Sur les articles de Henri Poincaré SUR LA DYNAMIQUE. Le texte fondateur de la Relativité en langage scientiþque moderne. par Anatoly A.

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1 Sur les articles de Henri Poincaré SUR LA DYNAMIQUE DE L ÉLECTRON Le texte fondateur de la Relativité en langage scientiþque moderne par Anatoly A. LOGUNOV Directeur de l'institut de Physique des Hautes Énergies (Protvino, Russie) Membre de l'académie des Sciences de Moscou Traduction française de Vladimir Petrov (Institut de Physique des Hautes Énergies, Protvino, Russie) Christian Marchal (Directeur de Recherches à l'ofþce National de Recherches Aérospatiales, Châtillon, France)

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6 B x B y B z (α, β, γ) = (f, g, h) = 1 2 (F, G, H) = 1 ψ = ϕ (ξ, η, ζ) = 1 (X, Y, Z) = 3 (X 1,Y 1,Z 1 ) = 1 u 1 ρ 3 = ρ 2 1 (u, v, w) = = + t = ε = β } 1 k = γ = 1 β 2 c : 1 µ : 4π ε : = µ ε c 2 =1 c =1 µε =1 β

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9 x = γl(x βt) ; y = ly ; z = lz ; t = γl(t βx) x y z t x y z t β 1 γ = 1 β 2 2

10 l β x = βt (x = y = z =0) x y z β x β Ox O x x y z t x l =1

11 ρ ρ v β γ =1/ 1 β 2 ρ = γl 3 (1 βv x ); ρ v x = γl 3 ρ(v x β) ; ρ v y = l 3 ρv y ; ρ v z = l 3 ρv z l =1. f f x = γl 5 (f x β ) ; f y = l 5 f y ; f z = l 5 f z β F F x = γl 5 (F x β ) ρ ρ ; F y = F y l 5 ρ ρ ; F z = F z l 5 ρ ρ l =1

12 x y z t x y z t t t x y z t x y z t v ρ ρ ξ = k 2 (ξ + ε) ; η = kη ; ζ = kζ ; ρ = ρ kl 3 l = 1 l = 1

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14 v 1 v 1 v v 1 v 1 v 2 v 2 /c 2

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16 t t

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20 4π ϕ ρ µ ε = ρ + t = = ; t = = ρ ε ϕ = ρ = t ϕ ρ + (ρ ) =0 t µε ϕ + =0 t = µρ = 2 µε 2 t = 2 2 x y µε 2 z2 t 2

21 = µ ; = ε =0 = 1/ µε = = µε τ = x y z τ = ρ( + ) x = γl(x βt) ; t = γl(t βx) ; y = ly ; z = lz l β γ = 1 1 β 2 γ β x 1 = γl(x βµεt) γ = 1 β2 µε = 2 µε 2 t 2 = l 2 ( t) 2 = r 2 4πr 3 /3 x = γ l (x + βt ); t = γ l (t + βx ); y = y l ; z = z l

22 γ 2 (x + βt v x t v x βx ) 2 + [ y γv y (t + bx ) ] 2 + [ z γv z (t + βx ) ] 2 = l 2 r 2 t =0 γ 2 x 2 (1 v x β) 2 +(y γv y x ) 2 +(z γv z x ) 2 = l 2 r πr3 l 3 γ(1 v x β) ρ ρ = γ l 3 ρ(1 βv x) v x v y v z v x = x (x βt) = t (t βx) = v x β v y = y t = v z = z t = y γ (t βx) = z γ (t βx) = 1 βv x v y γ(1 βv x ) v z γ(1 βv x ) ρ v x = γ l 3 ρ(v x β) ; ρ v y = 1 l 3 ρ v y ; ρ v z = 1 l 3 ρ v z ρ = 1 γl 3 ρ ; v x = γ 2 (v x β) ; v y = γ v y ; v z = γ v z ρ ρ + (ρ v )=0 t

23 λ D t + λρ ; x + λρ v x ; y + λρ v y ; z + λρ v z t x y z D = D 0 + D 1 λ + D 2 λ 2 + D 3 λ 3 + D 4 λ 4 D 0 =1; D 1 = ρ t + (ρ ) =0 λ = l 4 λ t + λ ρ ; x + λ ρ v x ; y + λ ρ v y ; z + λ ρ v z D D = D ; D = D 0 + D 1λ + + D 4λ 4 D 0 = D 0 =1; D 1 = D 1 l 4 =0= ρ t + (ρ v ) ρ ε ϕ = ρ ; A = µρ v ϕ = γ l (ϕ βa x); A x = γ l (A βµεϕ) ; A y = A y l ; A z = A z l E = A t ϕ ; B = A

24 = γ [ t l t + β ] ; x x = γ l [ x + β ] t ; = 1 y l y ; E x = 1 l E 2 x ; E y = γ l (E 2 y βb z ); E z = γ l (E 2 z + βb y ) B x = 1 l B 2 x ; B y = γ l (B 2 y + βe z ); B z = γ l (B 2 z βe y ) = 1 z l z ϕ t + A =0 D + ρ v = H B ; = E ; D = ρ t t f f f = ρ (E + v B ) f x = γ l 5 (f x β ) ; f y = 1 l 5 f y ; f z = 1 l 5 f z F u F u F u = ρ = + ; F u = f ρ = E + v B

25 F ux = γ l 5 ρ ρ (F ux βf u v) ; F uy = 1 l 5 ρ ρ F uy ; F uz = 1 l 5 ρ ρ F uz F ux = l 2[ F ux + β(v ye y + v ze z) ] F uy = l2 γ (F uy βv xe y) F uz = l2 γ (F uz βv xe z)

26 (, ϕ) (, ρ) x y z t =0 f =0 f ρ ρ E B

27 =0 =0 =0 =0 ρ =0 J = ( ) ε 2 t τ µ ε = ρ ; = = + ρ ; ( = ε ) t L = 2 µν 4 µ µ µ ( ϕ, ) F µν 0 E x E y E z E x 0 B z B y µν = µ ν ν µ = E y B z 0 B x E z B y B x 0

28 L = 2 µν/4 =0 = / t =0 = ρ J τ = x y z t t = t 0 t = t 1 J t = t 0 t = t 1 (B δc) t τa t C δc [ ] t1 τ AB δc t τ A t 0 t B δc δc =0 t = t 0 t = t 1 x y z A B x y z t = x AB y z t B A x y z t x x = ± A B τ t = x B A τ t x J

29 δa δb = δa ( ) δ δj = t τ δ =0 µ = µ ) δj = t τ [δ δ ] = t τ δ [ ] =0 δ = τ = τ = τ J = τ = τ = [ ] ε 2 t τ µ = τ 2 µ [ ] ε 2 t τ 2 2 2µ δj δ δ δ J δj = t τ(ε δ δ ) ε δ = δρ δj = t τ [ ε δ δ ψ(ε δ δρ) ] ψ δj

30 δ = ε (δ ) t + δ(ρ ) ψ = ϕ = [ δj = t τ εδ + ] t + ϕ + t τ [ ϕδρ ε δ(ρ ) ] δρ =0 δ(ρ ) =0 δj + + ϕ =0 t δj = t τ [ ϕδρ ε δ(ρ ) ] τ ξ τ δξ δξ τ δj = δξ τ t δj = r o + ξ α U δu = δα U α

31 x y z t α t α x y z t α t α v x = ξ x t = ξ x t + ξ x = x t x y z x y z (x, y, z) = (x,y,z ) = r o α x y z t t x y z x y z = t ; + = ( + ) r o 1+ ( + ) ( + t) = = 1 = (t) t (ρ ) =0 ρ t + ρ =0; ρ t = ρ t + ρ ; ρ t + (ρ ) =0 t α δξ = ξ α δα

32 1 α ρ ρ δα + ρ (δξ) =0; α δξ = ξ α +(δξ )ξ α ξ = α ; (ρ ) α α = ρ α + δρ + (ρ δξ) =0 =0 [ ] δξ δα ρ δξ =( ξ/ t)δα δρ =( ρ/ α)δα δξ t τ ϕδρ= t τ ϕ (ρ δξ) t τ ϕδρ= t τ ρδξ ϕ δ(ρ ) = (ρ ) α δα ρ x y z x y z ρ x y z 2 (ρ ξ) t α (ρ ) t = (ρ ) α = ( ρ ξ ) = ( ρ ξ ) α t t α =0

33 U 1 (U ) t 1 (U ) α i = x y z [ 1 ρ ξ ] i α t = U t + (U ) = U [ α + U = α α [ 1 ρ ξ ] i = t α t ( )] ξ α [ ρ ξ ] ( i + ρ ξ t α ξ ) i t [ ρ ξ ] ( i + ρ ξ α t ξ ) i α ξ t = ; δ(ρv i )+ (ρv i δξ) = δα ξ α = δξ ; (ρ δξ) t δα (ρ ) α = δ(ρ ) + (ρ δξ i ), i =(x, y, z) { [ ] (ρ δξ) t τ δ(ρ )= t τ + [ Ai (ρ δξ i ρv i δξ) ]} t i { [ t τ ρ δξ ] + [ ρ(δξ i v i δξ) ] } t i i = {[ t τ ρ δξ A ] } + ρ δξ [ ] t [ J = t τ ρδξ ϕ + ] t + = t τ ρδξ [ + ]

34 δξ = ρ( + ) e = e( + ) ρ [ ] ε 2 J = t τ 2 2 2µ t τ = l 4 t τ x y z t x y z t l 4 l 4 E 2 = E 2 x + γ 2 (E 2 y + E 2 z)+γ 2 β 2 (B 2 y + B 2 z)+2γ 2 β(e z B y E y B z ) l 4 B 2 = B 2 x + γ 2 (B 2 y + B 2 z)+γ 2 β 2 (E 2 y + E 2 z)+2γ 2 β(e z B y E y B z ) [ ] l 4 εe 2 B 2 = ε 2 B2 µ µ [ εe 2 J = t τ B 2 ] 2 2µ J = J

35 t t t 1 x y z + t = t 1 =+ J J δj = δξ τ t δj = f δξ τ t δξ δξ (x,y,z )= t = + ξ x = γl(x βt) y = ly ; z = lz r = r o + ξ t = γl(t βx) δξ t δt x y z δ = δξ + δt δr = δξ + v δt δx = γl(δx βδt); δy = lδy ; δz = lδz ; δt = γl(δt βδx) δt =0 δr = δξ + v δt = l(γδξ x,δξ y,δξ z ); δt = γlβ δξ x

36 x (δξ x ) t (δt =0) Oxyz t γl = l(1 β 2 ) 0,5 δξ x O x y z t β Oxyz t δt =0 δt = γlβδξ x v x = v x β 1 βv x ; v y = δt v y γ(1 βv x ) ; v z = γl(1 βv x ) δξ x = δξ x(1 βv x ) (v x β)γlβδξ x l(1 βv x ) δξ y = δξ y(1 βv x ) v y lβδξ x l(1 βv x ) δξ z = δξ z(1 βv x ) v z lβδξ x v z γ(1 βv x ) γ l δξ x = γ(1 βv x ) δξ x l δξ y = δξ y γβv y δξ x l δξ z = δξ z γβv z δξ x l δξ = δξ + δξ x[ (γ 1) γβ δξ x ] f δξ t τ = l 4 δξ t τ = f δξ t τ

37 δξ l 5 f x = γ(f x β ) ; l 5 f y = f ; l 5 f z = f z x y z t x y z t ε 2 ( 2 /µ) ε 2 +( 2 /µ) β γ =1 l =1 β 1/ µε E 2 = 2 2β( ) x B 2 = 2 2µεβ( ) x εe 2 + B 2 µ = ε µ 4εβ( ) x

38 x = γl(x βt) ; y = ly ; z = lz ; t = γl(t βx) x = γ l (x β t ); y = l y ; z = l z ; t = γ l (t β x ) γ 2 =1 β 2 ; (γ ) 2 =1 (β ) 2 x = γ l (x β t); y = l y ; z = l z ; t = γ l (t β x) β = β + β 1+ββ ; l = ll ; γ = γγ (1 + ββ )= 1 1 β 2 l β r = r + δr ; t = t + δt δx = βt ; δy =0; δz =0; δt = βx T 1 t ϕ x + x ϕ t = T 1ϕ β =0 l =1+δl δx = x δl ; δy = y δl ; δz = z δl ; δt = t δl ; T l β T ϕ = x ϕ x + y ϕ y + z ϕ z + t ϕ t y z x T 2 ϕ = t ϕ y + y ϕ t ; T 3ϕ = t ϕ z + z ϕ t

39 [T 1,T 2 ]ϕ = x ϕ y y ϕ x z T T 1 T 2 T 3 [T 1,T 2 ] [T 2,T 3 ] [T 3,T 1 ] x = lx ; y = ly ; z = lz ; t = lt x 2 + y 2 + z 2 t 2 x = γl(x βt) ; y = ly ; z = lz ; t = γl(t βx) l β P y P x x z z x = γl(x + βt) ; y = ly ; z = lz ; t = γl(t + βt) l β β P x = γ l (x + βt) ; y = y l ; z = z l ; t = γ (t + βx) l

40 P l =1 l = 1 l l ε 2 ( 2 /µ) x 2 +y 2 +z 2 t 2 L 2 T 2 L T ε ϕ = ρ ; = µρ ϕ(x, y, z, t) = 1 4πε ρ1 R τ 1 ; (x, y, z, t) = µ 4π ρ1 1 R τ 1 τ 1 = x 1 y 1 z 1 ; R = [ (x x 1 ) 2 +(y y 1 ) 2 +(z z 1 ) 2] 1/2 = r1 ρ 1 1 ρ x 1 y 1 z 1 t 1 = t R

41 r o =(x,y,z ) t r 1 =(x 1,y 1,z 1 )=r o + ξ t 1 ξ =(ξ x, ξ y, ξ z ) r o,t 1 x 1 = x + ξ x r o r o + v 1x t 1 y 1 z 1 t x y z t 1 = ( r 1) r 1 R r 1 + [ (r 1 ) r 1 ]v 1 R = r o + [ r o r (ξ) ] τ = x y z [ τ 1 I + v 1 r ] [ ] 1 (ro + ξ) = τ R r o v 1 [ I + v 1 r 1 r R ] =1+v 1 r1 r R =1+ω ω v 1 r 1 r t 2 t 1 = t 2 r 2 = r o + ξ 2 ξ 2 ξ t 2 t 1 t 2 x 2 = x + ξ 2x r r o

42 [ ] (ro + ξ 2 ) τ 2 = x 2 y 2 z 2 = τ r o e 1 = ρ 2 τ 2 = ρ 1 τ 2 ρ 1 τ 1 (1 + ω) = e 1 ϕ(x, y, z, t) = 1 e 1 µ ; (x, y, z, t) = 4πε R(1 + ω) 4π v 1 R(1 + ω) e 1 (x, y, z, t) R ω ϕ v 1 ω t 1 ϕ x y z t x t 1 v 1y = v 1z =0 β = v 1z v 1

43 ω =0; A =0; ϕ = e 1 4πεR e 1 R r r 1 r = (x,y,z ) r 1 =(x 1,y 1,z 1) B =0; E = e 1(r r 1 ) 4πεR 3 β l =1 = γβ(0, E z, +E y)= γβe [ 1 0, (z1 z), (y y 1 ) ] [ 4πεR 3 (x βt =(E x, γe y, γe z)=γe x1 + βt 1 ), (y y 1 ), (z z 1 ) ] 1 4πεR 3 x r 1 + v 1 (t t 1 )=(x 1 + βt βt 1,y 1,z 1 ) t E/B = c =

44 x 1 y 1 z 1 l =1 [ε 2 ( 2 /µ)] µεc 2 =1 E /B E/B c ( r 1 ) ( r 1 ) ( r 1 ) c =1 R = ( r 1 ); R = (r 1 r), R = r 1 = t t 1 γ 2 (1 β 2 )=1 l =1 E B = E B E B E (r r 1 )=0; B (r r 1 )=0 l =1 E (r r 1 )=γe ( r 1)+γβ [ E x (t 1 t)+b y (z z 1 )+B z (y 1 y) ] B (r r 1 )=γb ( r 1)+γβ [ B x (t 1 t)+e y (z 1 z)+e z (y y 1 ) ] E ( r 1 ) B ( r 1 )

45 ϕ r 1 v 1 = r 1 / t 1 x y z t x 1 y 1 z 1 t 1 ϕ x y z t 2 t t 1 = R = r 1 r 1 v 1 v 1 / t v 1 v 1 / t ( 1) 2 r 1 1 E/B = c = E B E B E = ϕ ; B =0

46 E x = l 2 E x ; E y = γl 2 E y ; E z = γl 2 E z ; B =(γβl 2, 0, 0) E =(β, 0, 0) E } x βt = x γl t (x βt) 2 + y 2 + z 2 = r 2 (β, 0, 0) x 2 γ + 2 y 2 + z 2 = l 2 r 2 r γlr ; lr ; lr r r γl ; A = 1 2 r l ; εe 2 x τ r l B = 1 2 ε(e 2 y + E 2 z) τ

47 C = 1 (By 2 + B 2 2µ z) τ B x = B x =0 A B C C =0; C = µεβ 2 B x βt y z } τ = (x βt) y z, τ = x y z = γl 3 τ A = γ l A ; B = 1 γl B ; A = l γ A ; B = γlb B =2A A A + B + C = γla (3 + β 2 ) A + B C = 3l γ A P = µε(e y H z E z H y ) τ = εβ(ey 2 + Ez) 2 τ =2βB =4βγlA E = A + B + C L = A + B C P E = L β L β P β = 1 E β β

48 β γ l β P = L β ; E = L + βp l =(1 β 2 ) 1 6 = γ 1 3 L = A + B C = l γ (A + B ) C =0 L = l γ L J = J J = L t ; J = L t x βt y z x y z J = J t = l γ t βx ; t = l t γ t / t x

49 r ; θr ; θr γlr ; θlr ; θlr A +B γlr θlr θlr A + B = 1 γlr f ( ) θ γ f r = θ =1 l =1; γr = ; θ = γ l = γ 1 3 ; γ = θ ; γlr = L = 1 ( ) θ γ 2 r f γ L = a 1 β 2 1+β r β 1 β a θ =1 ( ) 1 f = aγ 2 (1 β2 ) 1+β γ β 1 β = a 1+β β 1 β f r θ

50 θr = θ 2 r 3 = r = bθ m b L = 1 bγ 2 θ m f ( ) θ γ β L θ =0 mθ m 1 f + θ m f γ =0 f f = mγ θ θ = γ θ/γ =1 f m 1/γ β u =1/γ =(1 β 2 /2) ) f(u) =a (1+ β2 3 β f (u) = f/ u βf (u) = 2 3 aβ β =0 u =1 f = a ; f = 2 3 a ; f f = 2 3 m = 2/3 τ X τ x X

51 P t = X τ J = L t ; δj = X δu τ t = δu P t t δu Ox L θ r r θ ( ) L L δj = δβ + β θ δθ t δβ = (δu) t P δβ t = δu P t = δj t δj L β = P ; L θ =0 ( L/ β) θ β L β = L β + L θ θ β

52 θ r F (θ,r) J = [L + F (θ,r)] t (L + F ) =0 θ (L + F ) =0 r r θ r = bθ m r θ F θ L = 1 ( ) θ bγ 2 θ f ; m γ L θ = θf mf bγ 2 θ m+1 γ = θ θ/γ =1 F θ = (3m +2)a 3bθ m+3 (3m +2)a F = (3m +6)bθ m+2 m = 1 F = a 3bθ r θ L = 1 γ 2 r f ( ) θ ; γ L θ = f γ 2 r ; L r = f γ 2 r 2 γ = θ r = bθ m F r = a b 2 θ ; F 2m+2 θ = 2 a 3 bθ m+3

53 F = Kr α θ β K α β r = bθ m Kαb α 1 θ mα m+β = a b 2 θ ; 2m+2 Kβbα θ mα+β 1 = 2 3 a bθ m+3 α =3ζ ; β =2ζ ; ζ = m +2 3m +2 ; K = a αb α+1 r 3 θ 2 ζ m = 1 ζ =1 ζ = 2

54 L L = 1 ) (ε 2 2 τ 2 µ L =(β, 0, 0) r θ L r θ L r θ L r θ L r θ (L + F ) (L + F ) =0; =0 θ r L t = τ F F = Kr 3 θ 2 L = f(u)/γ 2 r u = θ/γ f(u) L v v = = vx 2 + vy 2 + vz 2 P t L ( ) L = P v = L v = P v = t P v 2 v t + v P v v t

55 v v t = t x =(v x, 0, 0) = (v, 0, 0) ; P x t P y t = t = t L = v x L = v y v x t = v t f x τ = P v x v t f y τ = P v v y t ( P / v) (P /v) P = ( L/ v) = = / t m m / 1 v 2 m θ = γ f(1) = a m = 1 L = a 1 v2 ; P = L b v = a v b 1 v 2 m a/b 3 1/4 a 3/4 K 1/4 P / v m (1 v 2 ) 3/2 P /v m (1 v 2 ) 1/2 m 1 v2 = h P = m v h ; P v = m h ; 1 P 3 v 2 v P v = m 3 h 3

56 = τ = 1 h t + 1 ( h ) = 3 t m = / t = m / 1 v 2 1 βv x = λ λv x = v x β ; λv y = v y γ ; λv z = v z γ h = 1 v 2 λh = h γ t = γλ t v x = 1 v x t γ 3 λ 3 t v y = 1 v y t γ 2 λ 2 t + v yβ v x γ 2 λ 3 t v z = 1 v z t γ 2 λ 2 t + v zβ v x γ 2 λ 3 t v v = 1 t γ 3 λ v 3 t βh2 v x γ 3 λ 4 t 1 v + 1 ) h t v (v v = F h 3 t m

57 F F l =1 F x = F x β λ ; F y = F y γλ ; F z = F z γλ ; {λ =1 βv x} F x m F x β m λ = 1 h v x t + v x h 3 v v t = 1 hγ 2 λ 2 v x t +(v x β) ( 1 h 3 λ v t β ) v x hλ 2 t = 1 v x hλ t + v x h 3 λ ( ) v 1 β t t hλ + v2 h 3 λ F y m = 1 h v y t + v y h 3 v v t = 1 v y hγλ t + βv y v x hγλ 2 t + v ( y λ h 3 γλ 2 t v ) βh2 x t F y m γλ = 1 v y hγλ t + v y h 3 γλ t F z F z

58 l =1 l F L h F θ r θ v L v =(v, 0, 0) L = P v x t v x v t = f x τ = F x L = P t v y v v y t = f y τ = F y P v = q(v) =q(v x); q(v x ) v x t = F x ; P v = s(v) =s(v x) s(v y ) v y t = F y ; q(v x) v x = F t x ; s(v y) v y t = F y l =(v, 0, 0) F x = 1 l 2 F x ; F = 1 l 2 γλ F ; (λ =1 βv x) v x = 1 v x t γ 3 λ 3 t ; v y = 1 v y t γ 2 λ 2 t v x v x v x = v x β, λ q(v x)=q v ( ) x β = γ3 λ 3 q(vx) ; s(v vx β λ l x)=s 2 λ = γλ l 2 s(v x)

59 Ω(v x )=s(v x )/q(v x ) l [ ] Ω(v x)=ω vx β 1 β 2 = Ω(v x ) 1 βv x (1 βv x ) 2 β v x Ω(v) =Ω(0) (1 v 2 ) v x =0 Ω(v) = s(v) q(v) = P v = A = m = 1 Ω(0) P v( P / v) P Ω(0) (v v 3 ) ( ) m v P = A 1 v 2 s(v) = P v = Avm 1 (1 v 2 ) m/2 (v β) m 1 (1 β 2 ) (1 m)/2 = vm 1 l 2 l β l v m =1; l =1

60 l =1 = = t { = = m 1 v 2 m = L L = m 1 v 2 (γ, γ ) (γt,γ ) T = (t, ) J = [ ε 2 t τ 2 2 2µ ]

61 F (F J 1 = ) t (F ) J + J 1 J J 1 F = ωτ = ω τ ω τ ω τ ω J 1 = ω τ t J 1 = ω τ t ω = ω F l =1; τ t = l 4 τ t = τ t. J 1 = J1

62 F F F F = Kr 3 θ 2 ζ =1 K = a 3b 4 K m L = a b 1 v 2 v L = a ) (1 v2 b 2 (a/b) a

63 t t t t r o t + t r o + r v 1 f(t,,, v 1 )=0 t

64 t t v 1 t = v 1 x v y = v z =0 β = v x v =0 F F = r l =1 x = γ(x βt) ; y = y ; z = z ; t = γ(t βx) λ =1 βv x =1 β 2 = 1 γ 2 ; r 3 F x = F x ; F y = γf y ; F z = γf z

65 x βt = x v x t ; r 2 = γ 2 (x v x t) 2 + y 2 + z 2 F x = γ(x v xt) r 3 ; F y = y γr 3 ; = V V = 1 γr F z = z γr 3, t x t x v x t y z t x y z t 1 x 2 + y 2 + z 2 t 2 x 2 + y 2 + z 2 t 2

66 = δ δt ; δ =(δx, δy, δz) ; v 1 = δ 1 δt ; δ 1 =(δ 1 x, δ 1 y, δ 1 z) δ δt δ 1 δ 1 t t x, y, z, t 1; δx, δy, δz, δt 1; δ 1 x, δ 1 y, δ 1 z, δ 1 t 1 P P P P P P x 2 + y 2 + z 2 t 2 ; x δx + y δy + z δz t δt P P P t v 1 δx δy δz δt δ 1 x δ 1 y δ 1 z δ 1 t 2 t 2 ; t 1 2 ; t v 1 1 v1 2 ; 1 v 1 (1 v2 )(1 v 12 ) ; f t = (l =1) f x = γ(f x βf t ); f y = f y ; f z = f z ; f t = γ(f t βf x ) f t t 2 f 2 t ; f t t ; δ f t δt ; δ 1 f t δ 1 t

67 f t T T = ρ = ; ρt = f t ρ ρf t f t δt δt δ, δt ρ /ρ δt /δt ρ ρ = γ(1 βv x)= 1 γ(1 + βv x) = 1 v 2 δt = 1 v 2 δt f t 1 Q O, P, P,P,Q f t δ δt δ 1 δ 1 t T v 1 2 T 2 1 v 2 ; Tt 1 v 2 ; v 1 T (1 v2 )(1 v 2 1) ; T 1 v 2 T x 2 + y 2 + z 2 t 2 (,t) (, ) (ρ, ρ) [ (, ) 1 v 2 ] = T [ ] m (, 1) 1 v 2 (m,m) m m / 1 v 2 (, ϕ)

68 r 2 t 2 =0; t = r t<0 t v 1 =0, t = v 1 1 v 2 t F r 4 r r 1 v 2 r v 1, 1 v 2

69 r r (1 v1) v 2 1, (r + v 1 ) 4 r + v 1 t = r v 1 0; r ; r v 1 ; 1 2 ; ( + r ) ; (v 1 ) ; 0 r o + t + t r = r o + r 1 t r 1 = r 1 r = r 1 + v 1 t v 1 r(r r 1 )= v 1 t t = r r 1 = r + v 1 0; r 1 + (v 1 ) ; r 1 ; 1 F 2 ; F [ r 1 + r 1 (v v 1 ) ] ; F (v 1 v) ; 0.

70 r 1 r (v v 1 ) 1 r 4 1 ; { r1 (v 1 v) 1 } ; r 1 r 2 1 r 1 (v v 1 ) r 3 1 ; 0. A B M N P M = 1 B 4 ; N = A B 2 ; P = A B B 2. C C 1 (A B) 2 (C 1) (A B) 2 A B C M N P T = γ 0 = 1 1 v 2 ; γ 1 = 1 1 v 2 1, γ = 1 1 β 2 r = t 0; A = γ 0 (r + r v) ; B = γ 1 (r + r v 1 ); C = γ 0 γ 1 (1 v v 1 ). (r,t); (γ 0, γ 0 T ); (γ 0, γ 0 ); (γ 1, γ 1 ) = a γ 0 + b + cγ 1 γ 0 v 1 ; t = r : T = ar γ 0 + b + cγ 1 γ 0

71 a b c T T =0 γ0 2 Aa + b + Cc =0 v 1 b =0; c = Aa C. } γ 0 = γ 1 =1; C =1; A = (v 1 v) r 1 ; B = r 1 ; r = r 1 + v 1 t = r 1 v 1 r = a( Av 1 ) Av 1 r 1 v 1 rv 1 = a( + rv 1 )=ar 1 = r 1 r 3 1 a 1/r 3 1 1/B 3 = C γ 1Av 1 γ 0 B 3 C ; T = Cr + γ 1A γ 0 B 3 C

72 1/B 3 1 B +(C 1)f 1(A, B, C)+(A B) 2 f 3 2 (A, B, C), f 1 f 2 A B C b a b c v 1 a b c = γ 1 [ (1 v1 )+v B 3 1 (r + ) ] C + rv 1 + (v 1 ), γ 1 Av 1 Cγ 1 Av 1 C = γ 1 ( + rv 1 ); = γ 1 [v 1 ]. C = + B 3 B 2 = e 2 b 2. /B 3 /B 3

73 v 2 v x 2 +y 2 +z 2 t 2 (,t) (, (m,m) m = m / 1 v 2 m = (ρ, ρ) (, ϕ) [(, )/ 1 v 2 ]

74 ε 2 2 µ (, ) = (m,m) t x y z t 1