d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r
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1 2.4 ( ) U(r) ( ) ( ) U F(r) = x, U y, U = U(r) (2.4.1) z 2 1 K = mv 2 /2 dk = d ( ) 1 2 mv2 = mv dv = v (ma) (2.4.2) ( ) U(r(t)) r(t) r(t) + dr(t) du du = U(r(t) + dr(t)) U(r(t)) = U x = U(r(t)) dr(t) U U dx(t) + dy(t) + y z dz(t) U(r(t)) du = U dx(t) + U dy(t) + U dz(t) = U dr(t) = F(r) v(t) (2.4.3) x x x 21
2 d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r B W W = rb r A F dr i F(r i ) r i, r i = r i+1 r i r A r B n r 1, r 2,, r n = r B F i = F i, s i = r i W i = F i s i cos θ i θ i U = W [ ] = [ ] [ ] m 22
3 (,, mg) F = (,, mg) z = z = h h W = F dr = F dz = mgh F = (,, mg) dr = (,, dz) ( ) O = (,, ) P = (x, y, z) W = P O F dr = z F z dz = mg z dz = mgz ( ) du(x, y, z) = U(x + dx, y + dy, z + dz) U(x, y, z) = U x U U dx + dy + y z dz F (x, y, z) x, y, z dx, dy, dz F df F (r) r 2 = x 2 + y 2 + z 2 x, y, z x df (r) = df (r) dr, dr d(r 2 ) = 2rdr dx 2 = 2xdx, dy 2 = 2ydy, dz 2 = 2zdz r 2rdr = 2xdx + 2ydy + 2zdz, x = x r, 2r F (r) x df (r) = df (r) dr 1 F (xdx + ydy + zdz), r x = F r r x = x F r r (2.4.6) f(x, y) 2 f (x, y) f(x, y) = f(x + dx, y + dy) f(x, y) df(x, y), df(x, y) = f f dx + x y dy df 1 df(x) = f (x)dx x df(x, y) 3 z = f(x, y) P= (x, y) U(r) U(r + dr) U(r) du(r) = U x 23 dx + U y U dy + dz = U dr z
4 du = U dr(t) = U v(t), dr(t) = dr(t) = v(t) U(r 1 ) U(r ) = = t1 t r1 r du t1 = t U dr = t1 U v(t) = r1 r F dr F = U t F v(t) U(x) = x F dr x > : (dx > ) (F = kx < ) F dr = kxdx > U = F dr = x (kx )dx = 1 2 kx2 x < : (dx < ) (F = kx > ) F dr = kxdx > U = F dr = x (kx )dx = 1 2 kx2 F = mg sin θ(cos θ, sin θ) (x, y) = (L sin θ, L(1 cos θ)) dx = L cos θdθ, dy = L sin θdθ F dr = mgl sin θdθ U(θ) U() = mgl θ 2 sin θ dθ = mgl[cos θ] θ = mgl(1 cos θ) = mgy 1 2 F = k/x 2 x > x = x x 1 W x1 x1 [ 1 W = F dx = k x x x 2 dx = k 1 ] x1 = k + k x x x 1 x x U(x) = k/x 24
5 U(r) r M m U = GMm 1 r, r = (x2 + y 2 + z 2 ) 1/2 r (2.4.6) du = du 1 dr r (xdx + ydy + zdz) = GMm 1 (xdx + ydy + zdz) r3 U F = U = GMm r 3 : (x, y, z) = GMm r 3 r y 1 2 mv2 + mgy = E, v = dy = ± E 2g mg y E dy ( ) dy E = 2 d E mg y mg y = ± 2g 2d x = dx/ x y t E mg y = ± g/2(t t ), y = E mg g 2 (t t ) y E = mgy 2.5 p = mv dp = F (2.5.1) 25
6 p(t 2 ) p(t 1 ) = t2 t 1 F(t) t 1 t F 1, F 2 F 1 + F 2 = dp 1 = F 1, dp 2 = F 2, dp 1 + dp 2 = F 1 + F 2 = (2.5.2) P = p 1 + p 2 2 1, 2 r 1, r 2 2 V (r 1, r 2 ) V (r 1, r 2 ) ( V F 1 =, V, V ) = 1 V (r 1, r 2 ), F 2 = 2 V (r 1, r 2 ) x 1 y 1 z 1 i (i = 1, 2) 2 V (r 1, r 2 ) = v(r 1 r 2 ) 1 v(r 1 r 2 ) + 2 v(r 1 r 2 ) = x F 1x = v( x, y, z) v( x, y, z) = x 1 x F 2x = x 2 v( x, y, z) = v( x, y, z) x d x v( x, y, z) = x 1 x x v( x, y, z) = x 2 x r = r 1 r 2 = ( x, y, z) x/ x 1 = 1, x/ x 2 = 1 F 1x + F 2x = (2.5.2) 1: d (r 1 r 1 + d, r 2 r 2 + d) r 26
7 ( ) ( ) 2: E = 1 2 m 1v V (r 1 ) m 2v V (r 2 ) + v(r 1 r 2 ) (2.5.3) de = m 1v 1 (t) a 1 (t) + 1 V (r 1 ) v 1 (t) + 1 v(r 1 r 2 ) + m 2 v 2 (t) a 2 (t) + 2 V (r 2 ) v 2 (t) + 2 v(r 1 r 2 ) = v 1 (t) [m 1 a 1 (t) F 1 F 1] + v 2 (t) [m 2 a 2 (t) F 2 F 2] dp 1 + dp 2 = F 1 + F 2 V (r) A, B (2.5.3) V (r) E E = 1 2 m AvA m BvB 2 + v(r A r B ) v(r A r B ), ( r A r B ) (scattering) ( target)
8 2 k x V (x 1 x 2 ) = 1 2 k(x 1 x 2 ) 2 = 1 2 kx2 x = x 1 x 2 (x 1 + x 2 )/2 V (x) x F 1 = = kx (+1) = k(x 2 x 1 ) x x 1 V (x) x F 2 = = kx ( 1) = k(x 1 x 2 ) x x 2 F 1 + F ( ) ( ) 2 S, S O, O O O R r, r r(t) = R (t) + r (t) (2.6.1) r(t) = x(t)u 1 + y(t)u 2 + z(t)u 3, r (t) = x(t) u 1 + y (t)u 2 + z (t)u 3 u i, u i (i = 1, 2, 3) z z y O R (t) y O x x 28
9 O O xy R (t) (,, Z(t)) (2.6.1) x(t) = x (t), y(t) = y (t), z(t) = z (t) + Z(t) O O u i = u i (i = 1, 2, 3) z Z(t) = z + V t + at 2 /2 O z(t) = v t 1 2 gt2 z (t) = z(t) Z(t) O z (t) = z + (v V )t 1 (g + a)t2 2 O g g + a (2.6.1) 2 R (t) = R () + v t R () = x(t) = v x t + x (t), y(t) = v y t + y (t), z(t) = v z t + z (t), t = t t = t 9 : 2 O, O t = t = x, x x 2 (ct) 2 = (x ) 2 (ct ) 2 2 x 2 (ct) 2 = (x ) 2 (ct ) 2 t t = iτ τ y = cτ 2 x 2 + y 2 = x 2 + y 2 29
10 x = r cos θ, y = r sin θ x = r cosh θ, ct = r sinh θ cosh θ = cos iθ = 1 2 (ex + e x ), sinh θ = i sin iθ = 1 2 (ex e x ) cosh 2 θ sinh 2 θ = O O z ω u 1(t) = cos(ωt)u 1 + sin(ωt)u 2, u 2(t) = sin(ωt)u 1 + cos(ωt)u 2, u 3(t) = u 3 (2.6.2) O y y x O x du 1(t) = ωu 2(t) = ω u 1(t), du 2(t) = ωu 1(t) = ω u 2(t) (2.6.3) ω = (,, ω) = ωu 3 ω u i ( ) r(t) = x(t)u 1 + y(t)u 2 + z(t)u 3 = x (t)u 1(t) + y (t)u 2(t) + z(t)u 3 = [x (t) cos(ωt) + y (t) sin(ωt)]u 1 + [ x (t) sin(ωt) + y (t) cos(ωt)]u 2 + zu 3 (2.6.4) 2 x, y x(t) = x (t) cos(ωt) y (t) sin(ωt), y(t) = x (t) sin(ωt) + y (t) cos(ωt) (2.6.5) (2.6.3) (2.6.4) v(t) = dr(t) = v x u 1 + v y u 2 + v z u 3 = v xu 1(t) + v yu 2(t) + v z u 3 + ω (x u 1 + y u 2) = v xu 1(t) + v yu 2(t) + v z u 3 + ω r (2.6.6) 3
11 v x(t) = dx (t)/, v y(t) = dy (t)/, v z(t) = dz (t)/ (2.6.6) 2 a(t) = dv(t) = a x (t)u 1 + a y (t)u 2 + a z (t)u 3 = a xu 1(t) + a yu 2(t) + a zu 3 + ω (v xu 1(t) + v yu 2(t) + v z u 3) + ω (v xu 1(t) + v yu 2(t) + v z u 3 + ω r) = a xu 1(t) + a yu 2(t) + a zu 3 + 2ω (v xu 1(t) + v yu 2(t) + v z u 3) ω 2 r u i (i = 1, 2, 3) a 2(v ω) ω 2 r (2.6.7) O ma = 2m(v ω) + mω 2 r + F 2 1. : mω 2 r (r = x 2 + y 2) 2. : 2mωv sin θ θ ω v 31
08-Note2-web
r(t) t r(t) O v(t) = dr(t) dt a(t) = dv(t) dt = d2 r(t) dt 2 r(t), v(t), a(t) t dr(t) dt r(t) =(x(t),y(t),z(t)) = d 2 r(t) dt 2 = ( dx(t) dt ( d 2 x(t) dt 2, dy(t), dz(t) dt dt ), d2 y(t) dt 2, d2 z(t)
Gmech08.dvi
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mugensho.dvi
1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim
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Gmech08.dvi
145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
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5. [. ] z = f(, y) () z = 3 4 y + y + 3y () z = y (3) z = sin( y) (4) z = cos y (5) z = 4y (6) z = tan y (7) z = log( + y ) (8) z = tan y + + y ( ) () z = 3 8y + y z y = 4 + + 6y () z = y z y = (3) z =
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13 13.1 O r F R = m d 2 r dt 2 m r m = F = m r M M d2 R dt 2 = m d 2 r dt 2 = F = F (13.1) F O L = r p = m r ṙ dl dt = m ṙ ṙ + m r r = r (m r ) = r F N. (13.2) N N = R F 13.2 O ˆn ω L O r u u = ω r 1 1:
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n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x
n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v 3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt
2.2 ( y = y(x ( (x 0, y 0 y (x 0 (y 0 = y(x 0 y = y(x ( y (x 0 = F (x 0, y(x 0 = F (x 0, y 0 (x 0, y 0 ( (x 0, y 0 F (x 0, y 0 xy (x, y (, F (x, y ( (
(. x y y x f y = f(x y x y = y(x y x y dx = d dx y(x = y (x = f (x y = y(x x ( (differential equation ( + y 2 dx + xy = 0 dx = xy + y 2 2 2 x y 2 F (x, y = xy + y 2 y = y(x x x xy(x = F (x, y(x + y(x 2
m dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d
m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m
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B 38 1 (x, y), (x, y, z) (x 1, x 2 ) (x 1, x 2, x 3 ) 2 : x 2 + y 2 = 1. (parameter) x = cos t, y = sin t. y = f(x) r(t) = (x(t), y(t), z(t)), a t b.
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1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2
1 1.1 R(x) = 0 y + P (x)y + Q(x)y = R(x)...(1) y + P (x)y + Q(x)y = 0...(2) 1 2 u(x) v(x) c 1 u(x)+ c 2 v(x) = 0 c 1 = c 2 = 0 c 1 = c 2 = 0 2 0 2 u(x) v(x) u(x) u (x) W (u, v)(x) = v(x) v (x) 0 1 1.2
5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { (
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p = mv p x > h/4π λ = h p m v Ψ 2 Ψ
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II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re
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) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
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1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
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II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {
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77
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n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1)
D d dx 1 1.1 n d n y a 0 dx n + a d n 1 y 1 dx n 1 +... + a dy n 1 dx + a ny = f(x)...(1) dk y dx k = y (k) a 0 y (n) + a 1 y (n 1) +... + a n 1 y + a n y = f(x)...(2) (2) (2) f(x) 0 a 0 y (n) + a 1 y
214 March 31, 214, Rev.2.1 4........................ 4........................ 5............................. 7............................... 7 1 8 1.1............................... 8 1.2.......................
chap03.dvi
99 3 (Coriolis) cm m (free surface wave) 3.1 Φ 2.5 (2.25) Φ 100 3 r =(x, y, z) x y z F (x, y, z, t) =0 ( DF ) Dt = t + Φ F =0 onf =0. (3.1) n = F/ F (3.1) F n Φ = Φ n = 1 F F t Vn on F = 0 (3.2) Φ (3.1)
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DVIOUT
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18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb
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r 1 r 2 r 1 r 2 2 Coulomb Gauss Coulomb 2.1 Coulomb 1 2 r 1 r 2 1 2 F 12 2 1 F 21 F 12 = F 21 = 1 4πε 0 1 2 r 1 r 2 2 r 1 r 2 r 1 r 2 (2.1) Coulomb ε 0 = 107 4πc 2 =8.854 187 817 10 12 C 2 N 1 m 2 (2.2)
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9 O y O ( O ) O (O ) 3 y O O v t = t = 0 ( ) O t = 0 t r = t P (, y, ) r = + y + (t,, y, ) (t) y = 0 () ( )O O t (t ) y = 0 () (t) y = (t ) y = 0 (3) O O v O O v O O O y y O O v P(, y,, t) t (, y,, t )
No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
( 12 ( ( ( ( Levi-Civita grad div rot ( ( = 4 : 6 3 1 1.1 f(x n f (n (x, d n f(x (1.1 dxn f (2 (x f (x 1.1 f(x = e x f (n (x = e x d dx (fg = f g + fg (1.2 d dx d 2 dx (fg = f g + 2f g + fg 2... d n n
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Chap11.dvi
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difgeo1.dvi
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