.

Figure 5 Phase space of a harmonic oscillator To begin, we must first define the canonical momentum P . For a Lagrangian L ()rv,,t the variation of the action is

b ⎛⎞∂∂LL δδδSdt=⋅+⋅∫⎜⎟rv (36) a ⎝⎠∂∂rv But the variation of the velocity is dδ r δ v = (37) dt Substituting this into (36) and integrating once, by parts, in the usual fashion, we find that

b ∂∂∂LLLb ⎛⎞d δδSdt=⋅rr − − ⋅ δ =0 (38) a ∫⎜⎟ ∂∂∂vvra ⎝⎠dt by Hamilton’s principle. But the variation δ r vanishes at the endpoints, and is arbitrary in between. Therefore, the first term vanishes, and from the second term we obtain the equations of motion d ∂LL∂ − = 0 (39) dt ∂∂vr These are known as the Euler-Lagrange equations, and they may be used to find the equations of motion for any Lagrangian L (rv,,t) . Returning to (38), we consider a translation δ r = ε of the trajectory. For a system that is translationally invariant we may use the arguments given previously to show that the quantity ∂L P = (40) ∂v is conserved. We call this the canonical momentum. It is a simple matter to confirm that for a particle in a potential Φ ()r,t , this definition agrees with the previous definition of the momentum of a particle. However, it is more general than that definition, which makes Hamiltonian (and, for that matter Lagrangian) mechanics so useful. In terms of the canonical momentum, the Euler-Lagrange equations of motion are expressed in the suggestive form dP ∂L − = 0 (41) dt ∂r When ∂∂=−∇=LU/ rF for some force F , (41) looks just like Newton’s equation of motion.

2.2. Legendre Transformation The change from the variables r , v and t to the variables r , P and t is accomplished by using a Legendre transformation. Legendre transformations are frequently used in thermodynamics to change from one set of variables to another. For example, a liquid or gas may be characterized by its p , volume V , and temperature T . From the definition of the S we see that the heat added to a system in a reversible process is dQ= TdS and the done by the system is dW= pdV . According to the First Law of Thermodynamics the change of the internal energy in a reversible process is dU=− dQ dW = TdS − pdV (42) This shows that the internal energy does not change in a process that occurs at constant entropy and volume. Therefore, the internal energy U is a function of the entropy S and volume V . For processes such as a change of state that take place at constant temperature and pressure, however, the Gibbs function is more useful. The Gibbs function is related to the internal energy by the Legendre transformation GU= +− pVTS (43) Since dG=+ dU pdV + Vdp − TdS − SdT = Vdp − SdT (44) we see that the Gibbs function is a function of the variables p and T . For example, if we boil water at constant pressure, the temperature remains constant as we add heat, but the steam expands at constant pressure. In this case the internal energy of the system increases, but the Gibbs function, which is a function of the variables p and T , remains constant as the water changes to steam. Returning to the equations of motion of a particle in a field, we see that to change the variables of the system from r , v , and t to r , P , and t we use a Legendre transformation and introduce the Hamiltonian function HL= Pv⋅− (45) Using the definition (40) of the canonical momentum, we find that ∂L ∂L dddddHL=⋅PvvP +⋅ − =⋅ vP − ⋅ ddt r − (46) ∂∂r t Evidently, the Hamiltonian is the desired function of r , P , and t .

2.3. Canonical Equations of Motion To derive the canonical equations of motion we begin, as before, with Hamilton’s principle,

bb δδSdt==⋅−=∫∫LH δ()Pv dt0 (47) aa Since the endpoints are fixed, we can take the variation inside the integral and get bbbddrrδ δδSdtdtdt=⋅∫∫∫PP +⋅ − δH =0 (48) aaadt dt But the variation of the Hamiltonian is ∂HH∂ δ H =⋅+⋅δδrP (49) ∂∂rP If we substitute this into (48) and integrate the second term by parts, remembering that the variation δ r vanishes at the endpoints, we find that the variation of the action is given by two terms,

bb ⎛⎞ddrP∂∂HH ⎛ ⎞ ∫∫⎜⎟−⋅δδPrdt − ⎜ +⋅= ⎟ dt 0 (50) aa⎝⎠dt∂∂Pr ⎝ dt ⎠ Now, in the Hamiltonian formulation the coordinates r and P of phase space are given equal standing, so the variations δ P and δ r are individually arbitrary. Therefore, the quantities in parentheses must individually vanish, and we arrive at the canonical equations of motion dr ∂H = (51) dt ∂P dP ∂H =− (52) dt ∂r We also see that the total of the Hamiltonian is dddH ∂∂HHrP ∂∂ HH =⋅+⋅+= (53) dt∂∂rP dt dt ∂∂ t t Therefore, unless the Hamiltonian is explicitly time dependent, it is a constant of the motion. Since it is conserved in a time-invariant system, the Hamiltonian must be the total energy, or something proportional to it. For a particle in a field we compute

⎛⎞1122 H =⋅−=PvLUUmmvmv vv ⋅−⎜⎟ − = + (54) ⎝⎠22 Note carefully, however, that when using the canonical equations of motion it is important that the Hamiltonian be expressed in terms of the coordinates and the canonical momenta, not in terms of the or, when they are different, the ordinary momenta. For example, p2 H = + U (55) 2m is the proper form of the Hamiltonian for a particle in a field.

2.4. Liouville’s theorem Hamiltonian mechanics is the natural framework for describing the motion of physical systems in phase space, and phase space is where we live in , , and the physics of particle beams. Fundamental to the discussion of systems in phase space is Liouville’s theorem.

We consider a system described by the phase-space coordinates ()Pqii, . For a single particle in one , the coordinates are ( x, px ) , and the phase space is shown in Figure 6. Within this phase space, we consider a volume V enclosed by the surface S . Inside this surface are a number N of systems. The surface S moves with

⎛⎞dqii dP the local v phase = ⎜⎟, . Since the of two particles cannot ⎝⎠dt dt cross in phase space, no particles can cross the surface S , and the number of particles inside remains constant.

Figure 6 Liouville’s theorem Interestingly, and importantly, as the surface S moves around in phase space, its volume V remains constant. To see this, we observe that the rate of change of the volume in phase space is dV =⋅vnˆdS (56) v∫ phase dt S where nˆ is a normal to the surface S . But by the divergence theorem, dV =⋅=∇⋅vnˆdS v dV (57) v∫∫phase phase phase dt SV where the divergence of the velocity in phase space is  ⎛⎞⎛∂∂qPii ∂∂HH ∂∂ ⎞ ∇⋅phasev phase =∑∑⎜⎟⎜ + = − ⎟ =0 (58) ii⎝⎠⎝∂qPii ∂ ∂∂ qppq iiii ∂∂ ⎠ when we use the canonical equations of motion. Therefore, we see that dV = 0 (59) dt which is called Poincare’s theorem. But since this is true for an arbitrarily small volume, and the number of points in the volume remains constant, we see that the ρ of points in phase space is a constant of the motion. That is, dρ = 0 (60) dt This is called Liouville’s theorem. Note that it is derived from the canonical equations of motion for a conservative system. It doesn’t apply when there are dissipative forces, such as or drag. For example, for a harmonic oscillator with , the volume of phase space that encloses any set of particles will eventually collapse to a point at the origin. All the particles will eventually come to rest.

Figure 7 Particle beams in real space, viewed at the indicated sections. We have already seen the usefulness of the phase plane for the theory of chaotic motion. To illustrate the use of Liouville’s theorem to describe particle beams, we consider the beam illustrated in Figure 7 and Figure 8. In Figure 7, both beams are presumed to have uniform, round distributions in transverse position and transverse momentum. Clearly, however, the beams are different; one is at a rough focus, while the other is emerging from a sharp focus. The differences are much more apparent in Figure 8. In the first beam, the momentum and position in the x direction (likewise the y direction) is uncorrelated, but in the second case the x -momentum is proportional to the x -position and the distribution in phase space is very different. In the case of a perfectly focused beam the occupied by the beam vanishes.

Figure 8 Particle beams in phase space, viewed at the indicated section.

The usefulness of the phase-space picture is shown in Figure 9. As the beam passes through the focus, the area in phase space occupied by the beam shears in the horizontal direction as the trajectories take the particles to new values of y , keeping py constant. From Liouville’s theorem we know that the area of the distribution remains constant during this evolution. At the focus, the ellipse circumscribing the particles is erect, and the width in the y -direction is a minimum. In fact, if we pass the beam through a focusing , we shear the beam vertically (the momenta py change but, at least for a thin lens, the vertical positions y are almost unchanged), but the area remains constant. As the beam drifts toward the focus, the distribution shears in the horizontal direction (the positions change, but the momenta are constant). When the beam reaches a focus, the ellipse is again erect, and the width in the y -direction is a minimum. Since the area is a constant, we can find the width in the y -direction by knowing the width in the

py -direction. This is established by the lens, which determines the shear in the vertical

( py ) direction in the phase plane. We can obtain a narrower focus by extending the distribution in the vertical direction, or we can collimate the beam (narrow the distribution in the py -direction), but only by spreading out the beam in the y -direction to keep the area constant.

Figure 9 Particle-beam evolution through a focus, viewed at the indicated sections. 3. VELOCITY-DEPENDENT POTENTIALS

3.1. Thus far, we have restricted our attention to forces that depend only on the position of the particle and not on its motion. This excludes magnetic forces on charged particles and friction and other dissipative forces in more complex systems. We shall not discuss friction, here, but magnetic forces are easy to include in the theory. To do this, we postulate a Lagrangian of the form 1 LU=−mv2 ()rv,, t (61) 2 where U need not be an energy, just an interaction term with the units of energy. We assume further that the potential U is linear in the velocity. If this doesn’t agree with experiment, we can correct it later. Since the Lagrangian is a , the vector velocity can appear only in a scalar product, so the Lagrangian must have the form 1 L ()rv,,tmvqtqt=−Φ+⋅2 () r , vAr () , (62) 2 where q is a parameter that we will later identify with the charge on the particle, Φ is called the , and A is called the magnetic vector potential. For motion in the x -direction, the Euler-Lagrange equation of motion (39) is d ∂LL∂ − = 0 (63) dt∂∂ vx x which becomes d ∂Φ∂A ()mv++ qA q −⋅= qv 0 (64) dtxx ∂∂ x x But the derivatives appearing here are dA∂∂∂∂ A A A A x =+++vvvxxxx (65) dtxyz∂∂∂∂ x y z t

∂A ∂A ∂A ∂A v ⋅=vvvx +y + z (66) ∂∂∂∂x xyzxxx If we substitute these into (64) and rearrange the terms, we get ⎡∂A ⎤ dvxxxx∂Φ ∂∂∂ A⎛⎞y A⎛⎞ A ∂Az mq=−⎢⎥ − + vyz⎜⎟ − − v⎜⎟ − (67) dt⎣⎦∂∂ x t⎝⎠ ∂∂ x y⎝⎠ ∂∂ z x If we define the electric and magnetic fields by ∂A E =−∇Φ− (68) ∂t BA= ∇× (69) then the equation of motion becomes dv mq= ()EvB+× (70) dt which is just the law for a particle in an . We see now that for a force linear in the velocity, this is the only possible equation of motion.

3.2. Hamiltonian Mechanics For a Lagrangian of the form 1 L ()rv,,tmvqtqt=−Φ+⋅2 () r , vAr () , (71) 2 the canonical momentum defined earlier by (40) is ∂L PvA==+mq (72) ∂v The Hamiltonian defined by (45) is then 1 HL=⋅−=Pv mv2 +Φ q (73) 2 which is the total energy of the particle. However, the Hamiltonian is correctly expressed in terms of the canonical momentum, not the velocity, so we write PA− q v = (74) m and the Hamiltonian becomes ()PA− q 2 H =+Φq (75) 2m The canonical equations of motion are now dqrPAp∂−H == ==v (76) dt∂P 2 m m

2 dP ∂∂H ()PA− q =− =− −q ∇Φ (77) dt∂∂rr2 m Using the vector identity ∇()ab ⋅ = a × ( ∇× b ) + b ×( ∇× a) +( a ⋅∇) b +( b ⋅∇) a (78) and the definitions (68) and (69) of the electric and magnetic fields, the second equation becomes dp = q()EvB+× (79) dt as before. Ho hum. Hamiltonian mechanics is useful for formal developments, such as Liouville’s theorem, but the canonical momentum itself is frequently useful for solving problems. For example, for a plane electromagnetic the field is necessarily to the direction of propagation (to satisfy Gauss’s law), and has the form

AA= ⊥ (,)x t (80) where A⊥ is independent of the coordinates in the y and z directions. We can ignore the potential Φ . Since the Lagrangian (or the Hamiltonian, for that matter) is invariant in the y and z directions, the transverse components of the canonical momentum are conserved. Thus, the first integral of the motion is immediately

Pp⊥⊥=+q A ⊥ =constant (81) and the transverse momentum is

pA⊥⊥=−q +constant (82) where the constant is the momentum of the particle before the wave arrives.

Figure 10 Vector potential in a uniform magnetic field As another example, we consider Larmor’s theorem for the behavior of in a magnetic field. For a uniform magnetic field B , the magnetic vector potential may be expressed in the form 1 ABr= × (83) 2 This consists of circles about the z axis, as shown in Figure 10. The Lagrangian

1 2 L =−Φ+⋅∑∑∑mvii q()rvAr,, t q () i t (84) iii2 consists of a central potential about the nucleus, plus the interaction of the with the nucleus and each other, represented by Φ , and the magnetic vector potential (83). Since all the terms in (84) are invariant under about the magnetic field, the total canonical angular momentum about the direction of the magnetic field is conserved:

∑∑rPii×= r i ×( p i −q A i) =constant (85) ii where the constant is the angular momentum of the in the absence of the magnetic field. Thus, the field causes the atom to rotate about the direction of the magnetic field. To see this, we change to a new rotating at the Ω about an axis parallel to the field B . The velocity of the particles in the new frame of reference is

vv''ii= −×Ω r i. (86) If we substitute this into (84) we get

N ⎡⎤112 L =+×⋅+×⋅−Φ∑∑⎣⎦22mv''''',iiiiii m()Ω rv q() Brv q () r t , (87) i=1 i to lowest order in Ω and B . If we choose the frequency of rotation to be qB Ω =− (88) L 2m the effect of the magnetic field vanishes in the rotating coordinate system. We therefore see that the motion of charges in a magnetic field is just the motion in the absence of the magnetic field plus a rotation of the entire system at the Larmor frequency ΩL . This is

Larmor's theorem. Note that the Larmor frequency ΩL is half the cyclotron frequency of the particles in the same magnetic field, and the direction of the rotation vector ΩL is opposite the direction of the magnetic field B for positive charges.