The Charge Distribution Function in Perturbation Theory of Classical Electrodynamics, Lecture notes of Electromagnetism and Electromagnetic Fields Theory

Download The Charge Distribution Function in Perturbation Theory of Classical Electrodynamics and more Lecture notes Electromagnetism and Electromagnetic Fields Theory in PDF only on Docsity! SLAC-PUB-441 June 1968 THE CHARGE DISTRIBUTION FUNCTION IN PERTURBATION THEORY OF CLASSICAL ELECTRODYNAMICS* E. L. Chu Stanford Linear Accelerator Center Stanford University, Stanford, California * Work supported by the U.S. Atomic Energy Commission. THE CHARGE DISTRIBUTION FUNCTION IN PERTURBATION TIiEORY OF CLASSICAL ELECTRODYNAhIICS* E. L. Chu Stanford Linear Accelemtor Center Stanford Universi.ty, Stmford, Ca!ifornizi I. IXTRODUCTION The theory of classical electrody-namics is formally complete. There is avail- able an abundance of mathematical methods in this field. Yet, it is by no means an easy task to solve a specific problem, because new problems are ever increasing in complexity. In attempt.&, 0’ to obtaain useful results, efficiently and accurately, one is often bewildered by his freedom of choice between several usable approaches. It seems advisable to study the salient features of different methods and the rela- tions that exist between them. For a given charge-current distribution and known boundary conditions, . Maxwell’s equations determine the electromagnetic field uniquely. Haking deter- mined the field intensities, one may obtain the motion of the charged particles from the Lorentz equation and certain initial conditions. This knowledge of the particle motion, in turn, determines the charge-current distribution. If the latter $-current (k P) were correct, the ?&xwell field (E, B) obtained from it woul$ be ya &?a$ the same field as was used in the corentz equation which yielded the same 4 -current. Instead of solving directly the Lorentz equation in 3-dimensional space, one may solve the so-called Bol tzmann-Vlasov equation I,2 fGr t.he charge distribu- tion function in 6-dimensiopal phase spcccc. The latter equation is obtained by substituting the Lorentz equation into the Liouville equation (also known as the collisionless Boltzmann equation). From the charge distribution function, the 4-current vector is determined rcac1il.y. The use of Mxwellts equations for de- termining the electromagnetic fie1.d from the 4-current is the same in both methods, and, in this paper, these equations wiil be assumed to have been solved whenever they need be. - 1 - The Lorentz equation, ‘which we are discussing, is the microscopic Lorentz equation. The other Lorentz equation, macroscopic, is a consequence of the Boltzmann-Vlasov equation. 3 To compare the microscopic Lorentz equation with the Boltzmann-Vlasov equation is, in effect, to compare the two forms of the Lore&z equat.ion. Hereafter, unless explicitly stated to be otherwise, the Lorentz equation is.meant to be the microscopic equation. The electromagnetic fields ( E, B) which appear in the Lorentz equation con- j mm sist of the fields applied externally and the fields induced by the electron beam itself. The induced fields are, supposedly, the microscopic fields; the charge- and the current-density of the beam are the sum of Dirac delta functions repr?senting the contribution of individual point charges, and In the Boltzma.nn-Vlnsov equation one also uses the microscopic particle- density and the microscopic fields, so that this equation and the set of Maxwell equ.at.ions constitute a closed system of eqnations for the beam-field problem. 4 This system of microscopic equation s may then be averaged stat.istically. The resulting system is not 3 closed one, unless the correlation effects arisirrg from the random parts of the fields are taken to be vanishingly small. This assumption is implied in Ref. 3 and will also be used in this paper, In other words, the micro- scopic fields will. be assumed to be no different from the corresponding macro- scopic fields; only the kinetic quantities (velocity, momentum etc. ) may have random flue tua tions. The equivalence between the Lorentz equation and the Boltzmann-Vlasov equa- tion has been discussed by Watson5 and Chandrasekhar, 6 Each spatial or momen- tum variable x1 of a particle may be exprc Jssed as a function of time t and the six integration constants o! k required for specifying the initial conditions. If these . k solutions x1 = x1 (CY , t) of the Lorentz equation are substituted into the expression of the charge distribution function 13 (x1, t) and if $ is such a function that, after the substitution, $ becomes a function of ak only, then d$/dt = 0, which is the . Boltzmann-Vlasov equation. The fact that I+!I should be totally independent of t in the absence of collisions is well-known, and is usually proved by using the Liouville theorem. 7 This simple discussion seems to convey the thought that, if there are numerous particles in a system, it would be very hard, if not impossible, to solve the Lorentz equation because of the great number of intcgra~tion constants. This thought, hov;ever, is often questionable. The well-known work of Pierce 8 on electron-beam tubes con- tains many examples of the judicious solution of the Lore&z equation. In this paper a new proof of the equivalence betJA;cen the two methods will be given, and their relative merits will be discussed. Let 5 ( r, t) be tile Lorentz force acting on a charged particle of charge 2 , A. rest mass m, and relativistic mass ST.‘, -- (1. la) Let pK (T, t) be the kinet,ic momentum of this particle, ^S pI’ = myu . #e-” \ C,% (1. lb) The Lorentz equa.tion is simply (1.2) -3- To obtain the corresponding form of the Lo’rer;iz equation we apply the Taylor operator on both sides of Eq. (1. 2). Since and dt ZZ -2 at + u(r’, t v+.% dp i.e., :“Ii = dt c fi + UoPy) * w. (1. lla) (1. lib) the resulting equation is the Lorentz equation in the Lagrangian approqch: (1.12) This equation may then be separated into different orders quite easily. The first- order equation is as given by Eq. (7. 1.0) in Section VII. This brie1 revieiv of the perturbation theory serves to shorn the effective use of the Taylor and Lagrange operators. Lrl succeeding sections we will use these operators in 6-dimensioml phase space to discuss the perturbation solution of the Boltzmann-Vlasov eyuaiion. We choose 6-space over 8-space, l3 7 l4 2 l5 becFLIse L 6-space is used in the majority of references, 16,17 iiElUding Hamilf,on’s early work. The use of 6-space does not hamper the relativistic treatmeilt. No new rnathernat- ical procedure I. ‘3 involved in ac’xplitig txo more diri:enSio!X , time nnd eneqgy. Speaking in general , either the cova~innt- or the con~a~variant-momentuin components may be used as the non-spatial coordinates in a phase space. We will ’ - 6 _ components are the so-called conjugate variables. It can readily be shown that, when these variables are the coordinates, the determinant of the 6 ‘x 6 metric tensor of the phase space is unity, regardles s of the spatial coordinate system, whether it be Cartesian or curvilinear. When a certain canonical transformation is applied to the phase-space coordinates, the two groups of new coordinates remain conjugate to each other; but no new coordinate.may retain its purely spatial or purely non- 16,127 spatial character. The determinant of the new metric tensor remains unchanged, 19 because the Jacobian of any canonical transformation of coordinates is unity. In the following discussion, this property of the metric tensor is not needed. I II. THE DISPLACEMENT VECTOR AX33 COORDINATE TRANSFOI>LIVL~TIONS Consider at the same time two simple systems of charged particles having one-to-one correspondence. One system is in the unperturbed state and the other in the perturbed state. An unperturbed particle is located at time t at the point 7 in the phase space, having coordinates x1 en referred to a given’ 6-dimensional coordinate system. The corresponding perturbed particle is located at the same time, t’ = t, at the point ?:‘, having coordinates x’l referred to the same co- ..“+I ordinate’system. These two phase points are related by the displacement vector as ‘follows: This may, alternatively, be given by x =x1+ J $(xk, t> * (2. lb) Here, t1 ‘is the difference between the two i-th coordinates. The displacement vector ?( Y, t ) is defined by the set of components i k *A aa- < (x , t.), and vice versa. . . . -. In a cartesian coordinate system, cl= el; in curvilinear coordinate systems, cl+ - t1 and {’ . . may not be the components of a vector. Howxer , {id51 in any reference system when <l-+ 0. .- The first three coordinates (xl, x2, x3) npy be desigxatetl to be the spatial coordinates, and the other three (x4, x5, x6) the momentuin coordinat~es, either canonical- or kinetic -momentum as the case may be. T!ILW, the first three com- ponents of any vector are the spatial components, and the other three the non - spatial coinponents. l?ach component of T is some function of the unperturbed -I+* coordinate- and nlome;ltmn-~~ariablc s and the ti.me. The three components of F *zA later. \ -6 - III. CHARGE DISTRIBUTION FUNCTIONS Let $. and II, decote, respectively, the utiperturbed and the perturbed charge . distribution function. A certain number of identical charged particles in a 6-space phase element dx1dx2 . . m dx6 bear an electric charge dq . The charge distri- bution function is defin.ed to be the ckrge density per unit phase element, Thus, in the unperturbed state, we have dq = $o(xi, t )&lx)’ . (3.1) Hese, (dx16 E d&x2 . . 0 dx6 . Ln the perturbed state, these same particles in the corresponding phase element must have the same amount of charge. It is assumed that there occurs no collision between particles. Thus, we also have dq = J,(xfi, t) (ck-@ . (3.2) Since i.e. , , , J@(x”, t) = (cb(x15 t ) , (3.3) we obtain from Eqs. (2. 7) and (3. 3) JC&( x1, t) = c,bo(Xi, t) , or simp: y , (334) c’i(xi, t) = G $o(x1’ t) , (3,5) beca.use Q J Z: =: 1. In other words, the perturbed charge distribution function may be represczted in general as the, result of operating on the u.nperturbecl dis - tribution function with the Lagrange operato?. Once + is represented in this form, it becomes a rOl:tj1lCJ 1Ilatier to sejj?!r?+ te 5; into diflerent orders of ma~~~itucle. For esample, usirLg Eq. (2. 9) and sep::r?->iEg {I into diflererlt orders, t1 = ti +tk i . . . , - 1: - we obtain (3.6a) (3.6b) etc. Equation (3.5) is the consequence of Eq. (3.3). In tensor language, the charge distribution function is a scalar density, not an absolute but a relative scalar of ~- 20 weight one. Scalar densities of all kinds behave in the same manner. . IV. THE EQUATION OF CONTINUITY Let us consider the time development of the charge distribution function. As the time increases from t to t’, a certain phase point moves ‘from x1 to x ” and a small phase element around this point changes from (TV to 6V’. In the absence of coll.isions , the number of charged‘particles under consideration remains the same. Thus ) J t&x, t)(dx)’ = s f$(x, t’)(dxf . 6V 6V’ = J 4(x’, t’)(dif . (411) 6V’ The last integral over 6V’ may be changed by a transformation of variables to an integral over 6V. Then Eq. (4.1) becomes Here, t is considered fixed and t’ 2 t + 6t an auxilizry parameter. The equations of transformation of variabl.es may be written as (4.3) which must exist because the solution to the initial-value problem of the physical sys tern exists. The Jacobian 1 ax'/a s i also contains t’ as a parameter. Since 6V is arbitrary, we must have +!I(x’~, t’) / ax’/ ax 1 = l!,(xi, t )’ . (4.4) From this equation, it then follows that - 13 - v . THE BOLT Z>L%NN-,VL4SOV EQUATION We differentiate Eq. (3.4) with respect to time. Since , -&nb= -$+k .i tk a ----&- ) $(xfi, t) = 22 -& ‘Ld, t) , ax Eq. (5.1) is the same as d'/1O 1 dJ d + -= ;~.G$~+JC~. (5.2) dt Operating on this equation with the Lagrange operator &? used in Section III and using S2 J 2 = 1, we obtain d+ = R “‘b. “0 dJ dt - - --i- dt ’ dt (5.3) The unperturbed state is supposedly a physically realizable state. Hence, in the absence of collisions, d4”o -z 0 (5.4) d t. according to the Liouville theorem. Thus, the first term on the right of Eq. (5.3 ) vanishes. If the perturbed state is also physically realizable, the last term of Eq. (5. 3) must also vanish. In other words, dJ o -ZZ dt should also follow from the principles o f dynamics. While this is obvious in view of the Liouville theorem : we IAll , neverthe? ~ss , gi-ve an explicit proof of Eq. (5. 5) to elucidate its inner contents , which arc iilost essential for the discussion of *3ur subject. - 16 - In this connection, it is more convenient to consider the transformation of co- ordinates in 7- rather than 6-space. ‘\5Te re-write Eq. (2. lb) as follows: Ii X = x1 + $($) , (5.6) Here, i and k range from 1 to 7 ; x 17 = t’; X7 = t; ‘7 6 = 0. This set of equations (5. 6) consists of the same six equations in the set (2. lb) and the 7th equation t’ = t . The Jacobian of ihhc transformation in 7-space is the same as the one in 6-space, because at’ i = 1, 2, . . . , 6. y= ax1 i at at = 1, i = 7. IJet g.. 1J denote the covariant element of the metric tensor of the x-coordinate sys tcm and kij the corresponding metric element of the x1-system. Then, ax1 .axJ -I_ ---- $ld 7 aI;‘l’ axrE gij * Taking the determinants of the matrices on both sides of this equation, we obtain where Hence, and 1 dJ LE= In 7-space, d/d t = x’ (a/ax’) in the x-system and d/d t = x”(a/ax”) in the x*-system. Thus, Eq. (5.5) may be nritten as ., (5.9) . - 17 - The first two terms on the right of this equation cancel each other, because both terms are equal to the absolute divergence of the 7-vector dg/dt . The vector dZ is an infinitesimal, directed, line element i.n the 7-space. w dz = zidx’ = aidx” : The time derivative of this. vector is’ dg/dt = ii? = ii;” . w+. Here, z. and ki are the covariant base vectors of the two coordinate systems. lNa1 WA The absolute divergences of the same vector dz/d t in two different coordinate systems are the same. Because of this, Eq. (5.9) becomes : - ,i 1 dJ ax aK’ 5 dt= --. - -- . axll ax’ (5.10) Since at'/at' ' = a t/at = 0, Eq. (5. 1.0) is valid in 7-space as well as in 6-SpaCe. Hereafter, the Latin indices will again be considered to range from 1 to 6. To corkinue our proof, it is helpful to recognize that the six coordinate variables c are three pairs of conjugate variables. Let x , CY twanging from 1 to 3, denote the three spatial (contravariant) coordinates. Let x 3-CCY denote the three momentum coordinates, canonical or kinetic as the case may be, x 3+cr = pQ! . Thus, a&’ akc ' ab, -~ = - + -.- . ax1 axQ ap, (5.11) Each pair of coordinates ( xQ, p,) or ( xQ, x 3+cY ) are conjugate variables. In Eq. (5. ll.), the Go! are the three velocity components and the b, the three force components (in the generalized sense) pertaining to a certain particle in unperturtled state. According to the canonkal equations of I-l’amilton, we have ‘C x = aJio/ap, - 1s - , the (5.12a) VI. FURTHER DISCUSSION OF TfiE DISPIL~CCEXIENT VECTOR To evaluate (l/J)(dJ/d t) in terms of the displacement vector, we first differ- entiate Eq. (-, 7 lb) with cl/d t, then with a/ax: rt , and then ipply cont.rac tion by putting Q = i. Thus, ai’l axk 8 . . +=-..-.---- 3Xf1 a x’l axk (2-t,+ . According to Eq. (5. 10) , where 6k i = axk/axl is the,Eironecker delta, Since, d a{' ai' a$ -- axk --- + - - = ;i% axk axk ax” ’ Eq. (6.2) becomes: d ati -- dt axk l The first term on the right of this cquatl’on vanishes, because ax*’ -. - 6; 4. -,e -2: = -__ - -__ --;- 27 0. ax’l ax ax ax" ax" ax1 Considering the x-coordinate system to be Cartesian, v:e may easily write Ey. (6.3) in its vector form, namely, (6.1) (6.2) (S.&j i.e. , (6: 4b) In these equations, the double dot product is performed as in the equation 2s WUYA :za~ = (2. zJ(E- ,g ; cl = ZiFi = %jZ! is the unity dyadic. Both Eq. (6.3) M S++k and Eqs. (6.4a) or (6.4b) are valid h any coordinate system. Equation (6.4b) can easily be separated into different orders of magnitude. The first two terms are as follows : (6.5a) To satisfy the condition [(l/J) dJ/d t], = 0 so that (d$/d t)l = 0, we must require that L6.6) L identically. Thus v” * f-1 must be a constant or zero. were a. non- tl,‘.’ k\ If F * y1 I U. vanishing constL%nt , then according to Eq. (3. 6s) @I would contain a sn~all constmt fraction of $/o equal to (a $/ax’) o. . This part is certain!y of no interest in the perturbation problem and may be excluded from consideration. Therefore, we may simply require., instead of Eq. (6.6), If this is satisfied, then the Jacobian J is equal to unity, to the first order. the set of Eqs. (2. lb) into its two Parts: lo! X = x*+ (” ; (6.8a) P:, = P, + xcr , These yield, on differentiation with d/d t , . (6.8b) to! ‘cl! U = 3 “+5 ; (6.9a) (6,9b) In Eq. (6.9a), u: = &” and u’~ = il(y are, respectively, the unperturbed and the corresponding perturbed Particle velocity, ,~ _/.^I pv The quantities ut andu@ are vector components in any coordinate system but [u are z&t. In Eq. (6.9b), foa!= l p, m,j f; +$f iJ ,-- The quantities f oa!, fh , ancJkU . .~- - --.-. ._ .” __ ., _a- .-- ’ 1.17 are not vector components except in a. Cartesian coordinate system. Later, when no ambiguity may arise, we will also use the notation fai =Ga for the time derivative of the momentum of a perturbed particle at the point (x, p, t) . Let g QP be the ‘metric tensor elements of the S-space evaluated at the point (;,I and g& the metric elements eval,uated at the point (r’), ’ Biti “CYp Then uoh P = gap “0 and II’ = ’ 1 U’F a “@ - By defir,ition, 52’ = mYO uoo! +en ; OCY (6.10~~) Pt, = my’ U’ + eA’ cz .a!’ Subst,ikting these relations in Eq. (6. Sh) we obtain (6. lob) Equations (6. Sa) and (6. Sb) have their corresponding 3-dimensiona vector forms , namely, (6.12aj - 23 - VII. THE I,ORE?IT% EQUATION The vector u’ is the perturbed velocity of a certain partjcle at the phase point (x’, p’) and the time t’ = t . The vector 2;’ is the perturbed force (canonical or kinetic) on this particle at the same phase point and the same time. Thus, Here, the Taylor operator Z is defined in the G-space ( xa!, p,) by Eq. (2. 6). Similarly, Using these relations we obtain from Eq. (6. 13a) and Eq. (6.13b) (7.2a) Equation (7.2a) serves to define the different orders of the velocity vector, (7.3a) (7.3b) etc. Similarly, Eq. (7.2h) defines the different orders of the vector F and, in ,v+ conjunction wit’i? Eq. (6. 151, repr L,~LI. *-Q -ts the Lorentz ecyuxtioil of motion:. Since Eq, (7.2b) becomes The 3-vectors F and d(ez/dt are evaluated in the usual manner. We have %#+A These yield d eA= $& - dt f&t, a% w &a Using this and Eq. (7.2a), we obtain from Eq. (7.5) (7; 6) The scalar product of this vector equation and the vector (J;‘~ + { ) is quite .s**’ simple. This is V-8) The Lorentz equation GI its present form may easily be separated into different orders, if y’ may be approximated by yO . In any case, * ziO << 1, and alternatively y’ = 2 2 I+1 c )I l/2 @a. . + . . . * 1 . . . 1 I (7.9a) , (7.9b) - 27 - Also, (7. SC) The first-order parts of Eqs. (7. 7) and (7.8) are as follows: (7.10) (7.11) These equations are, as they sl~ould be, precisel, ~7 the same as obtained before, 10 In Ref. 10, y’ is represented by Eq. (7.9a). When c Id satisfies Eq. ‘47. 10) and VI is obtained from Eq. (6. 15), the pair of Hamilton’s equations &!A CY U 1 = (x01)1 = a,?l,/apcl and cEajl = - a&(/axcu will also be satisfied.. Then, as proved eazlier, [ (l/J)(dJ/d t)] 1 will vanish. This ma.y also be shown directly by evaluating (d/d t) (f - r .) . This is h.1 i.e., where u se1 is as given by Eq. (7.3~) and $‘I is obtained from Eq., (7.2b), p, Sine e Therefore, (d r;;/ci t)l = !I . ’ - 2s _ Thus, i. e. , u(x, P, t> = ;,O’“, P, t) , jay, because LSY - $Vo = eV - eV0 and this is independent of p . In kinetic phase space, u = u = . . . . = 0. +A *it& w-%2 In this respect, the velocity components ua’ or uQL behave like coordinate variables x CY and p CY ’ and the kinetic phase space (xcy , p,) is akin to the velocity phase space ( xQ, ucr) . From Eq. (7.2a) we obtain, in view of Eq. (8.8), Thus, Eq. (6.16) becomes The Lorentz equation (7. 7) or (S. 6) mtry,then be written as (8.9) is. 10) (8.11) The first-order part of this equation is, a.s it should be, the same as Eq, (7. 10). In canonical phase space (p = pK +- e 4) , $fit, /: 9, c”,‘:! a. ( aillo a' aX$ a a% a XX - .1. --- ---- _ -_ .---^ \ ---4 -.--. at 3 ‘t; cy as c2 CY 3x ap, / $Jl. 1.. ( apcL ax cc ai?/ 1 a -.-_-- --. a2 ap, ) tie. (8.12) In kinetic phase space (p = p,) , fit M a a a a wo --fU - 0 +f - axa! CYO c’il + f al - , at a PO! . ap, where f = CYO and . f = C21 ( ) p, 1 = zQ, - e l3E + u X J& ( ,SO > are the first two orders of As discussed earlier, (d+/d t)l = 0 if $l is as given by Eq. (3.6a .), i.e. , (8.13) (8.14a) (8.14b) (8.15) and cl satisfies the condition required by the Liouville theorem [(l/J)(dJ/dtill = 0 . - 32 - IX AN EXAMPLE : MULTIPOLE OSCILLATIONS OF A BUNCHED BEAM Lee, Mills, and Morton 22 have described a self -cons is tent solution of multipole oscillations of a bunched electron beam in connection with their work on storage- ring beam instabilities. The bunched beam is circular in cross section with radius , a - and travels along the axial z-direction with a. constant velocity. For our present purpose, the beam may be assumed to be enclosed in a circular waveguide of radius b I , b >- a, and of infinite length in the z-direction. In the unperturbed state, the distribution of the charged particles in the beam is supposed to be axially symmetric, and the particles execute simple harmonic motion of small amplitudes ,in radial directions with a certain characteristic frequency w. determined by the electro- magnetic field acting on the beam. The transverse momentum of any particle is assumed to be negligibly small in comparison with its longitudinal momentum.Their problem is to determine whether such a beam may become unstable with respect to transverse oscillations of multipole symmetry. We will first describe their formu- lation of the unperturbed problem in kinetic phase-space, and then use a specific displacement vector to derive +l from which the charge and current densities, 4 and il , are obtained, As discussed in previous sections, if the displacement lw vector satisfies U1.e Lorentz equation, then IJ!J 1 satisfies the Boltzmann-Vlasov equation, and vice versa. The unperturbed state is characterized by the Hamiltonian ,::.:To and the charge distribution function tie. 0 ,3 - Now we consider a.n oscillatory perturbation of certain 2P-pole symmetry represented by the following displacement vector in s-space : & = E e-jut bK K cos 0 c$ - $& (2/l) sin P $I. ] . (9.9) In this equation, E denotes a small dimensionless parameter (E << 1) and w an unknown frequency to be determined: (h’ , $I, z) are the usual cylindrical coordi- &&es, and Es ‘.A%$) n+a e ) the three covariant base vectors. It is to be noted that t iwd is independent of the momentum variables and 2 . iJ = 0 . Taking the time-derivative of Eq. (9. 9) and denoting E e -W _ r , we obtain (9.10) The mom en turn -part Tla! of the displacement vector < does not contribute fi* to the charge density p . This may be seen easily by integrating the function fi over the momentum space to obta-in P . p(r, t) = 1 $0, P, WW3 = *ti 1 CtiO(x, p, t)(dp)3 = po !;; t) + 1 (Q - I)dJ, (x, p, t) (dp)3 . The p;?rt of the integra~rl ( Q - l)Go, which co8t.Gns ‘70, is a complete divergence in the momentum space. The volume integral of a divergence expression may bc tr:l.nsformed to a surface integral and therefore vanishes because tie vanishes on the surface at I 1 p =m. jyy Despite its null effect on p , the vector q< is required for the representation of ti. According to Eel. (S. lo), i.e. , (9. lla) i.e. , The secorxl term on the left of Eq. (9. lib).is negligible if (9. llb) . Under this condition, rl m.4 = my0 fJ, i.e. , w jwbl K COS .L!$ -!- PK COS j$ 3 sin 4!$ K 3 + e (3 2 2 jwiV K j- SiIlPQ - p+ cos mg - KPK ; sin J!$ . (9-. 12) *r\ The momentum-part of v” * -i is i, $) a a -- . q = --- r7 M.1 lo! * ap . p*v* bP, (9.13) This eqktion applies to any spatial coordinate system. From Eq. (9.1.2) we obtain a ap -2. * ,?ll = 0 . +'A~- Since Z %? $11 is independent of p &VI ’ Therefore, and [(l/J) (dJ/d t)] 1 - 0, whatever the unl;llown frequency u is. -37 - This result may easily be verified by transfqrming Z, and rl to refer to F-7 /4r4 Cartesian coordinates. We may also evaluate the divergence of the 6-vector Fl WA directly. The latter is -( e . f$;4 - j C&I K CbS E(l f PK COS !@ - II sin 1$~ K + )I . (9.14) Thus, because the B-dimensional metric determinant z is unity. Here, we may note that t :+a + f]Io except in Cartesian coordinates. The first-order distribution function is given in terms of f /w<J and ZI as follows: (9.15) Since dtio/cit = 0 and e. T= 0, this $I a*.\ A? 1 satisfies the Liouville equation I”. (d $/cl Ql = 0 which, however, should net be confused with the I?oltzmann-J’lasov equation. When and only when the Lorentz ec~~~ ati. is satisfied, will these t!vo equations be iclentic‘al. Both equations are given by Eq, (8. IS), but they diffe inthe coefficient f al’ i In the Bolt,zmann-Vlaso\r equati.on this coefficient is givcnby IQ. (8. 1Gb) ; in the ot’ner f and F 1 is given by Eel. ('7. 13) . ,9--ei- This expression of AI is the same as obtained by direct integration of the integral J $X0 $1 (dP)3 > in which $I is as given by Eq. (9.16) and uO is re- WA placed by p/&I . w Both Ey. (9.19) and Eq. (9.22) agree with the corresponding expressions considered by Lee et al. 22 - - The quadrupole case is particularly simple. When I P = 2, wehave (9.23a) *y1 = q$xPx - +a,~,) - @MA , (9.23b ) .til. .=L? eN/2r 2 a “> f (z - vot)d (P, - PzO) l .- [Mm2 (p% - pz) - o:( x2 - y”) + jwM-l(xpx - ypy)l * 4’ 1 + 1 2 Mu2 0 ( x+Y 2 2 -a 2 )I 2 (9.23c) P1 = ? (eN/m) f (z -vOt) cos 29 a(a - K) , / (9.23d) and i =- til j w PO $ + ~kzP1vo . n (9.2%) As discussed earlier, from PI and & one obtains I& and B . *,la The un- known complex frequency w, which determines the stability of the multipole. oscillation, may then be obtained by two alternative procedures, one using the Boltzmann-Vlasov equation and the other the Loreilk equation. In most problems there are some simplifying approximations. For the present example, e(;o -‘+.zo X ,BO)I 4 I, The two results obtained by these alternative methods shoulcl agree with each other to the desired degree of accuracy. If not, one or the other calculation may be im- proved upon, Whiche\7CY is more cor1vcnic3lt. - 41 - X. SUMMARY AND DISCUSSION The perturbed distribution function $(x, p, t) may always be represented by G!fio( x, p, t) in terms of the unperturbed function tie and-a 6-dimensional displace - ment vector f( x, p, t) or ti( x, p, t) which defines the Lagrange operator S? w4 according to Eq. (2. 9). The total time-derivative dlb /dt 0 should vanish, because $~6 represents a physically realizable state. When this is true and the displacement vector satisfies the Liouville theorem, Eq. (5.5), dfi/dt will also vanish. The solution of an electrodynamical problem must not only satisfy the Liouville equation d$/dt = 0, but also be consistent with the Lorentz equation. If y is such * that the latter equation is satisfied, then I,!J = !A tie will satisfy the former, but not conversely. On the other hand, the Boltzmnnn-Vlasov equation is a combination of the two; it may be used instead of the Lorentz equation, and vice versa. Using vector notation we may write the 6-dimensional displacement in several equivalent forms. In theCartesian’system (X /[, Fp) , . (10. la) Here, the base vectors %. -1 are orthogonal unit vectors Xi. = a i -i - - a. . a = ?+l *+t ’ ‘.>$\ 1 A+& In the curvilinear coordinate system (xcy , p,), xa! = XQ (2) (10. lb) Here, g..gk = i 6li and N -i N -i e. e = a.a wwl w.- T-++ 1 A.“. -cl?. -1 ii-%> It may further be noted that -9 -42 - These yield: . j& * &+A = 2 aPA a Pa! - =--- axa --A ' ax 23+cY mh *; =o, and e a& ’ ma3+h ax'l! (y &%3+o! L-z ape! -=e -zh. a3 - (10.2a) (10.2b) (9 2c) (10.2d) Corresponding to the two parts of ? are the 3-vectors [’ and q . %S+- WC P.* (,lO. 3a) These two vectors must satisfy a consistency relation given by Eq. (8.10) in the case of kinetic phase-space and by Eq. (6.15) or Eg. (6.16) in the case of canonical phase -space. The canonical space is conceptually simpler because xQ and p, are conjugate varia.bles used in the Hamilton equations. On the other hand, the kinetic space is algebraically simpler, because in this space u = LI Here, the vector potential ew. +‘;I 0’ A does not a.ppear in the Hamiltonian expression; at\ one of the canonical equations must be modified to account-for the presence of /&[See Eel. (5. Sa)] , Lagrange and Taylor operators are most convenient to use for perturbation . calculations o They gi\*e rise to the desired series expansions, the first feiv terms -43 - we obtain from Eq. (10.8) and the equatiqn corrksponding to Eq. (5.3) : . yO 6 dJ* 5L+;--- . yO J* dt . (10.9) Here J* is the Jacobian of transformation from the unperturbed to the perturbed coordinates in velocity phase spaces : Operating on both sides of Eq. (10.9) from the left with 2 22” and recalling that &*G?* = 1, we obtain, after canceling out the factor +!J: , As shown in Appendix B, Eq. (10.10) agrees with the general formula Eq. (5.10). In a velocity phase-space, the first-order displacement vector Fl should, therefore, s satisfy A $5 Xl [ . + 5(log$d = 0 in order that the Liouville equation (10. 8) may be satisfied to the first order. This equation. may further be transformed, thus : (10.11) The spatial and non-spatial variables in different phase spaces have, so far, been considered tacitly to be the genuine spas *;-1 and momentum or velocity coordinates. When canonical transformations are applied as in the Hamil.tonian treatment of dynamics, the new variables Xa! and Per are functions of both the spatial (xa) a and the non-spatial (pa) variables. No new coordinate, 2s or P 01’ is a pure TO spatial or a pure non-spatial one. To call X or P a spatial or non-spatial Q \ -46 - coordinate is a matter of nomenclature. 19 Let.the new kind of phase spaces be called non-separable and the other kind separable. The 6-dimensional analyses discussed in this paper are applicable, without restriction, to both kinds of phase spaces. On the other hand, the 3-dimensional formulas are applicable only in sep- arable phase spaces. Even in the latter spaces, any vector and any tensor must , obey the law of 6-dimensional point transformation of coordinates. According to r:*-. --- Eq.’ (~16. k&h e non-spatial components of a 6-vector in a separable curvilinear phase-space are not the same as those obtained from the non-spatialcartesian components by a 3-dimensional transformation of spatial coordinates. In separable phase spaces the momentum part of the displacement vector q does kr* not enter into the expression of the charge density p (r , t), nor of the current P.w density &(z t). The Lorentz equation (7. 7) is also independent of rl . As long as &AC no collision between particles is contemplated, one may, perhaps, question the advisability of using the Boltzmann-Vlasov’ equation of seven independent variables (X YPcr’ / t) instead of the Lorentz equation of four space-time variables. This question can hardly be answered because each method has its merits and shortcomings. We may, however, note that the Etiltzmann-Vlasov equation is a scalar equation which may represent an easier approach for certain problems than to solve the vector Lorentz equation directly. A deeper insight into the physical aspects of a certain problem may always be gained by considering’the charge distribution function and the various equations based on it. Such knowledge would be very helpful in justifying the use of certain simplifying approximations which may be needed. Strictly speaking, the spalial part of the displacement vector < in a separable w&f phase-space may depend on both the spatial and th,e momentum variables (xo and p,). In Set tion IX, eI is taken to be independent of p o. This assumption is, perhaps, generally possible if the random part of the unperturbed velocity ( LI ,+:*.o - $0 > ) is at most a first-order small quantity. Then, it-is permissible to substitute <z,> “.Zo (2 t) for z. in Eqs. (7.10) and (7.11). Then &> which is determjned by these equations must be a function of ( r *, t ). In fact, L = {(r , t) if the velocity m?w function u h-0 of the system of particles may properly be described by an Eulerian velocity-field ,“ro(,rLt) . I The usual equation of continuity (4.8) is shown to be the limiting form of a general formula’, Eq. (4. ‘7). Thti latter formula bears some resemblance in form to the --.. Fokker -Planck equation . 6, 23, 24 Thisis not sur$sini. i fa& we believe ..__ _. ~. ----- ___ . -_ - .- _ ---.-I-- .~ ~. that the Fokker-Planck equation may be derived by statistically averaging the gen- era1 equation of continuity (4.7). In the absence of collisions Eq. (4.8) is valid, as long as $J represent,s the den- sity distribution of something which is conserved, No other dynamical principle is ‘involved‘. About two decades ago, Vlasov’ noted the generality and the flexibility of this equation and applied it in h.is well-known work to varied subjects, such as the theory of crystals, electron plasma, striations in metals, and high frequency electron-beam tubes. He tool; the equation of continuity in 6-space as an obvious extension of the one in 3 -space, and has even indicated the feasibility of further extending it to 9- and higher 3P-dimensional spaces (x CY ‘o! “a , x , x , . . . . ) . h the presence of collisions caused. by sharply varying inter-partic1.e forces, the homogeneous Liouville eqtmtion equation becomes invalid. One must then use the celebrated Boltzmalul equation, or t’ne Fokker-Planck equation, or some neiver gener- ! i L.?,) I alized version of t-he Boltzmaan equation. They may,, be written in one abbreviated & ,’ \ form, namely de & ( ) dt = at coll ’ D - 48 - Eq. (5. lo), thus Using these relations we obtain (A. 3) This proves that the density divergence indeed transforms like a’scalar density from the unperturbed to the perturbed state. If I+!I~ satisfies the equation of contin- uity, so must ti = SIti0 satisfy the corresponding equation a ($u’)/ax’ = 0 regardless of the vanishing or non-vanishing of dJ/d t. - A-2 - APPENDIX B LIOUVILLE’S THEOREM IN VELOCITY PHASE-SPACE In canonical and kinetic phase spaces, Liouville’s theorem states that dJ/dt = 0, J = a (x’, ~‘)/a (x,p) being the Jacobian of transformation from the unperturbed to the perturbed coordinates. In the velocity phase-space, d?/d t is given by Eq. (10.1.0) and, in general, does not vanish. Here J* = a ( x1, u ), 8 (x,u). It ’ / seems instructive to derive Eq. (10.10) from Eq. (5.10) in curvilinear coordinates. The latter equation gives the general expression of dJ/d t in any phase-space. * ra In the velocity phase-space, x = u Ia! l ,3+cY and x = ;h . Thus ,P al? a$ -------;-= +- to! l W 1) ax’l axla U; au; x The last term in this equation may be evaluated by using the following relation: (B- 2) This equation is based on the definition ph. = my’u’ . CY Differentiating &k with respect to u; and contracting, we obtain from: Eq. (B. 2) 1 3 +---- - my’ au; -.mrtu; I - B-l. - Since we have and We also have y’ Z’ 1 -g ( ‘/a- 1 ’ up u,/c 2 ) - l/2 , (B. 4a) ay'/a$ = ( y'3/c2) P , (p’L= eE:-u th a a --- ,A eA’ - - e A’ ax (T axtu A and a&)./au; = 0 . On substitution of these relations Eq. (B. 3) becomes: i.e. , Thus , a kll/axfl = - 5 i/!/y’ D (B. 4b) (B. 4c) (B.5a ) (B. 5b) (B. 6) - B-2 - 14. J. L. Synge, The Relativistic Gas (North-Holland Publishing Co. ,Amsterdam, 195’7). 15. R. W. Lindquist, Ann. Phys. (N. Y. ) E, 487 (1966). 16. E. T. Whittaker, Analytical Dynamics, 4th Ed. , Chap. X, XI, and ,ylI (Dover Publications, New York, 1944). 17. L. Nordheim and E. Fues , Die Hamilton-Jacobische Theorie der Dynamik, Handbuch der Phys. , Vol. 5, Chap. 3, pp. 91-130 (verlag von Julius Springer, Berlin, 1927). 18. H. Goldstein, Classical Mechanics, Chap. 8 (Addison-Wesley Press, Inc. , Cambridge, Mass. , 1951). 19. L. D. Landau and E. M. Lifshitz, Mechanics, translated from the Russian by J. B. Sykes and J. S. Bell, Chap. VII (Addison-Wesley Publishing Co. , Inc. , Reading, Mass., 1960). 20. 0. Veblen, Invariants of Quadratic Differential Forms, Cambridge Tracts in Math. and Math. Phys. , No. 24, First Ed.(1927), Chap. II, III, and IV (Cambridge University Press, 1962). 21. E. Madelung, Die Mathematischen Hilfsmittel des Physikers , Third German -e--w -- Ed. , pp. 135-138 (Dover Zbblications , New York, 1943). 22. M. J. Lee, I?. E. Mills, and P. L. Morton, ” Throbbing Beam Transverse Resistive Instabilities in Circular Accelerators and Storage Rings ,I’ SLAC Report No. 76, August 1967, Stanford Linear Accelerator Center, Stanford University, Stanford, California. 23. S. A. Rice and P. Gray, The Statistical Mechanics of S?&gle Liquids, ----w--e--. - Chap. 4 (Interscience Publishers, a division of John Wiley and Sons, Inc. , New York, 1965). 24. Ta-You Wu, Kinetjc Equations of Cases and Plasmas, pp. 51-54 (Adclison- _-IL ___-_ . . -. I__-___ -----_.. Wesley Pitblishing Co. , Palo Alto, California, 1966). - K-2 “- 9 *. . . L. 25. S. Chapman and T. G. Cotvling, The Mathematical Theory of Non-Uniform -- - Gases (Cambridge University Press, 1952); first printed in 1939. --- 26. A. Sommerfeld, Thermodynamics and Statistical Mechanics, translated from the German by J. Kestin, Chap. V (Academic Press,Inc. , New York, 1956). I LIST OF SYMBOLS (for the use of the printer) Symbol First Appeared on Page Author: E. L. Chu Description 2K !?KO Rx1 . . . 111 . . . 111 lower case Greek alphabet psi lower case bold Greek alphabet zeta with a . . . 111 1 1 1 1 2 2 3 4 4 tilde bold gradient vector with a tilde lower case bold English alphabet i lower case Greek alphabet rho , bold upper case English alphabet E bold upper case English alphabet B lower case Greek alphabet delta bold lower case English alphabet r with lower case subscript q bold lower case English alphabet p with upper case subscript K bold lower case English alphabet p with subscripts upper case K and zero bold lower case English alphabet p with subscripts upper case K and Arabic number one bold lower case Greek alphabet zeta with subscript Arabic number two upper cask English alphabet J with ‘a bar -S-l-