LCR e ix LC AM m k x m x x > 0 x < 0 F x > 0 x < 0 F = k x (k > 0) k x = x(t)

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1 LCR e ix LC AM m k x m x x > 0 x < 0 F x > 0 x < 0 F = k x k > 0 k x = xt

2 m 2 x t 2 = k x 2 x t 2 = ω 2 0 x ω0 = k m ω α x t = xt 3 t sin t = cos t 2 sin t = sin t, 2 t t +α y = gt t sinω 0 t + δ t f gt = f y y = ω 0 cosω 0 t + δ, cos t = sin t 2 cos t = cos t 2 t y t cosω 0 t + δ t = f gt gt gt t = ω 0 sinω 0 t + δ x t = ω 2 0 xt 2 2 xt = A sinω 0 t + δ xt = a sin ω 0 t + b cos ω 0 t A, δ a, b x0 x 0 x0 = x 0, x 0 = v 0 xt = v 0 ω 0 sin ω 0 t + x 0 cos ω 0 t 2 x y = yx y k = k y x k y n + a n 1 y n a 1 y + a 0 y = b k = 1, 2,, n 1 n b n xt t 2

3 vt = x t F F = bv b > 0 F + F m 2 x = k x bv 2 x = ω 2 t 2 t 2 0 x 2µv b 2µ = m µ x = v, 2 x = v t t 2 t t x = v x 0 1 x = t v = ω2 0 x 2µv t v ω µ v 0 1 A = ω 2 0 2µ v s s A u x v P v e x 2 v p p + O e 1 x x u x v x v = xe1 + ve 2 e 1, e 2 x v OP x OP = = xe v 1 + ve 2 p + = Pe 1, p = Pe 2 OP = s p + + u p = spe 1 + upe 2 = sp 1 + up 0 0 = P 1 s u

4 x s = P v u P p+ p = Pe1 Pe 2 = P e1 e 2 = PI = P P = p+ p p +, p A Ap + = λ + p +, Ap = λ p λ ± Ap = λp p 0 A λi = 0 A λi = λ 1 ω 2 0 2µ λ = 0 λ2 + 2µλ + ω 2 0 = 0 λ = µ ± µ 2 ω 2 0 = λ ± λ ± p ± = p q 0 λ± 1 p A λ ± I p ± = = 0 λ 2µ λ ± q ± p + q = 0 p ± = p q 1 ω 2 0 λ ± P = 1 1 p + p = λ + λ, D = P 1 λ+ 0 AP = 0 λ P P = a b t c a b s = as + bu as = t + bu t c u t cs + u cs t + u t a b s = t a b s c u = t c t u x v = P s u t P s = P u t s u

5 342 7 x v = P s u x x = A P s s = AP s s = P 1 AP t v v t u u t u u s s = D s λ+ s = t u u t u λ u t s = λ +s t u = λ u s, u e t at = a t log e a, t et = e t, t eλt = λe λt a a = e log e a e e s t = λ + st u t = λ ut st = ae λ +t, ut = be λ t a, b λ 1 ω 2 0 2µ λ 2 2 x t 2 + 2µ x t + ω 2 0 x = 0 = λ 2 + 2µλ + ω 2 0 = 0 x = e λt 2 e λt t 2 + 2µ eλt t + ω 2 0 eλt = λ 2 + 2µλ + ω 2 0 eλt = 0 λ = λ ± λ

6 v = x t x, v 1 3 x s 1 1 ae λ+t ae = P = v u λ + λ be λ t = λ +t + be λ t aλ + e λ +t + bλ e λ t exp x = e x e λ ±t = exp µ ± µ 2 ω 2 0 t xt = ae λ +t + be λ t µ ω 0 > 0 µ ω 0 < 0 0 > µ 2 ω 2 0 = ω e λ ±t = e µ±iω t = e µt e ±iωt ω = xt = ae λ +t + be λ t = e µt cos ωt ± i sin ωt ω 2 0 µ2 = e µt {a + b cos ωt + ia b sin ωt } 3 t x x x = A A t v v t I v = t 1 ω 2 0 2µ t x = 0 v x = e λt λ 1 e λt ω 2 0 2µ λ λe λt = 0 e λt λe 0 λt y = yx n y k = y k x k = 1, 2,, n 1 y, y 1, y 2,, y n 1 1 n y = e λx n

7 344 7 x0 = x 0, v0 = x 0 = 0 a, b v0 = aλ + + bλ λ ± = µ ± iω x0 = x 0 = a + b v0 = 0 = µa + b + iωa b a = x 0 2 i µx 0 2ω, b = x i µx 0 2ω i x0 = x 0, v0 = 0 xt = x 0 e µt {cos ωt + µ ω sin ωt } LCR 2 R V V R V I

8 V I = V/R V = RI 0 V V RI = 0 2 Q V C Q = CV C C Q = CV C Q V = Q/C I V I I L V = L I t t I L L R C LCR S V V 0 C Q/C L L I RI t R I S L V 0 I C

9 V 0 Q C L I t RI = 0. S I Q I = Q = Q = I t = t t Q 2 L 2 Q t 2 + R Q t + Q C = 0 L 2µ = R/L ω 2 0 = 1/LC 2 Q t 2 + 2µ Q t + ω 2 0 Q = x µ < ω 0 Q0 = Q 0 I0 = x Q Qt = Q 0 e µt {cos ωt + µ sin ωt } ω µ = R 2L, ω = ω 2 0 µ2 = 1 L 1 c 2 R LC LCR LCR V 0 cos ω t 2 V 0 cos ω t Q C L I t 2 Q t 2 + 2µ Q t RI = 0 + ω 2 0 Q = v 0 cos ω t 2µ = R/L, ω 2 0 = 1/LC, v 0 = V 0 /L R I L I C

10 Q = Qt Q 1, Q 2 2 Q 1, Q 2 Q 1 Q 2 Q 1, Q 2 : Q t = ae λ +t + be λ t λ ± = µ ± µ 2 ω 2 0. Qt ω Q t = a 1 cos ω t + b 1 sin ω t a 1, b 1 L 2 Q t 2 + R Q t + Q C = V 0 cos ω t a 1, b 1 Qt Q t Q t = Qt + iq t Qt iq t L 2 iq t t 2 + R iq t t + iq t C = V 0 i sin ω t Q t L 2 Q t t 2 + R Q t t + Q t C = V 0 e iω t Qt Qt = Re Q t Q t Q t = ce iω t

11 348 7 Lω 2 + irω + 1 ce iω t = V C 0 e iω t c = c Q t = ce iω t = V 0 e iω t Lω 2 + irω + 1 C V 0 Lω 2 + irω + 1 C Qt Q t I = Q t I t = I t = = iω V 0 e iω t V Lω 2 + ir + 1 = 0 e iω t R + i Lω C 1 V 0 R i Lω 1 cos ω t + i sin ω t R 2 + Lω 1 2 V 0 R 2 + Lω 1 2 Z = R + i Lω 1, Z = R = Z cos ϕ, Lω 1 R cos ω t + Lω 1 sin ω t R 2 + Lω 1 2 = Z sin ϕ I t = V 0 Z R 2 + Lω 1 2 cos ϕ cos ω t + sin ϕ sin ω t = V 0 Z cosω t ϕ Z Z

12 AM LCR ω I t I t Z = R + i Lω 1 R LC I t = V 0 Lω 1 cosω t ϕ Lω = 1 LCω 2 = 1 I t LC LC AM AM LC L C v t = V + v t cos ω t C variable conenser LC ω 2 = 1 L C ω = 1 LC LC v

x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin

2 2.1 F (t) 2.1.1 mẍ + kx = F (t). m ẍ + ω 2 x = F (t)/m ω = k/m. 1 : (ẋ, x) x = A sin ωt, ẋ = Aω cos ωt 1 2-1 x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ