1 1 1.1 (Isaac Newton, 1642 1727) 1. : 2. ( ) F = ma 3. ; F a 2 t x(t) v(t) = x (t) v (t) = x (t) F 3 3 3 3 3 3 6 1 2 6 12 1 3 1 2 m
2 1 x 1.1: v mg x (t) = v(t) mv (t) = mg 0 x(0) = x 0 v(0) = v 0 x(t) = x 0 + v 0 t 1 2 gt2 v(t) = v 0 gt t x = x 0 + v2 0 2g v2 2g 1.1 (x, v) 2 1 1 θ m m l ( 1.2) lθ = v mv = mg sin θ θ sin θ θ lθ = v mv = mgθ
1.1. 3 θ mg θ mg sin θ 1.2: g ξ = lθ + i l v g ξ = i l ξ ( ) g θ(t) = A sin l t + ω 0 A ω 0 1.3 2 1.3: sin θ = θ (x, v) (x, v )
4 1 1.3 1.4: θ sin θ θ 1.4 θ 1.3 1.2 2 = f(x, y) = g(x, y) (x(t), y(t)) x (t) = f(x(t), y(t)) y (t) = g(x(t), y(t)) (x 0, y 0 ) (x 0, y 0 )
1.2. 5 = f(x 0, y 0 ) = g(x 0, y 0 ) t = 0 (x 0, y 0 ) x(t) x 0 ) = f(x 0, y 0 )t y(t) y 0 = g(x 0, y 0 )t f(x 0, y 0 ) = g(x 0, y 0 ) = 0 = f(x 0, y 0 ) + ax(t) + by(t) = ax(t) + by(t) = g(x 0, y 0 ) + cx(t) + (t) = cx(t) + (t) a = f x (x 0, y 0 ), c = g x (x 0, y 0 ), x x = y A = ( a c b = f y (x 0, y 0 ), d = g y (x 0, y 0 ) ) b d d x = Ax ( ) A λ 0 0 µ = λx = µy
6 1 2 t = 0 (x 0, y 0 ) x(t) = x 0 e λt y(t) = y 0 e µt λ µ A = µ λ r = x 2 + y 2 tan θ = y x = λx µy = µx + λy dr = d (x2 + y 2 ) 1/2 ) = 1 ( 2 (x2 + y 2 ) 1/2 2x ) + 2y = 1 (x(λx µy) + y(µx + λy)) r = 1 r λ(x2 + y 2 ) = λr d tan θ = d y x/ y/ = x x 2 x(µx + λy) y(λx µy) = x 2 = µ x2 + y 2 x 2 d tan θ = 1 cos 2 θ dθ = x2 + y 2 x 2 dθ = (1 + tan2 θ) dθ dθ = x2 + y 2 d tan θ x 2 = x2 x 2 + y 2 + y 2 µx2 x 2 = µ dr = ar, dθ = µ 0 r = r 0 θ = θ 0 r(t) = r 0 e λt θ(t) = θ 0 + µt
1.2. 7 µ r λ < 0 λ > 0 λ < 0 λ > 0 λ = 0 λ 1 1 0 λ = λx + y = λy y(t) = y 0 e λt = λx + y 0e λt x(t) = C(t)e λt = C(t)λeλt + C (t)e λt = λx(t) + C (t)e λt C (t) = y 0 C(t) = x 0 + y 0 t x(t) = (x 0 + y 0 (t t 0 ))e λt y(t) = y 0 e λt 1. λ, µ > 0 2. λ, µ < 0 3. λ > 0, µ < 0 λ < 0, µ > 0 4. λ > 0 5. λ < 0 λ, µ 0 1.5 (1) (2) (4) (5)
8 1 1.5: (λ > 0 λ < 0) 1.6: (λ > 0 λ < 0) (3) (5) (1) (4) (2) 1.8 λ > 0 λ < 0 λ = 0 1.9 µ > 0 µ < 0 1 det(xe A) = 0 A 2 x 2 (Tr A)x + det A = 0 E det A A Tr A A
1.2. 9 1.7: (Jordan λ > 0 λ < 0) 1.8: (λ > 0 λ < 0) ( ) 2 2 λ, µ a = b = b 1 b 2 a b 0 Aa = λa Ab = µy a 1 b 1 2 P = a 2 b 2 P 1 λ 0 AP = 0 µ a 1 a 2
10 1 1.9: (λ = 0 µ > 0 µ < 0) P 1 P 1 x = y = = Ax dp 1 x = P 1 AP P 1 x y 1 y 2 = λ 0 y 0 µ ( 2 λ ± µ 1/ 2 i/ ) 2 U = i/ 2 1/ 2 (P U) 1 A(P U) = U 1 P 1 λ µ AP U = µ λ λ P 1 λ 0 AP = 0 λ P 1 λ 1 AP = 0 λ
1.2. 11 1 = x xy = y + xy 1.10: 2 (0, 0) (1, 1) (0, 0) (1, 1) 1.10 x y 2 = x2 + y 2 y = x 2xy A 0 3 = y = εy(1 y)2 x
12 1 ε = 0 x 2 + y 2 ε > 0 1 1.3 2 M m m M F = G Mm r 2 e e M m 2 1kg M r = 6400km g = 9.8m/s 2 2 1.11: 2 1.11 G Q, q (Charles Coulomb, 1736 1806) F = k Qq r 2 e 1
1.3. 13 1 1.11 2 1.12 2 1.12: 1.11 2 1.12 N S E B 1 Faraday(Michael Faraday, 1791 1867) Maxwell(James Maxwell, 1831 1879) v q F = q(e + v B) ρ 1 ρv j
14 1 1. div E = ρ ε 0 2. rot E = B t 3. div B = 0 4. c 2 rot B = E t + j ε 0 ε 0 = div f(x, y, z) f(x, y, z) ( f div g(x, y, z) = g(x, y, z) = x + g y + h ) (x, y, z) z h(x, y, z) h(x, y, z) rot h f(x, y, z) f(x, y, z) y g z f rot g(x, y, z) = g(x, y, z) = (x, y, z) h(x, y, z) x y z h(x, y, z) f(x, y, z) grad f(x, y, z) = f(x, y, z) = z g x g x f y f x f y f z (x, y, z) D D D div E = ρ ε 0 0 div B = 0 div E dz = D D E n ds
1.3. 15 n D ds D ρ ε 0 dz D ε 0 D 2 S C S rot E n ds = C E ds s C S B t n ds S C c 2 E rot B n ds = t n ds + j n ds ε 0 S S C S 1. div E = 0 S 2. rot E = B t 3. div B = 0 4. c 2 rot B = E t (4) c 2 rot B t = 2 E t 2 c 2 rot rot E = c 2 grad div E + c 2 E = c 2 E 2 E t 2 = c2 E
16 1 B = 2 x 2 + 2 y 2 + 2 z 2 rot rot = grad div div rot = 0 c sin(x + ct) E = sin(y + ct) sin(z + ct) 2