m d2 x dt 2 = K( x), d 2 x j dt 2 = K i.

Transkript

1

2 P

3

4 m d2 x dt 2 = K( x), m δ ij d 2 x j dt 2 = K i.

5 C W C = C K i dx i δ ij δ ij λδ ij, m m λ d v dt K BA = K AB

6 R 4

7 E 3 R Σ Σ x = R x a, R T R = I, R... E 3 T 1, = 7

8 E 3 = O 3 T 3,... E 3 O 3 T 3 P 1 P 2 ((t 1, x 1 ) ((t 2, x 2 ) t = t 2 t 1 x = x 2 x 1 t x i 2 ψ(t, x) = ψ(t, x) t 2m t = t t t, x i = x k x i x k = Rk i x k i t ψ (t, x ) = 2 2m ψ (t, x ) ψ (t, x ) = ψ (t(t ), x( x ) R E 3 T 3 x = R x a vt t = t b

9 x = x vt t = t (O 3 (T 3 G 3 )) T 1, = 10 G 3... t t x t=0 d2 x dt t d 2 x 2 dt t 2 λt, x ±λ 2 x

10 c c c I I v I I v I I I v I I v

11 I I v v I I v = f ( v ) v v v v = v f ( f ( v ) ) = v g f R + g(0) = 0 g = f ( v ) = ± v, I I L I l < L l < L I I u g g =

12 l = L vt, L = l + vt, L w = T u = l T L < l T l l = L L =: α α = α( v ) l L α(0) = 1 α = L vt l = l + vt L vt = L αl vt = αl l T T = αl l L αl 1 α 1 u = l T, v = αl l T u + v = αl T, u v = l αl l w = L T = L αl l T L αl = αl T αl l α(l αl) = u + v 1 + (1 α 2 ) u, v 1 + (1 α 2 ) u v = 1 + (1 l α2 ) αl l = αl α2 l αl l) α = 1 L = L T = T w = u + v α > 1 w u α < 1 v α 2 1

13 w = ϕ(v, u) w w = ϕ( u, v) w = ϕ(v, u) (1 α 2 ) u v, α = α(v) (1 α 2 (v)) u v = (1 α2 (u)) u v 1 α2 (v) v 2 = 1 α2 (u) u 2 K 1 α 2 = Kv 2, K 1 c2 K = 0 c c α = 1 v2 c 2 α = L L L = 1 v2 c 2 L < L P 1 P 2 I l = 0 T = αt ( ) =

14 w = u + v 1 + (1 α 2 ) u v = u + v 1 + uv c 2 u = c w = c v c v < c α u > c u = c2 v w = v > 0 u > c w = v = c2 u w = w = P 1 P 2 L = l u = w = c2 v L > l > l P 2 P 1 λν = c v c E = c v ν E = c E /λ = c λ (1 v c ) = ν S(1 v c )

15 λ E = (c + v)τ S = c+v ν S ν E = c E /λ E = ν S 1+ v c v v r v δ ν E = ν S 1 v2 c v r c δ ν S v = v r v r = 0 ν E = ν S 1 v c 1 + v c ν E = ν S 1 v2 c 2

16 P 1 0 I I P 1 P 2 L = l + vt α T = L αl v = l + vt α 2 l αv = T + 1 v ( 1 α 2 ) l α α = 1 v2 c 2, t T, x L, t T, x l x = x + vt, t = 1 v2 c 2 γ := 1 α = 1 1 v2 c 2 t + v c 2 x 1 v2 c 2

17 x = γ (x vt) ( t = γ t vx ) c 2 ( ) x = f x + at + b x c t = gt d x a, b, c, d v v y = f(v)y f(v) = f( v) y = f( v)y f 2 = 1 f = 1 y = y z = z c v x = x + x x = x v v 2 v x = x x ( x v = 0)

18 (v, 0, 0) v, (x, 0, 0) x, (0, y, z) x ( t = γ t ) v x c 2 x = x + x = γ ( x vt ) + x = x + (γ 1) x γ vt = x + γ 1 v 2 ( v x) v γ vt v c d x dt = c d x dt = c c 2 ( t) 2 ( x) 2 = c 2 ( t ) 2 ( x ) 2 v = (v, 0, 0) t = γ ( t v ) c 2 x x = γ ( x v t) c 2 ( t 2) ( x ) [ 2 (c = γ 2 v 2) ( ) ] ( t) 2 v 2 + c 2 1 ( x) 2 = c 2 ( t 2) ( x) 2 ( s) 2 := c 2 ( t 2) ( x) 2 ( t) 2 = 1 c ( s)2 x=0

19 ( l) 2 = ( s) 2 t=0 ds 2 = c 2 dt 2 (d x) 2 ds ds 2 ds 2 { } }{{} { } }{{} = }{{} x 0 = ct c = c = / c = 1 c = 1 t = γ (t vx) t = vx x = γ (x vt) x = vt v = δ t 2 x 2 = t 2 x 2

20 t = iτ τ 2 + x 2 = τ 2 + x 2 τ, x τ = τ φ x φ i x = τ φ + x φ φ ϕ = iχ t = t χ x χ x = t χ + x χ χ = v φ = iχ

21

22 Q I I P v = 1 γ v > 1 x = iξ, t = iτ ξ τ v 1 v > 1 v > 1 v s = v s = 1 + ϵ c =

23 ω dω k dk > 1 ω(k) = k k 1 P = {Q : ( s) 2 = 0} P P P =

24 P P I x t = 0 x = γ(x vt) x = γ x x = γ 1 x = 1 v 2 x L = x > x = L y = y, z = z c

25 t = l c v = y, 1 v 2 = y y = ( v c ) I x = 0 t = γ t t = t 1 v 2 I x = 0 t = γ t t = t 1 v 2 T = T + T = T 1 v 2 T T v2 2 v 1

26 v T = d v T d 2 v 0f.v 0 d A B A B t A t B = t B T A t B = 1 2 (t A + t A ) ds 2 = dt 2 d x 2 = dt 2 d x 2 = dt 2

27 P ds ds ds = 1 v 2 dt s 1 < s 2 T 2 = T 1 1 v 2 T 2

28 v = 0.994c µ γ = 29.3, b g m/s γ = γ SR (1+αβ 2 +α 2 β 4 ) α , α v v t = γ(t + v x ) x = x + γ 1 v 2 v( v x ) + γ vt

29 x = ut [ x = u + γ 1 ] v 2 ( v u) + γ v w = x t = u γ + γ 1 γv 2 v( v u) + v 1 + v u ( γ 1 γv 2 = γ ) γ + 1, γ2 1 = y 2 v 2 v, u, w u, v, w w R w R w v u v u w = w( v, u) =: v u u v w = u+ v 1+ v u u v w = v + 1 γ u t

30 δ v v δ u v + δ v = v δ u v = (v, 0, 0) I δ u, δ v I v, δ v, v + δ v γ( v + δ v) γ( v) I I (1 γ)δθ I δ u I I I I I δθ T = (γ 1)δΘ I I

31 ω T = dθ T dt = (γ 1) dθ dt I I I I δθ = v ( v + δ v) v v + δ v = v δ v v 2 δ Θ = v δ v v 2 γ 1 (v 0) v2 2 w T (v 0) 0 v v ω T = (γ 1) v 2 w T

32 P x = (x 0, x 1, x 2, x 3 ) = (x i ) = (x 0, x) P, Q x, y y x P Q ( s) 2 = ( x 0 ) 2 ( x) 2 ( x) = η ik x i x k i=0 k=0 x = η ik x i x k +1 0 η = (η ik ) = u, v uv = 1 [ (u + v) 2 u 2 v 2] = η ik u i v k = u 0 v 0 u v = vu 2 uv = 0 v u = 0 x i = L i k xk a i x i = L i k xk

33 u = (u i ) u i = L i k uk u 2 = u 2 u v = uv u, v uv = 0 u u 2 > 0 u 2 = 0 u 2 < 0 u 0 > 0 u u 0 < 0 u t t u 0 = γ ( u 0 v u ) > 0, u 0 > 0, v u v u < u u 0 u ( u = ± ) u 2, 0 t u u = u ( 0, ) u 2, 0, 0 x u = (±1, 1, 0, 0)) (±a, a, 0, 0) u i = λu i, λ = γ(1 v) = 1 v 1 ± v ) v u u v v u u v v = λu

34 P P = {f x i = f i (x), η ik dx i dx k = η ik dx i dx k} η mn f m x i f n x k = η ik x j ( ) f m x i = ±1 f ( 2 f m f n η mn x j x i x k + f m x i ( 2 f m f n η mn x i x k x j + f m x k ( 2 f m f n η mn x k x j x i + f m x j f η mn = η nm m n 2 2 f n ) x j x k = 0 (ijk) ( ) 2 f n ) x i x j = 0 (kij) () 2 f n ) x k x i = 0 (jki) ( ) ( 2 f m f n η mn x j x i x k + f n 2 f m ) x k x i x j = 0 f n 2 f m η mn }{{ x k } x i x j = 0 A mk ( ) f n (A mk ) x k 2 f m x i x j = 0 f i x k Li k x i = f i (x) = L i k xk + c i

35 ( γ L = (L i k ) = b ) a T M η mn L m i L n k = η ik L T ηl = η ( ) ( γ b T 1 0 T ) ( ) ( γ a T 1 0 a M T 0 1 = T ) ( 1 0 T ) b M }{{} b T γ a M T ( ) L 1 γ b T = a M T ( ) ( ) ( γ a LL 1 T γ b T 1 0 = b M a M T = T ) 0 1 γ 2 a 2 = 1, γ b + M a = 0 L 1 x = 0 x 0 = γx 0 x = ax 0 I I v = x x 0 = a γ γ 2 a 2 = 1 v = a = a 1 + a 2 < 1 v γ = ± 1 1 v 2 1 v 2 L = L R L v ( ) ( ) γ γ v T γ a T L v = γ v 1 + γ2 = v v T a 1 + a at 1+γ 1 + γ }{{} γ 1 v 2 L R = LL 1 v = LL v ( ) ( ) γ a T γ a T = b M a 1 + a at 1+γ ( ) γ 2 a 2, γ a T a T a2 1+γ = at γ b + M a, b a T + M + M a at 1+γ ( 1 0 = T ), 0 R

36 a γ = γ 1 M a = γ b ( 1 + γ ) b a T = b a T 1 + γ 1 + γ R = M b a T 1 + γ L R x 0 x 0 ds 2 d x 2 L 1 I I b γ I I R a = M a b γ a2 1 + γ b = γ b γ γ b = b = R v L u = ( γu γ u u T γ u u 1 + γ2 u 1+γ u u u T L u L v L w( v, u) u v ), L v = ( ) γ a T L := L u L v = b M ( γv γ v v T γ v v 1 + γ2 v 1+γ v v v T ) γ = γ v γ u (1 + v u) γv 2 a = γ v γ u v + γ u u + γ u ( u v) v = γ v + 1 γ v u γ v γv 1+γ v ( v u) v } 1 + v u {{ } w( v, u)

37 b = γ w( u, v) M = 1 + ( ) γ2 v v v T + γ2 u u u T γ v γ u + γ v γ u 1 + u v u v T 1 + γ v 1 + γ u (1 + γ v )(1 + γ u ) L u v L = L R L w( v, u) ( = LR w( v, u) L R = L w( u, v) L R ) R = M b a T 1 + γ u, v R R = v T +... u T v u v u = v γ γ u w, a = γ u u +... v = γ w( v, u) v w u 1 u γ u γ2 v 1 + γ v v v T + γ v u v T b = a( v u) γv ( u + v) ( b a T γv u v 2 T + γ v v γ v v T + u T + = γ 2 v u v T + γ v v u T + γ 2 v R = M 1 ( 1 + γ b a T 1 + γ v ( γ γ v (1 + v u) }{{} u 1+γ u 1+γv ) γ2 v ( u v) v T 1 + γ v ) v v T ( 1 + γ v u v 1 + γ v γ2 v 1 + γ v ) u v T γ v v u T 1 + γ v ( ) ) 1 u v 1 + γ v (1 + v u) γ2 v 1 + γ v = γ2 v 1 + γ v 1 + γ v

38 = 1 + γ v 1 + γ v ( u v T v u T ) R x = x a n x x + γ v 1 + γ v [ u( v x) v( u x)] }{{} ( v u) x γ v α = α n v u γ2 v w v 1 + γ v 1 + γ v w v γ2 v v I a = v v ( w = v + v) 1 + γ v t 0 : ω T = γ2 v v d v 1 + γ v dt γ < 0 γ 1 R = 1 T = P = L L = ±1 L + L = +1 ( ) ( ) L L 0 0 > 0 L0 0 1 (L0 0 )2 L α 0 Lα 0 = 1 L + L + := L + L T P

39 x(t) v = d x dt v < 1 : s x i (s) u i := dxi ds dxi ds = = u = ( dt ds, d x ) = dt ( 1, d x ) = γ(1, v) v α = uα ds ds dt u 0 s = t u = (1, 0) v 1, γ 1 u (1, v) u 2 = η ik dx i ds dx k ds = ds2 ds 2 = 1 u u dt ds > 0 a i := dui ds = d2 x i ds 2 0 = d ds (η iku i u k ) = 2η ik u i a k ua = 0 a a = (0, b), b = d v dt b a 2 = b 2 = b 2 b = (b, 0, 0) u(s) = ( f(s), f(s), 0, 0) a 2 = f 2 (s) = b 2 f = ±b(s s 0 ) = bs

40 x 0 (0) = 0, x(0) = 0 s ( 1 x(s) = u(s )ds = b (bs), 1 ) ((bs) 1), 0, 0 b = = m p = mu (m, m v) p (p 1, p 2, p 3 ) = γm v γm p 0 p 0 = m 1 v 2 v 1 m + mv2 2 p 0 E = m 1 v 2 = m

41 E 0 = mc 2 = p 0 v 1 p 2 = (p 0 ) 2 p 2 = m 2 p 0 = m 2 + p 2, v = p p 0 = p m 2 + p 2 = p E dp ds = ma = K Ku = 0 K = d p ds = γ d p dt K 0 = dp0 ds = γ de dt = γ

42 N A=1 p A = N p 1 + p 2 = p 3 + p 4 ( ) p 0 1 A = m A + T A, T A = m A 1 1 v 2 m 1 + T 1 + m 2 + T 2 = m 3 + T 3 + m 4 + T 4 p 1 + p 2 = p 3 + p 4, p A = m A v A 1 va 2

43 m 3 = m 1, m 4 = m 2 T 1 + T 2 = T 3 + T 4, T A = 1 2 m A v A 2 p A = m A v A p = q 1 + q 2 p 2 = q q q 1 q 2 q 0 1 q0 2 q 1 q 2 > 0 q A = m 2 = m m (q 0 1q 0 2 q 1 q 2 ) (q 0 A )2 m 2 A < q0 A A m2 A < m2

44 m = 0 p i = m dxi ds = 0 m 0 s 0 v 1 u x p p 2 = 0 p = ( p, p) 0 p = k, k = (ω, k) ω = k = 2π λ = 2π λν = c k I (v,0,0) I ϑ 1 k = ω ϑ ϑ 0

45 1 γ k = ω ϑ ϑ = ω γv 0 γv γ 1 1 ϑ ϑ 1 0 ω = γω(1 v ϑ) ω ϑ = γω( v + ϑ) ω ϑ = ω ϑ) ( ) () ( ) ω = δ ω, δ = 1 v 2 1 vcosϑ / : ϑ = ϑ v 1 v ϑ / : ϑ = ( ϑ 2 = ϑ v γ(1 v ϑ) ) 1 + v 1 v ϑ 2 2 ϑ + 2 ϑ = 1 v ϑ ϑ ϑ = π 2 ϑ = ( π 2 ϑ ) α = v α 20 α = k ϕ k 0, 3 α = v c α SRT > α Newton c 1 v 2 v 1 ϑ = 180 v, k

46 e γ p + q = p + q m 2 = q 2 = (p p + q) 2 = p 2 + p 2 2pp + +2q(p p ) + q 2 q(p p ) = pp e q = (m, 0), p = (ω, k) m (ω ω ) = 2 ωω (1 ϑ) ( ϑ = ( p, p ) ) 2π ωω ω = 2π λ λ λ = h (1 ϑ) m

47 h mc mc Q B p L e n p + p p + p + π 0 e + + e 2γ π 0 2γ, n p + + e + ν e = (E pot + E kin ) > 0 p + e + 2γ E B = m p + m e m 1 2 m eα 2 = 13, 56 (α = e2 c 1 ) 137 M M = E B m p + m e E B E B m p = 1 2 α2 m e m p 10 8 M M M M , 4

48 X? m A x A X P ( ) d x a d x a p A = m A m A ds A dt X(t) = p 0 A x A (t) p 0 A C X = P P 0 C v, v C p 0 2 > p0 1 I CM : A p A = 0 I CM l ik = x i p k x k p i x i l α = ϵ αβγ x β p γ l i = ϵ ijmn u i l mn ϵ ijkl ϵ u A lik A

49 S[C] L C = P2 P 1 ds C τ2 S[x(τ)] = m ẋ2 dτ τ 1 Ldτ ( d ) dτ τ S τ f(τ) x x f ẋ2 ẋ 2 f S f τ 2 > τ 1 τ 2 < τ 1 x 0 τ :, x 0 τ : τ = t L = mc 2 1 v2 c 2 mc2 + m 2 ( ) d x 2 dt S[x(λ)] = 1 2 λ2 λ 1 ( ) dx 2 dλ dλ λ aλ + b

50 d2 x dλ 2 = 0 m > 0 λ s m S = 1 2S λ π i = L ẋ i = mη ikẋ k dx k ẋ2 }{{} = mη ik ds ẋ2 dτ=ds ( πα = p α, π 0 = p 0) V = n 1 = 1 f : V R, v ±ϵ v f(αv + βw) = αf(v) + βf(w) ( 0, ϵ 0) f(v) = f i v i

51 V := { } (αf + βg)(v) = αf(v) + βg(v) V = V = v uv u : u (v) = uv = η ik u k v i = u i v i V V u i := η ik u k ϵ u u u i..., u i... u uv = u i v i = u i v i (u i ) = (η ik u k ) = (u 0, u) u i = η ik u k, η ik (η 1 ) ik = η ik η 2 = η il η lk = δk i u i = η ik u k = η ik L k l ul = η ik L k l ηlm u m = ( (L T ) 1) m u i m =: L m i u m L T ηl = η (L T ) 1, η ηlη = (L T ) 1 x i x i