m 1, m 2 F 12, F 21 F12 = F 21 r 1, r 2 r = r 1 r 2 = r 1 r 2 ê r = rê r F 12 = f(r)ê r F 21 = f(r)ê r f(r) f(r) < 0 f(r) > 0 m 1 r1 = f(r)ê r m 2 r2 = f(r)ê r
r = r 1 r 2 r 1 = 1 m 1 f(r)ê r r 2 = 1 m 2 f(r)ê r ( r 1 r 1 2 = + 1 ) f(r)ê r m 1 m 2 µ 1 µ = 1 m 1 + 1 m 2 µ = m 1m 2 m 1 + m 2 µ r = f(r)ê r µ r R = m 1 r 1 + m 2 r 2 m 1 + m 2 V = R = m 1 v 1 + m 2 v 2 m 1 + m 2
P = m 1 v 1 + m 2 v 2 = (m 1 + m 2 ) V r, R m 2 r 1 = R + r m 1 + m 2 r 2 = R m 1 r m 1 + m 2 K = 1 2 m 1 r 1 2 + 1 2 m 2 r 2 2 = 1 ( ) ( ) 2 m m 2 m 2 1 R + r R + r m 1 + m 2 m 1 + m 2 + 1 ( ) ( ) 2 m m 1 m 1 2 R r R r m 1 + m 2 m 1 + m 2 = 1 2 (m 1 + m 2 ) R 2 + m 1m 2 m 1 + m 2 ṙ 2 M = m 1 + m 2 V = R, v = r K = 1 2 M V 2 + 1 2 µ v 2 µ v = r L = r 1 m 1 v 1 + r 2 m 2 v 2 L = R (M V ) + r (µ v)
R = 0 V = 0 L = r µ r µ r = r 1 r 2 L r, v W 1 2 = 2 1 F d r F = f(r)ê r U(1) U(2) = U(r) = r 2 1 f(r)dr r 0 f(r)dr E = 1 2 µ v 2 + U(r) L = r µ r r F = 0 L = C L = 0
r = rê r v = r = ṙê r + r ϕê ϕ K = 1 2 µ (ṙê r + r ϕê ϕ ) (ṙê r + r ϕê ϕ ) = 1 2 µ(ṙ2 + r 2 ϕ2 ) E = 1 2 µṙ2 + 1 2 µr2 ϕ2 + U(r) L = µr 2 ϕ µ r, ϕ ϕ E = 1 2 µṙ2 + L2 + U(r) 2µr2 r ṙ µ V (r) = L2 + U(r) 2µr2 E = 1 2 µṙ2 + V (r) µ V (r) 1 2 µṙ2 1 2 µṙ2 = E V ṙ = dr/dt dr dt = 2 µ (E V )
dt = dr 2 µ (E V ) t t 0 = r r 0 dr 2 µ (E V ) U(r) t = t(r) r = r(t) V (r) r(t) r(ϕ) ϕ = dϕ = L µr 2 dϕ = L µr 2 dt L dr µr 2 2 µ (E V ) ϕ ϕ 0 = L µ r r 0 dr r 2 2 µ (E V ) m 1, m 2 v 1, v 2 r = r 1 r 2 v = v 1 v 2 U(r) = 0 V = 0 + E = 1 2 µ v 2 L2 2µr 2 = L2 2µr 2
E = 1 2 µ ṙ2 + θ r, v L = r µ v = µrv θ = µvd V = 1 d2 µv2 2 r 2 L2 2µr 2 E = 1 2 µv2 = r2 d 2 V E = c r = d E = V (d) r < d r = d 1 2ṙ2 = E V ṙ = 0 r = d r > d r 2 > 1 V (r) < E d 2 K = E V > 0 ṙ > 0 1 r 2 2 2 1 1 d r U(r) = G m 1m 2 r V (r) = G m 1m 2 r + L2 2µr 2
E,V r 10 8 6 4 2 0.5 1.0 1.5 2.0 r 1/r 2 1/r r 0 V + r 1/r dv /dr = 0 r 0 = L 2 Gm 1 m 2 µ, V min = L2 2µr 2 0 E > 0 r min E = V (r min ) 1 2 µṙ2 = E V 0 r min ṙ = 0 r min E > 0 E = V (r min ) V min < E < 0 r min, r max E = V (r)
E,V r E 0, E 0, E Vmin 4 2 0.2 0.4 0.6 0.8 1.0 1.2 r 2 4 6 V min E < 0 E = V min ṙ = 0 L 0 ϕ 0 µ r = f(r)ê r f(r) = µ r µr ϕ 2 0 = µ(r ϕ + 2ṙ ϕ) L = µr 2 ϕ ϕ = L µr 2 µ r = f(r) + L2 µr 3
f(r) f(r) = du dr µ r = d dr ) (U + L2 2µr 2 = dv dr V = U + L2 2µr 2 ṙ ṙ = dr dr = ϕ dt dϕ ϕ 1 dr r 2 dϕ = d 1 dϕ r ṙ = L ( ) d 1 µ dϕ r r = d dtṙ = ϕ d ( dϕṙ = ϕ L µ ) d 1 dϕ r ϕ r = L2 d µ 2 r 2 dϕ 1 r u = 1 r d 2 u dϕ 2 + u = µ 1 L 2 u 2 f( 1 u )
f(r) = k r 2 = ku2 µ 1 L 2 u 2 f( 1 u ) = kµ L 2 d 2 u dϕ 2 + u = kµ L 2 ũ = kµ L 2 A (ϕ ϕ 0 ) u(ϕ) = kµ L 2 + A (ϕ ϕ 0) = kµ L 2 (1 + ϵ (ϕ ϕ 0)) ϵ r = 1/u r(ϕ) = L 2 /(kµ) 1 + ϵ (ϕ ϕ 0 ) ϵ ϵ < 1 ϵ = 1 ϵ > 1 ϵ
ϵ ϵ r min, r max E = k + r min L2 2µr 2 min E = 1 ( ) L 2 2k 2r min µr min r min = r L2 0 1 + ϵ = µk 1 + ϵ E = k2 µ 2L 2 (ϵ2 1) ϵ = 1 + 2L2 E k 2 µ ϵ < 1 ϵ = 0 E = k2 µ 2L 2 r = r 0 = L2 µk ϵ = 0 r = r 0 dv dr = 0 r 0 = L2 kµ V min = E = k2 µ 2L 2
ϵ < 1 ϵ < 1 ϕ ϕ ϕ 0 = π, 0 r max = r 0 1 ϵ r min = r 0 1 + ϵ r min, r max r(ϕ) = (ϕ + 2π) ϵ < 1 ϕ 0 = 0 r(1 + ϵ ϕ) = r 0 r = ϵr ϕ = r 0 r 2 = x 2 + y 2, x = r ϕ (x + c) 2 a 2 + y2 b 2 = 1 a = r 0 1 ϵ 2 r 0 b = 1 ϵ 2 c = r 0ϵ 1 ϵ 2 b a = 1 ϵ 2 ϵ
c a c = ϵa m 1, m 2 µ = m 1m 2 m 1 +m 2 m 1 = M, m 2 = m p M m p µ = M m p M + m p m p R = M R + m p r p M + m p R L = µr 2 dϕ dt L 2µ dt = 1 2 r2 dϕ da = 1 2 r2 dϕ dt = T A = πab L 2µ T = πab
b = a 1 ϵ 2 L 2 4µ 2 T 2 = π 2 a 2 b 2 = π 2 a 4 (1 ϵ 2 ) r 0 = a(1 ϵ 2 ) L 2 4µ 2 T 2 = π 2 a 3 r 0 r 0 = L 2 /(kµ) T 2 = 4π 2 µ k a3 k = Gm 1 m 2 = Gµ(m 1 + m 2 ) T 2 = 4π 2 G(m 1 + m 2 ) a3 m 1 = M A= a b O r+dr r d 1 2 da= r d 2 m 2 = m M m T 2 = 4π2 GM a 3
n R s = nr m v2 R s = G Mm R 2 s v2 = G M R 2 R s R 2 Rs 2 = g R2 R 2 s T v = ωr s = 2π T nr T R T = 2π g n3/2 R = 6371 Km 6371 10 3 T = 2π s 5063 s 84 min 9.81 n T = 24 3600 s T 2/3 3 g R (2π) 2/3 = 6.63 R s 60R 6371 10 3 T = 2π s60 3/2 2.35329 10 6 s 27.3 days 9.81 ω = 2π/T ω
Log Ω 1.0 0.5 0.5 1.0 1.5 Log R 0.5 1.0 T = 88/365 0.39 365π 44 0.72 146π 45 1 2π 1.52 730π 687 5.2 146π 867 9.5 146π 2151 19.19 π 42 30.97 73π 6019 ϵ = 1 ϵ > 1 r = r 0 1 + (ϕ ϕ 0 )
ϕ ϕ 0 = ±π ϵ = 1 ϕ 0 = 0 r = r 0 r ϕ = r 0 x x 2 + y 2 = r 2 0 2r 0 x + x 2 y 2 2r 0 x = r 2 0 ϵ > 1 r = r 0 1 + ϵ (ϕ ϕ 0 ) (x c) 2 a 2 y2 b 2 = 1 a = r 0 ϵ 2 1 b = c = r 0 ϵ 2 1 r 0ϵ ϵ 2 1 E > 0 v 0 b b L = µv 0 b E = 1 2 µv2 0 ϕ 0 = π r 0 r = 1 ϵ ϕ
r cosϕ α = 1 ϵ ϵ E ϵ = 1 + 2EL2 µk 2 ( ) 2Eb 2 ϵ = 1 + k r 0 r 0 = L2 µk = 2Eb2 k r = 1 2Eb 2 k 1 + ( 2Eb k ) 2 ϕ ϕ = π r min = 1 + 2Eb k 1 + ( 2Eb k ) 2 b 0 b r min
E r min b 0 < r min < b ϕ α = 1 ϵ = 1 1 + ( ) 2Eb 2 k Θ Θ = π 2ϕ α Θ 2 = ϕ α = 1 ϵ Θ 2 = 1 ϵ b ( ) 2Eb 2 ϵ 2 = 1 + = 1 k Θ 2 b = k 2E Θ 2
v 0 Θ ϵ = 1 Θ, r 0 = k Θ 2E 2 2 2 U(r) = k r, k > 0 F (r) = k r 2 r = r 0 1 + ϵ (ϕ ϕ 0 ) ϕ 0 = 0 ϕ = 0 r min = r 0 ϵ 1 r ϕ α = 1 1 ϵ Θ = π 2ϕ α
b θ b = b(θ) θ = θ(b) b N t A n t N t = n t A R σ = π R 2 Σ = n t σ A
p = n tσ A A = n tσ N π N σ N σ = n t σn π R π = N π /( t) R σ = n t σr π 1 = 10 28 m 2 N π d σ ρ m n N σ = n t σn π n t N t m n M = N t m n A V = A d ρ = M V = N tm n A d = N t A mn d = n m n t d n t n t = ρ d m n
N σ = ρ d m n σn π N π = 10 4, σ = 1.5barns, ρ = 2.7 10 3 Kg/m 3, m n = 2.7 1.66 10 26 Kg, d = 10 4 m R 1, R 2 b < R 1 + R 2 σ = π(r 1 + R 2 ) 2 σ tot = i σ i N σ = σ tot n t N π z θ, ϕ
θ, ϕ Ω = A r 2 A r dθ, dϕ θ, ϕ da da = r 2 θ dθ dϕ dω = θ dθdϕ θ, ϕ Ω = dω = 4π N π dω N σ (dω) = N π n t dσ(dω) dσ(dω) dω dω dσ(dω) = dσ(θ, ϕ) dω dω
dσ(θ,ϕ) dω σ = dσ(θ, ϕ) π 2π dω dω = θdθ 0 0 dσ(θ, ϕ) dϕ dω z z ϕ b b + db dσ = 2πb db dω = 2π θdθ b = b(θ) dσ dω = b db θ dθ b = b(θ) dσ dω > 0 R b K K K = K v = v L = L µrv α = µrv α = µbv
α = α b = b(θ) θ = π 2α b = R α = R θ 2 db = R 2 θ 2 dθ dω = 2π θdθ dσ dω = R2 4 σ = 2π 0 π dϕ 0 R 2 dθ = πr2 4 db b d z d bdb dd
F = k qq r 2 = λ r 2 ˆn ϕ ±ϕ a θ = π 2ϕ a b b = b(θ) p = p p p, p p = m v θ p = 2 p θ 2 p = F dt p ˆn p p = F ˆndt = F n dt
F n = λ r 2 ϕ λ = kqq λ p = ϕdt r2 ϕ = ϕ(t) dt = dϕ = dϕ ϕ dϕ dt L = µr 2 ϕ = µbv 1 ϕ = r2 bv dt p = λ bv dt = dϕ ϕ = r 2 dϕ bv ϕa ϕ a ϕdϕ = 2λ bv θ 2 θ = π 2ϕ a b = 2λ µv 2 θ 2 θ db dθ = λ 1 2µv 2 2 θ 2 dσ dω = ( kqq 4E 2 θ 2 ) 2 λ E = 1 2 µv2
dϕ = ± 1 dr r 2 2µE + 2µk 1 L 2 L 2 r 1 r 2 q 2 = 2µE L 2 + µ2 k 2 L ( 2 1 ω 2 = r µk ) 2 L 2 dϕ = ± 1 dr r 2 q 2 ω = dω 2 q 2 ω 2 ω = q θ q θdθ dϕ = ± q 2 q 2 2 θ = ± θdθ θ dϕ = dθ ϕ ϕ 0 = θ ω = q (ϕ ϕ 0 ) ω, q 1 r µk 2µE L 2 = L 2 + µ2 k 2 L 4 (ϕ ϕ 0 ) r r = L 2 µk 1 + 1 + 2EL2 (ϕ ϕ µk 2 0 )
F (r) t d 2 r dt 2 = L2 mr 3 + F (r) ϕ d 2 u + κu + λf(u) = 0 dϕ2 κ f(u) F (r) ( 1 F (r) = k r 2 l ) r 3 k, l r(ϕ) ϵ v v R e, R a R e = 1 A.U, R a = 1.5 A.U. d 2 u + u = f(ϕ) dϕ2 f(ϕ) v = u du(ϕ) dϕ = v(ϕ) dv(ϕ) = u(ϕ) + f(ϕ) dϕ
w = ( u v ) ( 0 1, J = 1 0 ) ( 0, w 0 = f(ϕ) d dϕ w(ϕ) = J w(ϕ) + w 0(ϕ) ϕ ] w(ϕ) = e [a Jϕ 0 + e Jϕ w 0 (ϕ )dϕ ) 0 = ( c0 d 0 ) ( e Jϕ ϕ ϕ = ϕ ϕ ) (, e Jϕ ϕ ϕ = ϕ ϕ ) ϕ ϕ ( ) ( e Jϕ w 0 (ϕ )dϕ ϕ ϕ 0 = ϕ ϕ f(ϕ) ( ) ϕ ϕ = f(ϕ )dϕ + ϕ ϕ f(ϕ )dϕ ) dϕ ( Is (ϕ) I c (ϕ) ) u(ϕ) = c 0 ϕ + d 0 ϕ I s ϕ + I c ϕ
v 0 z = v2 0 2g g 2v 2 0 (x 2 + y 2 ) v 0 c v 2 v = g(1 + v 2 /v 2 τ ) d/dt = vdv/dz ( ) z max = v2 τ 2g 1 + v2 0 vτ 2 t = v τ g (v 0/v τ ) M v = (v x, v y ) m 1, m 2 M = m 1 +m 2 K δ = 1 2 µv2 0 µ = m 1m 2 /(m 1 + m 2 ) v 0 m 1, m 2 fracdmdt = k z(t) = v α t 1 m 2 gt2 v α k m 0 m
y M m 1 m 2 x r 0 r = 1 + ϵ ϕ ϵ = 1 ϵ > 1 m r = k ϕ ϕ k ϵ = 0.618 F C F d r = 0 F R xy z P x dl = R dϕ dm = σrdϕ Q OQ = (R ϕ, R ϕ, 0) OP = (x, 0, z)
QP r = OP OQ = (x R ϕ) 2 + R 2 2 ϕ + z 2 = x 2 + R 2 2xR ϕ + z 2 dm P du = G N dm r = G NσR dϕ r 2π dϕ U = G N σr x 2 + R 2 2xR ϕ + z 2 = 2G N σr 0 π 0 dϕ x 2 + R 2 2xR ϕ + z 2 P x a 2 = (x + R) 2 + z 2 b 2 = (x R) 2 + z 2 a 2 + b 2 = x 2 + R 2 + z 2, a2 b 2 = 2Rx 2 2 r r 2 = a2 + b 2 2 a2 b 2 2 = a 2 2 ϕ 2 + b2 2 ϕ 2 ϕ ϕ = π 2θ θ = [0, π 2 ] r r 2 = a 2 2 θ + b 2 2 θ
0 U = 2G N σr = 4G N σr a π/2 π/2 0 2dθ a 2 2 θ + b 2 2 θ dθ 2 θ + b2 a 2 θ 2 b/a b/a A, B A, B (xz) x AP B C R C A, B C R 2 C = (CA)(CB) z a = b U = G N 2πσ a = G N m R 2 + z 2 m 0 m a g v a m r = k ϕ ϕ k M V m M v
M ω T 3π ρg l ω T l/2 F C C F d r = 0 F M 0 dm/dt = λ, λ > 0 g = g(t) U m E = T + U nabla = ê1 h 1 q 1 + ê2 h 2 q 2 + ê3 h 3 q 3 q 1, q 2, q 3 h 1,2,3 h i = r q i q 1 = r, q 2 = θ, q 3 = ϕ V `
ê 1 ê r, ê 2 ê θ, ê 3 ê ϕ h 1 = 1, h 2 = r, h 3 = r θ divergence [ A 1 = (A 1 h 2 h 3 ) + (A 2 h 3 h 1 ) + ] (A 3 h 1 h 2 ) h 1 h 2 h 3 q 1 q 2 q 3 A = A 1 ê 1 + A 2 ê 2 + A 3 ê 3 F θ = F ϕ = 0 F ( r) = F r (r)ê r θ x 3 z ϕ x 1 x x 1 x 2 xy
curl rot A A 1 h 1 ê 1 h 2 ê 2 h 3 ê 3 = h 1 h 2 h 3 q 1 q 2 q 3 h 1 A 1 h 2 A 2 h 3 A 3 A 1 ê 1 rê 2 r θê 3 = r 2 θ r θ ϕ A r ra θ r θa ϕ F ( r) = F r (r)ê r r(ϕ) = r 0 (1 + a ϕ) F = b r p + c r q r 0, a a = 1 k, a U(r) = k e r/a r r = r 0 1 + ϵ ϕ ϵ = 1/2 v v v = 2 3 v
2 v = 3 v v = 8.5 10 3 m/s Σ 0 Σ a m v Σ r 0 = r + R F = m r 0 r 0 = r + R = r0 + a
m r 0 R 0 R Σ a = R Σ0 m r m r = F m a F α = m a O O O ω = ωˆn ω O v = ω r
r O ê dê dt = ω ê Σ 0 Σ Σ ω Σ 0 A ) ( da dt ( da dt : Σ 0 ) 0 : Σ ê 1, ê 2, ê 3 Σ A = A i ê i d A dt = da i dt êi ê i
A ( da ) = da i dê i dt dt êi + A i dt 0 ( da ) dt A i dê i dt = A i( ω ê i ) = ω (A i ê i ) = ω A 0 = d A dt + ω A ( da m ) = F dt ( ) d r = d r + ω r dt dt 0 0 ω = 0 ( d 2 ) ( ) ( ) r d d r dt 2 = + ω r 0 dt 0 dt = d2 r dt 2 + 2 ω r + ω ( ω r) d2 r dt 2 2 ω r Coriolis ω ( ω r)
F = m d2 r dt 2 + 2m ω r + m ω ( ω r) m d2 r dt 2 = F 2m ω r m ω ( ω r) = F + F C + F φ F C = 2 ω r Coriolis F φ = m ω ( ω r) F C, F φ r = a(1 + 3/2 ϕ) m m F = k/r 3, k < 0 m r ϕ = C r 120 0 d = 1µm N π = 10 10 A = 0.1mm 2 120 0 ρ = 10.5 10 3 Kg/m 3 ω z = ω2 2g ρ2 z ρ
r(0) = 0, v(0) = v 0 h = 180 θ = π 6 R ω m O r m ω M v = (v x, v y ) m 1, m 2 M = m 1 + m 2 m 1, m 2 K δ = 1 2 µv2 0 µ = m 1m 2 /(m 1 + m 2 )
v 0 m 1, m 2 r(ϕ) = k ϕ n, µϵ k, n > 0 F = F (r) n U(r) U u + u = µ/(ul) 2 F (1/u) ẍ+2αẋ+ω 2 0 x = 0 x(t) : i) (a + bt)e αt, ii) e αt (ω 1 t δ), iii) e αt (βt), a = 1, b = 1, δ = π 2, α = ω 0 2 ω 2 1 = ω2 0 α2 = β 2 r(0) = 0, v(0) = v 0 y M m 1 m 2 O r x ( (
x t,v t 0.4 0.2 2 4 6 8 t 0.2