A Gauge-invariant Hamiltonian Description of the Motion of Charged Test Particles

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1 A Gauge-nvarant Hamltonan Descrpton of the Moton of Charged Test Partcles Darusz Chruścńs Insttute of Physcs, Ncholas Coperncus nversty ul. Grudz adza 5/7, Toruń, Poland and Jerzy Kjows Centrum Fzy Teoretycznej PAN Aleja Lotnów 32/46, Warsaw, Poland Abstract New, gauge-ndependent, second order Lagrangan for the moton of classcal, charged test partcles (cf. [] s used to derve the correspondng hamltonan formulaton. For ths purpose a (relatvely lttle nown hamltonan descrpton of theores derved from second order Lagrangans s presented. nle n the standard approach, the canoncal momenta arsng here are explctely gauge-nvarant and have a clear physcal nterpretaton. The reduced symplectc form obtaned ths way s (almost equvalent to the Sourau s form (cf. [5]. Ths approach llustrates a new method of dervng equatons of moton from feld equatons (cf. [2] and [3]. Introducton In [] a new method of dervng equatons of moton from feld equatons was proposed. The method s based on a new dea of renormalzaton n classcal feld theory and a deep analyss of the geometrc structure of generators of ts symmetry group. It may be appled to any specal-relatvstc, lagrangan feld theory. When appled to electrodynamcs, t leads unquely to a manfestly gauge-nvarant, second order Lagrangan L for the moton of charged test partcles: L = L partcle + L nt = 2 (m a µ u ν M nt µν (t, q, v, ( where u µ denotes the (normalzed four-velocty vector (u µ = (u 0, u := 2 (, v, (2 and a µ := u ν ν u µ s the partcle s acceleraton (we use the Heavsde-Lorentz system of unts wth the velocty of lght c =. The sew-symmetrc tensor Mµν nt (t, q, v s equal to the amount of the angular-momentum of the feld, whch s acqured by our physcal system, when the Coulomb feld f µν (y,v accompanyng the partcle movng wth constant velocty v

2 through the space-tme pont y = (t, q, s added to the bacground (external feld. More precsely: the total energy-momentum tensor correspondng to the sum of the bacground feld f µν and the above Coulomb feld decomposes n a natural way nto a sum of terms quadratc n the bacground feld, 2 terms quadratc n the Coulomb feld 3 mxed terms descrbng ther nteracton. The quantty Mµν nt s equal to ths part of the total angularmomentum M µν, whch we obtan ntegratng only the mxed terms of the energy-momentum tensor (see Secton (3 for detaled dscusson. The above result s a by-product of a consstent theory of nteractng partcles and felds (cf. [2], [3], called Electrodynamcs of Movng Partcles. We have proved n [] that the new Lagrangan ( dffers from the standard one L = L partcle + L nt = 2 (m eu µ A µ (t, q, (3 by (gauge-dependent boundary correctons only. We have a smlar stuaton n General Relatvty, whch may be derved ether from the gauge-nvarant, second order Hlbert Lagrangan or equvalently from the frst order, coordnate-dependent Ensten Lagrangan, quadratc wth respect to the connecton coeffcents. Both Lagrangans generate, therefore, the same equatons of moton for test partcles n a gven external feld. In the present paper we explctly derve these equatons and construct the gauge-nvarant hamltonan verson of ths theory. Standard Hamltonan formalsm, based on the gauge-dependent Lagrangan (3, leads to the gauge-dependent Hamltonan H(t, q, p = m 2 + (p + ea(t, q 2 + ea 0 (t, q, (4 and the gauge-dependent momentum p := p n ea (t, q = mu ea (t, q (5 canoncally conjugate to the partcle s poston q. Ths gauge-dependence leads to serous conceptual dffcultes, f we want to descrbe quantzed partcles n a tme-dependent feld (e.g. a plane wave, and have no prvleged gauge (e.g. tme-ndependent at our dsposal. As was observed by Sourau (see [5], we may replace the non-physcal momentum (5 n the descrpton of the phase space of ths theory by the gauge-nvarant quantty p n. The prce we pay for ths change s, that the canoncal pre-symplectc form, correspondng to the theory of free partcles: Ω = dp n µ dq µ, has to be replaced by ts deformaton: (6 Ω S := Ω e f µν dq µ dq ν, (7 where e s the partcle s charge. Both Ω and Ω S are defned on the mass-shell of the netc momentum,. e. on the surface (p n 2 = m 2 n the cotangent bundle T M over the space-tme M (we use the Mnowsan metrc wth the sgnature (, +, +, +. These forms contan the entre nformaton about dynamcs: for free partcles the admssble trajectores are those, whose tangent vectors belong to the degeneracy dstrbuton of Ω. Sourau notced that replacng (6 by ts deformaton (7 we obtan the theory of moton of the partcle n a gven electromagnetc feld f µν. 2

3 The new approach, proposed n the present paper s based on Lagrangan (. It leads drectly to a perfectly gauge-nvarant Hamltonan, havng a clear physcal nterpretaton as the sum of two terms: netc energy mu 0 and 2 nteracton energy equal to the ammount of feld energy acqured by our physcal system, when the partcle s Coulomb feld s added to the bacground feld. When formulated n terms of pre-symplectc geometry, our approach leads unquely to a new form Ω N : where Ω N := Ω e h µν dq µ dq ν, (8 h µν := 2(f µν u [µ f ν]λ u λ (9 (bracets denote antsymmetrzaton,. e. we prove the followng Theorem One-dmensonal submanfolds of the partcle s mass shell, whose tangent vectors belong to the degeneracy dstrbuton of the form Ω N restrcted to the shell, are precsely the trajectores of test partcles movng n the external electromagnetc feld f µν. It s easy to see that both Ω S and Ω N, although dfferent, have the same degeneracy vectors, because h and f gve the same value on the velocty vector u ν : u ν h µν = u ν f µν. (0 Hence, both defne the same equatons of moton. We stress, however, that our Ω N s unquely obtaned from the gauge-nvarant Lagrangan ( va the Legendre transformaton. The paper s organzed as follows. In secton 2 we setch brefly the (relatvely lttle nown Hamltonan formulaton of theores arsng from the second order Lagrangan. In secton 3 we prove explctly that the Euler-Lagrange equatons derved from L are equvalent to the Lorentz equatons of moton. Fnally, Secton 4 contans the gauge-nvarant Hamltonan structure of the theory. The most nvolved proofs are shfted to the Appendx. 2 Canoncal formalsm for a 2-nd order Lagrangan theory Consder a theory descrbed by the 2-nd order lagrangan L = L(q, q, q (to smplfy the notaton we wll sp the ndex correspondng to dfferent degrees of freedom q ; extenson of ths approach to hgher order Lagrangans s straghtforward. Introducng auxlary varables v = q we can treat our theory as a -st order one wth lagrangan constrants φ := q v = 0 on the space of lagrangan varables (q, q, v, v. Dynamcs s generated by the followng symplectc relaton: d L(q, v, v = d (p dq + π dv = ṗ dq + p d q + π dv + π d v. ( dt where (p, π are momenta canoncally conjugate to q and v respectvely. Because L s defned only on the constrant submanfold, ts dervatve dl s not unquely defned and has to be understood as a collecton of all the covectors whch are compatble wth the dervatve of the functon along constrants (cf. [6]. Ths means that the left hand sde s defned up to 3

4 µ( q v, where µ are Lagrange multplers correspondng to constrants φ = 0. We conclude that p = µ s arbtrary and ( s equvalent to the system of dynamcal equatons: π = L v, ṗ = L q, π = L v p. (4 The last equaton mples the defnton of the canoncal momentum p: p = L L π = v v d ( L. (5 dt v Its tme dervatve ṗ = d ( L dt v d2 dt 2 ( L v s equvalent, due to the second canoncal equaton (3, to the Euler-Lagrange equaton: ( L d2 L := d ( L + L q dt 2 v dt v q = 0. (7 To obtan hamltonan descrpton (see e. g. [4] we smply apply the Legendre transformaton to formula (: dh = ṗ dq q dp + π dv v dπ, (8 where H(q, p, v, π = p v + π v L(q, v, v. In ths formula we have to nsert v = v(q, v, π, calculated from equaton (2. Let us observe that H obtaned ths way s lnear wth respect to the momentum p. Ths s a characterstc feature of the Hamltonans obtaned from a 2-nd order theory. In generc stuaton, Euler-Lagrange equatons (7 are of the 4-th order. The correspondng 4 hamltonan equatons descrbe, therefore, the evoluton of q and ts dervatves up to thrd order. Due to Hamltonan equatons mpled by symplectc relaton (8, the nformaton about succesve dervatves of q s carred by (v, π, p: v descrbes q: q = H p v, and the constrant φ = 0 s reproduced due to lnearty of H wth respect to p, π contans nformaton about q: v = H π, p contans nformaton about... q (2 (3 (6 (9 (20 π = H v = L v p, (2 the true dynamcal equaton reads: ṗ = H q = L q. 4 (22

5 3 Equatons of moton from the varatonal prncple In ths secton we explctly derve the partcle s equatons of moton from the varatonal prncple based on the gauge-nvarant Lagrangan (. The Euler-Lagrange equatons for a second order Lagrangan theory are gven by ṗ = L q, where, as we have seen n the prevous secton, the momentum p canoncally conjugate to the partcle s poston q s defned as: and Now, p := L v π π := L v = 2 uν M nt ν (t, q, v. (25 u ν M nt ν = u 0 M nt 0 + u l M nt l = u 0 r nt (23 (24 + u l ɛl m s nt m, (26 where r nt and s nt m are the statc momentum and the angular momentum of the nteracton tensor. They are defned as follows: we consder the sum of the (gven bacground feld f µν and the boosted Coulomb feld f µν (y,u accompanyng the partcle movng wth constant four-velocty u and passng through the space-tme pont y = (t, q. Beng b-lnear n felds, the energy-momentum tensor T total of the total feld fµν total := f µν + f µν (y,u (27 may be decomposed nto three terms: the energy-momentum tensor of the bacground feld T feld, the Coulomb energy-momentum tensor T partcle, whch s composed of terms quadratc n f µν (y,u and the nteracton tensor T nt, contanng mxed terms: T total = T feld + T partcle + T nt. (28 Interacton quanttes (labelled wth nt are those obtaned by ntegratng approprate components of T nt. Because all the three tensors are conserved outsde of the sources (. e. outsde of two trajectores: the actual trajectory of our partcle and the straght lne passng through the space-tme pont y wth four-velocty u, the ntegraton gves the same result when performed over any asymptotcaly flat Cauchy 3-surface passng through y. In partcular, r nt and s nt may be wrtten n terms of the laboratory-frame components of the electrc and magnetc felds as follows: r nt (t, q, v = d 3 x (x q (DD 0 + BB 0, (29 Σ s nt m (t, q, v = ɛ mj d 3 x (x q (D B 0 + D 0 B j, (30 Σ 5

6 where D and B are components of the external feld f, whereas D 0 and B 0 are components of f (y,u,.e.: D 0 (x; q, v = e 2 4π x q 3 ( 2 + ( v(x q 2 3/2 (x q, (3 x q B 0 (x; q, v = v D 0 (x; q, v. (32 It may be easly seen that quanttes r nt followng condton: and s nt m are not ndependent. They fulfll the s nt = ɛl m v l rm nt. To prove ths relaton let us observe that n the partcle s rest-frame (see the Appendx for the defnton the angular momentum correspondng to T nt vanshes (cf. []. When translated to the language of the laboratory frame, ths s precsely equvalent to the above relaton. Insertng (33 nto (26 we fnally get ( π = l + vl v r nt 2 l. (34 The quantty r nt depends upon tme va the tme dependence of the external felds (D(t, x, B(t, x, the partcle s poston q and the partcle s velocty v, contaned n formulae (3 (32 for the partcle s Coulomb feld. Now, we are ready to compute p from (24: p = ( v π ( π t + π vl q l mv 2 + vl π l = p n t + π vl + v l π q l v l v l ( π v l π l v (33. (35 Observe, that the momentum p depends upon tme, partcle s poston and velocty but also on partcle s acceleraton. Hovewer, usng (29 one easly shows (see Appendx Lemma π v l π l v = 0. Hence, the term proportonal to v l vanshes. Moreover, the followng lemma can be proved (see also Appendx: Lemma 2 (36 π t + π vl q l = p nt, (37 6

7 where we denote p nt (t, q, v = We see that p nt p nt µ (t, q, v = Σ Σ d 3 x (D B 0 + D 0 B. (38 s the spatal part of the nteracton momentum : Tµν nt dσ ν, (39 where Σ s any hypersurface ntersectng the partcle s trajectory at the pont (t, q(t. The above ntegral s fnte (cf. [2] and nvarant wth respect to changes of Σ, provded the ntersecton pont wth the trajectory does not change. It was shown n [] that p nt µ s orthogonal to the partcle s four-velocty,.e. p nt µ u µ = 0. Fnally, the momentum canoncally conjugate to the partcle s poston equals: p = p n + p nt (t, q, v. (40 It s a sum of two terms: netc momentum p n and the amount of momentum p nt whch s acqured by our system, when the partcle s Coulomb feld s added to the bacground (external feld. We stress, that contrary to the standard formulaton based on (3, our canoncal momentum (40 s gauge-nvarant. Now, Euler-Lagrange equatons (23 read dp n dt + dpnt dt = L q, or n a more transparent way: ( ( d mv p nt = dt 2 t + v l pnt q l ( p v l nt v l π l q (4. (42 Agan, usng defntons of π l and p nt one shows the followng Lemma 3 p nt v l π l q = 0, (for the proof see Appendx. Hence, the term proportonal to the partcle s acceleraton vanshes. In the Appendx we show that the followng denttes hold: Lemma 4 p nt t (43 + v l pnt = e q 2 u ν f l ν (t, q = e(e (t, q + ɛ lm v l B m (t, q. (44 We conclude that the Euler-Lagrange equatons (23 for the varatonal problem based on L are equvalent to the Lorentz equatons for the moton of charged partcles: ( d mv = e(e dt 2 (t, q + ɛ lm v l B m (t, q. (45 7

8 4 Hamltonan formulaton By Hamltonan formulaton of the theory we understand, usually, the phase space P of Hamltonan varables (q, p endowed wth the symplectc 2-form ω = dp dq and the Hamltonan functon H (the energy of the system defned on P. However, for tmedependent systems t s more convenent to replace ths framewor by the so called homogeneous formulaton. For ths purpose we consder the evoluton space P R endowed wth the pre-symplectc 2-form (.e. closed 2-form of maxmal ran: ω H := dp dq dh dt, (46 (ts potental p dq Hdt s called the Poncaré-Cartan nvarant. Obvously, ω H s degenerate on P R and the one-dmensonal characterstc bundle of ω H conssts of the ntegral curves of the system n P R (they may be consdered as generated by the superhamltonan whch vanshes dentcally on the evoluton space. Ths descrpton may be called the Hesenberg pcture of classcal mechancs: physcal states are not ponts n P but partcle s hstores n P R (see [5]. Let us construct the Hamltonan structure for the theory based on our second order Lagrangan L. Let P denote the space of Hamltonan varables,.e. (q, p, v, π, where p and π stand for the momenta canoncally conjugate to q and v respectvely. Snce our system s manfestly tme-dependent (va the tme dependence of the external feld we pass to the evoluton space endowed wth the pre-symplectc 2-form Ω H := dp dq + dπ dv dh dt, (47 where H denotes the tme-dependent partcle s Hamltonan. To fnd H on P R one has to perform the (tme-dependent Legendre transformaton (q, q, v, v (q, p, v, π,.e. one has to calculate q and v n terms of Hamltonan varables, usng formulae: p = L q π, π = L v. (48 Ths transformaton s sngular due to lnear dependence of L on v and gves rse to the tme-dependent constrants, gven by equatons (25 and (40. The constrants can be easly solved.e. momenta p and π can be unquely parameterzed by the partcle s poston q, velocty v and the tme t. Let P denote the constraned submanfold of the evoluton space P R parametrzed by (q, p n, t. The reduced Hamltonan on P reads: H(t, q, v = p v + π v L = Due to dentty u µ p nt µ = 0 (cf. [] we have H(t, q, v = Hence, we obtan m 2 + v p nt (t, q, v. (49 m 2 pnt 0 (t, q, v. (50 Theorem 2 The partcle s Hamltonan equals to the p 0 component of the followng gauge-nvarant, four-vector p µ := p n µ + p nt µ (t, q, v = mu µ + p nt µ (t, q, v. (5 8

9 sng the laboratory-frame components of the external electromagnetc feld we get: p nt 0 (t, q, v = d 3 x (DD 0 + BB 0. (52 Now, let us reduce the pre-symplectc 2-form (47 on P. Calculatng p = p (q, p n, t and π = π (q, p n, t from (25 (40 and nsertng them nto (47 one obtans after a smple algebra: Ω N = dp n µ dq µ e h µν dq µ dq ν, (53 where q 0 t and h µν s the followng 4-dmensonal tensor: e h µν (t, q, v := pnt ν q µ pnt µ. (54 q ν sng technques presented n the Appendx one easly proves Lemma 5 p nt ν q µ where by = eπ λ µ f λν, (55 Π λ µ := λ µ + u µ u λ (56 we denote the projector on the hyperplane orthogonal to u µ (.e. to the partcle s rest-frame hyperplane, see the Appendx. Therefore h µν = Π λ µ f λν Π λ ν f λµ = 2(f µν u [µ f ν]λ u λ. (57 where a [α b β] := 2 (a αb β a β b α. The form Ω N s defned on a submanfold of cotangent bundle T M defned by the partcle s mass shell (p n 2 = m 2. Observe, that the 2-form (53 has the same structure as the Sourau s 2-form (7. They dffer by the curvature 2-forms f and h only. However, the dfference h f vanshes dentcally along the partcle s trajectores due to the fact that both f µν and h µν have the same projectons n the drecton of u µ (see formula (0. We conclude that the characterstc bundle of Ω N and Ω S are the same and they are descrbed by the followng equatons: q = v, v = 2 e m (gl v v l (E l + ɛ lj v B j, (59 whch are equvalent to the Lorentz equatons (45. From the physcal pont of vew these forms are completely equvalent. (58 9

10 Appendx Due to the complcated dependence of the Coulomb feld D 0 and B 0 on the partcle s poston q and velocty v, formulae contanng the respectve dervatves of these felds are rather complex. To smplfy the proofs, we shall use for calculatons the partcle s restframe, nstead of the laboratory frame. The frame assocated wth a partcle movng along a trajectory ζ may be defned as follows (cf. [3], []: at each pont (t, q(t ζ we tae the 3- dmensonal hyperplane Σ t orthogonal to the four-velocty u µ (the rest-frame hypersurface. We parametrze Σ t by cartesan coordnates (x, =, 2, 3, centered at the partcle s poston (.e. the pont x = 0 belongs always to ζ. Obvously, there are nfntely many such coordnate systems on Σ t, whch dffer from each other by an O(3-rotaton. To fx unquely coordnates (x, we choose the unque boost transformaton relatng the laboratory tme axs /y 0 wth the four-velocty vector := u µ. Next, we defne the poston of the y µ /x axs on Σ t by transformng the correspondng /y axs of the laboratory frame by the same boost. The fnal formula relatng Mnowsan coordnates (y µ wth the new parameters (t, x may be easly calculated (see e. g. [3] from the above defnton: y 0 (t, x l := t + 2 (t xl v l (t, y (t, x l := q (t + ( l + ϕ(v 2 v v l x l, (A. ( where we denote ϕ(z := z z = z(+ z. Observe, that the partcle s Coulomb feld has n ths co-movng frame extremely smple form: D 0 (x = ex 4πr, B 0(x = 0, (A.2 3 where r := x. That s why the calculatons n ths frame are much easer than n the laboratory one. Let D and B denote the rest-frame components of the electrc and magnetc feld. They are related to D and B as follows: D (x, t; q, v = B (x, t; q, v = [( 2 l 2 ϕ(v 2 v l v Dl (y ɛ j v B j (y ], (A.3 [( 2 l 2 ϕ(v 2 v l v Bl (y + ɛ j v D j (y ], (A.4 (the matrx ( l 2 ϕ(v 2 v l v comes from the boost transformaton. The feld evoluton wth respect to the above non nertal frame s a superposton of the followng three transformatons (cf. [], [2], [3]: tme-translaton n the drecton of, boost n the drecton of the partcle s acceleraton a, purely spatal O(3-rotaton around the vector ω m, 0

11 where a := ω m := ( 2 l + ϕ(v 2 v v l v l, (A.5 2 ϕ(v2 v v l ɛ lm. (A.6 Therefore, the Maxwell equatons read (cf. [2], [3]: Ḋ n = 2 x m [( ɛ m D n ɛ n D m ω x ɛ mn ( + a x B ], B n = [( 2 ɛ m x m B n ɛ n B m ω x + ɛ mn ( + a x D ], (A.7 (A.8 (the factor 2 s necessary, because the tme t, whch we used to parametrze the partcle s trajectory, s not a proper tme along ζ but the laboratory tme. On the other hand, the tme dervatve wth respect to the co-movng frame may be wrtten as d dt = t + v q + ( v v = + v t v. (A.9 Therefore, tang nto account (A.7 and (A.8 we obtan: ( D n = t 2 ɛ nm m B, (A.0 ( B n = t 2 ɛ nm m D, (A. and D n = [ ω v 2 l m v l B n = 2 m ] ( ɛ m v l D n ɛ n D m x a v l ɛmn x B, (A.2 [ ] ω ( ɛ m v l D n ɛ n D m x + a v l ɛmn x B. (A.3 To calculate the dervatves of D and B wth respect to the partcle s poston observe, that Therefore y = v + ( 2 + ϕ(v 2 v v x. q Dn = v 2 ɛnm m B + ( + ϕ(v 2 v v D n, v q Bn = 2 ɛnm m D + ( + ϕ(v 2 v v B n. Now, usng (A.0 (A.3 and (A.5 (A.6 we prove Lemmas 4.. Proof of Lemma : (A.4 (A.5 (A.6

12 Observe, that nteracton statc moment (29 n the partcle s rest frame reads: R nt := x (D 0 D + B 0 B d 3 x = e x x Σ t 4π Σt r 3 D d 3 x. (A.7 Tang nto account that r nt = ( 2 2 ϕ(v 2 v v R nt we obtan the formula for π n terms of R nt : π = 2 (A.8 ( + ϕ(v 2 v v R nt. (A.9 Now, usng (A.2 one gets: R nt v l = { a m 2 v X l m ωm v ɛ l } j m Rj nt, (A.20 where X m = e 4π ɛ x j x m j B d 3 x. (A.2 Σ t r 3 Therefore where π π l v l v = A lr nt Bl m X m, A l = v + ɛ jm [ 2 ( ] l + ϕ(v 2 v v l [ v l 2 [ ( j + ϕ(v2 v j v ω m ( j v l l + ϕ(v2 v j v ω m l v ( + ϕ(v 2 v v ] ] (A.22, (A.23 Bl m = ( + ϕ(v 2 v v a m ( v l l + ϕ(v 2 v v a m l ( v a = ( 2 a m a a m. (A.24 v v l v l v sng the followng propertes of the functon ϕ(z: 2ϕ(z ( z + zϕ 2 (z = 0, (A.25 2ϕ (z ( z ϕ(z ϕ 2 (z = 0, (A.26 one easly shows that A l 0. Moreover, observe that Bl m defned n (A.24 s antsymmetrc n (m. Therefore, to prove (36 t s suffcent to show that the quantty X m s symmetrc n (m. Tang nto account that B = ɛ lm l A m, where A m stands for the rest-frame components of vector potental, one mmedatelly gets: x j x m ɛ j B d 3 x = r 5 (A Σ t r 3 x (3x x m r 2 g m d 3 x, Σ t (A.27 2

13 whch ends the proof of ( Proof of Lemma 2: To prove (37 observe that π t + π ( vl = π q l = t 2 Now, usng (A.0 we obtan ( R nt = x t 2 x Σ t r 3 ɛjm j B m d 3 x = ( x x Σ 2 j ɛjm B m t r 3 ( ( + ϕ(v 2 v v R nt. (A.28 t d 3 x + 2 ɛ m Σ t x r 3 B m d 3 x. (A.29 Due to the Gauss theorem ( x x j Σ t r 3 ɛjm B m d 3 x j x x x = ɛ jm B Σ t r 4 m dσ 0, (A.30 where dσ denotes the surface measure on Σ t. Moreover, observe that nteracton momentum n the partcle s rest-frame reads: P nt := ɛ m (D0B m + D B0 m d 3 x = e Σ t 4π ɛ x m Σt r 3 Bm d 3 x. (A.3 Therefore ( R nt = t 2 P nt. (A.32 sng the relaton between p nt and P nt p nt = ( + ϕ(v 2 v v P nt (A.33 we fnally get ( Proof of Lemma 3: sng (A.3 and (A.5 we obtan: P nt v l = { ω m 2 ɛ v l R nt v l = where Y j := e 4π j m Pj nt } am v Y l m, (A.34 v l 2 P nt + ( j l + ϕ(v2 v j v l Yj, (A.35 Σ t x r 5 (3x x j r 2 j D d 3 x. Now, tang nto account (A.9 and (A.33 we have (A.36 p nt v l π l q = C lp nt, (A.37 3

14 where C l := ( v l + ϕ(v 2 v v 2 + ( j + ϕ(v2 v j v ω m v l v 2 ( l + ϕ(v 2 v v l. (A.38 One easly shows that due to propertes (A.25 (A.26 C l 0, whch ends the proof of ( Proof of Lemma 4: Fnally, to prove (44 let us observe that p nt ( + v l pnt = t q l t Now, due to (A. we get ( P nt t = e 2 4π = e 2 4π lm r 0 0 p nt = ( + ϕ(v 2 v v Σ t r 2 D dσ S(r 0 r 2 D dσ = 2 ed (t, 0, ɛ jm ( P nt. (A.39 t (A.40 where we choose as two peces of a boundary Σ t a sphere at nfnty and a sphere S(r 0. sng the fact that n the Heavsde-Lorentz system of unts D = E and tang nto account the formula (A.3 we fnally obtan (44. References [] D. Chruścńs, J. Kjows, Equatons of Moton from Feld equatons and a Gaugenvarant Varatonal Prncple for the Moton of Charged Partcles, Journal of Geometry and Physcs 20, p (996 [2] J. Kjows, Electrodynamcs of Movng Partcles, Gen. Rel. and Grav. 26, p (994 J. Kjows, On Electrodynamcal Self-nteracton, Acta Phys. Polon. A85, p (994 [3] J. Kjows, D. Chruścńs, Varatonal Prncple for Electrodynamcs of Movng Partcles, Gen. Rel. and Grav. 27, p (995 [4] R. Abraham, J. E. Marsden, Foundaton of Mechancs, Benjamn, New Yor, 978 [5] J. M. Sourau, Structures des Systemes Dynamques, Dunod, Pars 970 S. Sternberg, Proc. Nat. Acad. Sc. 74, p (977 A. Wensten, Lett. Math. Phys. 2, p. 47 (977 [6] J. Kjows, W. M. Tulczyjew, A symplectc framewor for feld theores, Sprnger Lecture Notes n Physcs, vol. 07 (979, 257 pages. 4