Size: px
Start display at page:

Download ""

Transcription

1

2 i

3 A 24 A A A A A B 3 B B C 31 C C C C D 33 D D D ii

4 natural science Newtonequation of 3 motion 5 universe Schrödinger quantum mechanics classical mechanics theory of elasticity 1

5 1 1.1 point mass rigid bodyelastic body fluid dynamics statics 1.2 force f 5 f 4 f 1 f 3 f 2 f 2 f 3 f 1 f 4 f moment of force torque N N N O AOA N O r A f f 1 f 2 OA O O 2

6 N = r f (1.1) r f r f sin θ θ r f r f 1.4 F 1 F 2 F 1 + F 2 = F 1 = F 2 (1.2) F h1 F h2 F v1 F v2 F h1 + F h2 = F v1 + F v2 = 1.5 truss rahmen beam theory 3

7 1.6 frictional force tribology F s N F s = µ s N (1.3) µ s static coefficient of friction F d F d = µ d N (1.4) N F µ d dymanic coefficient of friction m 4

8 1.8 f 1,f 2,f 3,,, δx 1, δx 2, δx 3,,, δx i f i δx i = (1.5) principle of virtual work 5

9 2 2.1 mass (inertial x f x p x mass) r = y, f = f y, p = p y,,, m f z f z p z r ṙ = dr dt, r = d2 r dt 2 (2.1) f = m r, (2.2) momentum p = mṙ f = ṗ (2.3) ẋ ṙ = ẏ, ḟ = f ẋ, ṗ = f y p y ż f z p z p ẋ,,, 2.2 t f(t) t = p(t + t) p(t) (2.4) f t impulse (2.4) 2.3 (2.3) t t + t t+ t fdt = p(t + t) p(t) t t F (2.4) t = t 1 x = x 1, 6

10 ẋ = v 1 t = t 2 x = x 2, ẋ = v 2 x 2 f dx = 1 x 1 2 mv mv 1 2 (2.5) E k = 1 2 mv2 (2.6) kinetic energy (2.5) (2.5) x 2 x 1 f dx = t 2 t 1 f dx dt dt = t 2 t 1 f vdt = t 2 t 1 mẍ vdt = t 2 t 1 m v vdt = t 2 t 1 mv vdt = t 2 t 1 mv dv dt dt = v 2 v 1 mvdv = [ 1 2 mv2 ] v2 v 1 (2.5) m g mg ẍ = m ÿ (2.7) z x x = v x t (2.8) y = (2.9) z = v z t gt 2 /2 (2.1) (2.8) (2.1) z x t = r = r v = v r = r + v t 1 2 t 2 g (2.8) (2.1) t z = g 2v x 2 + vz x 2 v x x pendulum m T L sin θ r = L L cos θ, f = T sin θ T cos θ mg, (2.11) L z O m x Lsin T mg T θ f = m r g sin θ = L θ (2.12) 7

11 θ sin θ θ gθ = L θ (2.13) ( ) g θ = A sin L t + α (2.14) A α kr = m r (2.14) 8

12 3 3.1 r r ξ a ξ r = η a =,,, ζ a η a ζ 3.2 ξ, η, ζ x, y, z r = r + r (3.1) (2.2) f = f v = r f m v = m r (3.2) (3.2) x z O r y r r' (3.1) x x ξ y = + η y z z ζ O' 9

13 v v 3.3 r r R r = Rr (3.3) z O R R 1 = R T ṘRT. ṘRT a ṘR T a = ω a (3.4) ω ṙ = R(ω r + ṙ ) (3.5) r = R{ω (ω r ) + ω r + 2ω ṙ + r } (3.6) x A 11 A 12 A 13 A = A 21 A 22 A 23, A 31 A 32 A 33 R xξ R xη R xζ R = R yξ R yη R yζ,,, R zξ R zη R zζ y f mω (ω r ) m ω r 2mω ṙ = m r (3.7) centrifugal force Coriolis force (3.4) R ω ω (3.7) cos α sin α α β sin α cos β sin β cos α cos β R = sin α sin β cos β cos α sin β (3.8) z O α β R ω (3.4) α sin β sin β ω = α cos β = β + ( α) cos β (3.9) β 1 x y 1

14 ω β cos α cos α ω = α = β + ( α) 1 (3.1) β sin α sin α (3.9) (3.1) z η ω ω R O z z x y α(rad) ζ ξ ω α ω R ω ω cos α ω = ω = ω (3.11) ω sin α (3.7) R ṘR T = β α cos β β α sin β α cos β α sin β (3.4) ω R ω x t sin α cos ωt sin ωt cos α cos ωt R = sin α sin ωt cos ωt cos α sin ωt z O cos α sin α y m r =, f = (3.12) ζ mg R E ζ R E g (3.11) sin α mω (ω r ) = m(ζ cos α)ω 2 (3.13) cos α 11

15 2mω ṙ = 2mω ζ cos α (3.14) ζ < Foucault 2m 3.4 r = r + Rr (3.15) 3.2 (3.15) f = Rf, v = Rv, ω = Rω, 3.3 R ω f mω (ω r ) m ω r 2mω ṙ x O ṙ = R(ω r + ṙ + v ) m(ω v r = R{ω (ω r ) + ω r + 2ω ṙ + (ω + v ) = m r (3.16) v + v ) + r } ( ) (3.7) ( ) f = m r m(ω v + v ) (3.16) (3.2) z r r r' O' y 12

16 4 4.1 O ω ω ω r dm df = rdm (4.1) r r df = V V r rdm = V d {r (ω r)}dm (4.2) dt N N = d ( ) r (ω r)dm = d (Iω) (4.3) dt V dt (y 2 + z 2 ) xy zx I = xy (z 2 + x 2 ) yz V zx yz (x 2 + y 2 ) dm (4.4) I inertia tensor angular momentum (4.3) L = Iω (4.5) N = L (4.6) (4.6) f = ṗ P Q v dm r O ρ dv dm = ρdv dm v = ṙ = ω r d dt {r (ω r)} = d dt {r ṙ} = r r a 11 a 12 a 13 V a 21 a 22 a 23 dm a 31 a 32 a 33 V a 11dm V a 12dm V a 13dm = V a 21dm V a 22dm V a 23dm V a 31dm V a 32dm V a 33dm r (ω r) = r 2 ω (r ω)r x ω x r = y ω = z ω z (4.3) (4.4) ω y 13

17 p = mv m L = Iω I (4.6) t N(t) t = L(t + t) L(t) (4.7) N(t) t (4.7) 4.3 de = 1 2 dm v 2 1 E = de = 2 ω r 2 dm = 1 2 (Iω) ω = 1 2 L ω (4.8) V V ω r 2 = ω T (y 2 + z 2 ) xy zx xy (z 2 + x 2 ) yz zx yz (x 2 + y 2 ) ω (2.6) 1 2 m v 2 (2.6) (4.8) J Ns Nms 14

18 4.4 (4.6) (4.4) I = I x I xy I zx I xy I y I yz I zx I yz I z (4.9) I x, I y, I z (moment of inertia)i xy, I yz, I zx r r G r r = r G + r (4.1) M = V dm (4.11) (yg 2 + z2 G ) x Gy G z G x G I = M x G y G (zg 2 + x2 G ) y Gz G z G x G y G z G (x 2 G + y2 G ) (η 2 + ζ 2 ) ξη ζξ + ξη (ζ 2 + ξ 2 ) ηζ dm (4.12) V ζξ ηζ (ξ 2 + η 2 ) (4.12) M 3.2 x x G ξ y = + η y G z z G ζ (4.12) V r r Gdm = V r dm = V ξdm = V ηdm = ζdm = V I z = V (x2 + y 2 )dm = (x G 2 + y G 2 )M + V (ξ2 + η 2 )dm. 4.5 (4.6) (4.5) N 15

19 ω N N N ω I Iµ = λµ (4.13) µ (4.13) µ λ I eigenvectoreigenvalue E(I λe)µ = µ( ) det(i λe) = 1 µ = µ i, λ = I pi, (i = 1 3) (4.14) E = 1 1 µ i (4.13) µ ix µ i µ i = µ iy N pi ω pi (4.6) N pi = I pi ω pi (4.15) µ iz i 1 3 µ i I pi (4.15) µ 1x µ 2x µ 3x I p1 P = µ 1y µ 2y µ 3y, I p = I p2 (4.16) µ 1z µ 2z µ 3z I p3 P I = P I p P T (4.17) (4.17) I p P I I P T I p P 123- xyz- Iµ 1 = I p1 µ 1, Iµ 2 = I p2 µ 2, Iµ 3 = I p3 µ 3 IP = P I p (4.17) 16

20 I p (= P T IP ) µ 1 µ 2 µ P T P T µ 1 = 1, P T µ 2 = 1, P T µ 3 = 1 (4.18) I P T N L N p (= P T N) L p (= P T L) N = L N p = d dt (L p) = d dt (I pω p ) = İpω p + I p ω p (4.19) ω µ i I p I p1 I p3 ω p i ω pi I p i N p N ( = L ) N p = P T N = P T L = d dt P T L ( ) ( ) P T Iω P T IP P T ω = d dt = d dt = d dt (I pω p ) i İpω p = i = 3 N p = I p ω p ω p = N pi µ i = I pi ω pi µ i (4.2) (4.15) ω p3 3 θ I p I p1 cos 2 θ + I p2 sin 2 θ (I p1 I p2 ) cos θ sin θ (I p1 I p2 ) cos θ sin θ I p1 sin 2 θ + I p2 cos 2 θ I p3 I p İp θ İpω p = N = I θ N = mgl sin θi = ml 2 g sin θ = L θ I p ω p ω p 17

21 5 5.1 N = L 3.3 L L R r = Rr 3.3 L = R(µ L + L ) (5.1) µ a ṘRT a = µ a µ R (5.1) N = µ L + L (5.2) L = d dt (RL ) = ṘL + R L = RR T ṘL + R L = R(µ L ) + R L = R(µ L + L ) (5.2) µ (3.7) ω ω µ L = Iω ω 5.2 E = 1 2 L ω = 1 2 L ω (5.3) 1 2 L ω = 1 2 (RL ) (Rω ) = 1 2 (RL ) T Rω = 1 2 L T R T Rω = 1 2 L T ω = 1 2 L ω 5.3 gyro 18

22 ω µ 3.3 α = π/2 z ζ ξ x y µt ξ ω (5.2) L L = L µ N = µ L (5.4) x N g = N = µ L (5.5) µ ζ L ξ η 5.4 N g = µ N g z t O I ξ ω = µi ξ ω y (precession) O ζ < α < π/2 I λ I = I ξ I ξ I ζ, λ = λ, (5.6) x z O y Mg z H O I ξ I ζ λ ζξ x β µ α β µ = µ α + µ β µ α = α, µ β = β, µ = β cos α α β sin α (5.7) α β 19

23 H g N = HMg cos α ω = λ + µ ω = λ + µ µ L = L = I ζ αλ (I ζ I ξ ) α β sin α I ζ βλ cos α + (Iζ I ξ ) β 2 sin α cos α I ξ ( β cos α + α β sin α) I ξ α I ζ ( λ + β sin α + α β cos α) (5.8) (5.9) (5.1) (5.2) I ζ αλ + (I ζ 2I ξ ) α β sin α + I ξ β cos α = (5.11) HMg cos α = I ζ βλ cos α + (Iζ I ξ ) β 2 sin α cos α I ξ α (5.12) λ + β sin α + α β cos α = (5.13) α. (3.7) (5.2) λα, β ω µ (5.11) (5.13) t = α = α β = β λ = λ α = α β = β λ = λ α = α = const. µ µ α β = β = const.λ = λ = const. (5.14).pdf (I ζ λ) 2 4HMg(I ξ I ζ ) sin α β = I ζ λ 1 1 4HMg(I ξ I ζ ) sin α/(i ζ λ) 2 2(I ξ I ζ ) sin α (5.15) α = α = const. λ (5.15) 2

24 (5.15) α β HMg λi ζ (5.16) λ β H > λ β A 2T H I ζ = ρπt A 4, M = 2ρπT A 2 β 2Hg A 2 λ (5.17) G A 2T H O λ = α = β = (5.11) (5.13) t λ = α β{(i ζ 2I ξ ) sin 2 α I ξ cos 2 α} = (5.18) {} α β = β = λ = λ = const., β = β = const., HMg cos α = I ξ α (5.19) α 2.3 θ α sin cos 21

25 F F F (Friction) F λ λ > F = F (5.8) N = HMg cos α F (5.2) (5.8) (5.2) α α = αλ (5.22) I ζ αλ + I ξ β cos α = (5.21) HMg = I ζ βλ + (Iζ I ξ ) β 2 sin α (5.22) F = I ζ ( λ + β sin α) (5.23) β = I ζ λ 1 1 4HMg(I ξ I ζ ) sin α/(i ζ λ) 2 2(I ξ I ζ ) sin α (5.24) β (5.23) (5.23) λ = λ F t (5.25) I ζ λ I ζ F (5.24) λ β β β > (5.21) β (5.24) α β HMg λi ζ α = I ξ I ζ λ β cos α (5.26) α < (5.25) 22

26 (5.24) (5.26) (5.26) λi ξ /I ζ 23

27 A A.1 raar raar r = x y, a = a x a y z a z A 11 A 12 A 13 A = A 21 A 22 A 23, r =, R = ξ η, a = a ξ a η ζ a ζ R xξ R xη R xζ R yξ R yη R yζ,,,, A 31 A 32 A 33 R zξ R zη R zζ ṙ = ẋ ẏ ż V Adm =, ȧ = ȧ x ȧ y ȧ z, ṙ = ξ η ζ,,, V A 11dm V A 12dm V A 13dm V A 21dm V A 22dm V A 23dm V A 31dm V A 32dm V A 33dm,,, A.2 T r T = ( x y z ), a T = ( a x a y a z ), A T = 1 A 11 A 21 A 31 A 12 A 22 A 32 A 13 A 23 A 33 24

28 1 A 1 A = AA 1 = E = 1 E 1 A.3 A 11 A 12 A 13 x Ar = A 21 A 22 A 23 y A 31 A 32 A 33 z = a b = a x b x + a y b y + a z b z =a T b A 11 x + A 12 y + A 13 z A 21 x + A 22 y + A 23 z A 31 x + A 32 y + A 33 z e x e y e z a y b z a z b y a b = a x a y a z = a z b x a x b z b x b y b z a x b y a y b x A.4 ABC ab R d d (Aa) = Ȧa + Aȧ, dt d d (a b) = ȧ b + a ḃ, dt A(BC) = (AB)C = ABC (AB) T = B T A T R 1 = R T R(a b) = (Ra) (Rb) a (b a) = a 2 b (a b)a (AB) = ȦB + AḂ dt (A.6) (a b) = ȧ b + a ḃ dt (A.7) (A.1) (A.2) (A.3) (A.4) (A.5) A a r = Rr, r = R T r, (A.8) 25

29 A = RA R T, A = R T AR (A.9) A.5 R R 1 = R T ab a = Ra b = Rb a b a b = a b (Ra ) (Rb ) = a b (Ra ) T (Rb ) = a b (a T R T )(Rb ) = a b a T R T Rb = a b a (R T Rb ) = a b a (R T Rb b ) = E a (R T R E)b = ab R T R = ER T = R 1 R ṘR T RR T = E ṘR T + RṘT = { ( } ṘR T = RṘT = RṘT) T T { (ṘT ) T T = R T} 26

30 T = {ṘR } T ṘR T A anti s. r A anti s. r = ω r ω A anti s. A 12 A 13 A anti s. = A 12 A 23 A 13 A 23 ω x x ω = r = y ω y ω z A anti s. r = (A 13 )z ( A 12 )y ( A 12 )x ( A 23 )z ( A 23 )y (A 13 )x ω x y ω y x A anti s. ω ω = A 23 A 13 A 12 z ω r = ω y z ω z y ω z x ω x z r ω R r ṘR T r = ω r R T Ṙr = ω r r = Rr ṘRT r = ω r R T R T ṘR T r = R T (ω r) R T Ṙr = ω r r r R T Ṙr = ω r 27

31 I λ µ λ λ µ µ Iµ = λµ (1) µ Iµ µ = λµ µ (Iµ) T µ = λµ µ µ T I T µ = λµ µ (2) (1) I Iµ = λ µ z z µ z 1 z 2 (z 1 z 2 ) = z1z 2 µ Iµ = λ µ µ z 1 = x 1 + iy 1 z 2 = x 2 + iy 2 µ T Iµ = λ µ µ (3) (z 1 z 2 ) = (x 1 x 2 y 1 y 2 ) i(x 1 y 2 + x 2 y 1 ) (1) I (2) (3) z1z 2 = (x 1 iy 1 )(x 2 iy 2 ) = (x 1 x 2 y 1 y 2 ) i(x 1 y 2 + x 2 y 1 ) (2) λ = λ λ (1) (2) (z 1 z 2 ) = z1z 2 µ 1 µ 2 λ 1 λ 2 Iµ 1 = λ 1 µ 1 (1) Iµ 2 = λ 2 µ 2 (2) (1) (2) µ λ 1 λ 2 (1) µ 2 µ 2 (Iµ 1 ) = λ 1 µ 2 µ 1 µ T 2 Iµ 1 = λ 1 µ 2 µ 1 (1 ) (2) µ 1 µ 1 (Iµ 2 ) = λ 2 µ 1 µ 2 µ T 1 Iµ 2 = λ 2 µ 1 µ 2 28

32 (µ T 1 Iµ 2 ) T = λ 2 µ 1 µ 2 µ T 2 I T µ 1 = λ 2 µ 1 µ 2 (2 ) I I = I T (1 ) (2 ) (λ 1 λ 2 )µ 1 µ 2 = λ 1 λ 2 µ 1 µ 2 = µ 1 µ 2 = 29

33 B B.1 abc x t aẍ + bẋ + cx = (B.1) aλ 2 + bλ + c = λ = λ 1, λ 2 x(t) = C 1 e λ1t + C 2 e λ 2t (B.2) λ x(t) = C 1 e λt + C 2 te λt (B.3) C 1 C 2 λ e iθ = cos θ + i sin θ (B.4) C 1 C 2 B.2 ẍ + cx = (B.5) λ = ±i c (B.6) (B.8) x(t) = C 1 e +i ct + C 2 e i ct (B.6) = B 1 cos( ct) + B 2 sin( ct) (B.7) = A sin( ct + α) (B.8) 3

34 C C.1 A r r 2 r = m r (C.1) A N = L z xy z = xy r = r cos θ r sin θ (C.2) A = mr 2 ( r r θ 2 ) (C.3) = 2ṙ θ + r θ (C.4) (C.4) L I z = mr 2 ω z = θ L = mr 2 θ L θ = L mr 2 (C.5) (C.5) (C.3) r = L 2 /(ma) 1 B cos(θ + β) (C.6) B β cos θ sin θ r r = sin θ cos θ 1 cos θ sin θ r r θ 2 r = sin θ cos θ 2ṙ θ + r θ 1 u = 1/r du/dθ r θ (C.5) r = L d 2 u mr 2 dθ (C.3) 2 d 2 u dθ L 2 m u = A L u = C 1 cos(θ + C 2 ) ma L 2 31

35 C.2 xy r(θ) = ±l 1 ± ϵ cos θ (C.7) (C.7) ± θ + l ϵ ϵ ϵ = < ϵ < 1 ϵ = 1 1 < ϵ y/l x/l C ϵ =.,.5, 1,, 2. r(θ) = l 1±ϵ cos θ C.4 32

36 D D.1 z R E z ω α (rad) ξη ζ d ζ d ξ d φ O ζ = R E m L P T L R E g x t O y D.2 T sin θ cos φ f = T sin θ sin φ T cos θ mg (D.1) O d P L m R E O Lsin d z cos α ω =, ω = ω, ω sin α (D.2) (3.7) T sin θ cos φ ξ sin 2 α + ζ sin α cos α T sin θ sin φ + mω 2 η T cos θ mg 2mω ξ sin α cos α + ζ cos 2 α ξ = m η ζ η sin α ξ sin α + ζ cos α η cos α (D.3) 33

37 r = ξ η ζ = L sin θ cos φ L sin θ sin φ R E + L L cos θ (D.4) (D.3) ξ, η, ζ (D.4) θ, φ, T D.3 x ẋ ẍ r ξ = L( θ cos θ cos φ φ sin θ sin φ) η = L( θ cos θ sin φ + φ sin θ cos φ) ζ = L θ sin θ ξ = L( θ cos θ cos φ θ 2 sin θ cos φ θ φ cos θ sin φ) L( φ sin θ sin φ + φ 2 sin θ cos φ + θ φ cos θ sin φ) η = L( θ cos θ sin φ θ 2 sin θ sin φ + θ φ cos θ cos φ) +L( φ sin θ cos φ φ 2 sin θ sin φ + θ φ cos θ cos φ) ζ = L( θ sin θ + θ 2 cos θ) (D.3) 1.pdf (D.3) ζ T ξ η L R E R E T θ φ ω φ 2 sin θ = θcos θ = 1 3 θθ 2, φ 2 θ, ω φθ, ω φθ 2 (gθ + L θ) cos φ L(2ω θ sin α + 2 θ φ + φθ) sin φ = (gθ + L θ) sin φ L(2ω θ sin α + 2 θ φ + φθ) cos φ = (gθ + L θ) 2 + L 2 (2ω θ sin α + 2 θ φ + φθ) 2 = () 4 φθ φ φ φ φ 34

38 L θ + gθ = φ = ω sin α (D.5) (D.6) (D.5) (D.6) φ = ω φ = θ θ c 215 TAKAHASHI Kunio, 35

sec13.dvi

sec13.dvi 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:

More information

B ver B

B ver B B ver. 2017.02.24 B Contents 1 11 1.1....................... 11 1.1.1............. 11 1.1.2.......................... 12 1.2............................. 14 1.2.1................ 14 1.2.2.......................

More information

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 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

More information

2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................

More information

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

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 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

More information

K E N Z U 01 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.................................... 4 1..1..................................... 4 1...................................... 5................................

More information

) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)

) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) 4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7

More information

d dt P = d ( ) dv G M vg = F M = F (4.1) dt dt M v G P = M v G F (4.1) d dt H G = M G (4.2) H G M G Z K O I z R R O J x k i O P r! j Y y O -

d dt P = d ( ) dv G M vg = F M = F (4.1) dt dt M v G P = M v G F (4.1) d dt H G = M G (4.2) H G M G Z K O I z R R O J x k i O P r! j Y y O - 44 4 4.1 d P = d dv M v = F M = F 4.1 M v P = M v F 4.1 d H = M 4.2 H M Z K I z R R J x k i P r! j Y y - XY Z I, J, K -xyz i, j, k P R = R + r 4.3 X Fig. 4.1 Fig. 4.1 ω P [ ] d d = + ω 4.4 [ ] 4 45 4.3

More information

Gmech08.dvi

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

More information

: 2005 ( ρ t +dv j =0 r m m r = e E( r +e r B( r T 208 T = d E j 207 ρ t = = = e t δ( r r (t e r r δ( r r (t e r ( r δ( r r (t dv j =

: 2005 ( ρ t +dv j =0 r m m r = e E( r +e r B( r T 208 T = d E j 207 ρ t = = = e t δ( r r (t e r r δ( r r (t e r ( r δ( r r (t dv j = 72 Maxwell. Maxwell e r ( =,,N Maxwell rot E + B t = 0 rot H D t = j dv D = ρ dv B = 0 D = ɛ 0 E H = μ 0 B ρ( r = j( r = N e δ( r r = N e r δ( r r = : 2005 ( 2006.8.22 73 207 ρ t +dv j =0 r m m r = e E(

More information

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi) 0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()

More information

9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ0 ψ n (r) ω n Schrödinger Ĥ 0 ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ0 + Ĥint (

9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ0 ψ n (r) ω n Schrödinger Ĥ 0 ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ0 + Ĥint ( 9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) 2. 2.1 Ĥ ψ n (r) ω n Schrödinger Ĥ ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ + Ĥint (t)] ψ (r, t), (2) Ĥ int (t) = eˆxe cos ωt ˆdE cos ωt, (3)

More information

p = mv p x > h/4π λ = h p m v Ψ 2 Ψ

p = mv p x > h/4π λ = h p m v Ψ 2 Ψ II p = mv p x > h/4π λ = h p m v Ψ 2 Ψ Ψ Ψ 2 0 x P'(x) m d 2 x = mω 2 x = kx = F(x) dt 2 x = cos(ωt + φ) mω 2 = k ω = m k v = dx = -ωsin(ωt + φ) dt = d 2 x dt 2 0 y v θ P(x,y) θ = ωt + φ ν = ω [Hz] 2π

More information

ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d

ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9

More information

i

i 009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3

More information

dynamics-solution2.dvi

dynamics-solution2.dvi 1 1. (1) a + b = i +3i + k () a b =5i 5j +3k (3) a b =1 (4) a b = 7i j +1k. a = 14 l =/ 14, m=1/ 14, n=3/ 14 3. 4. 5. df (t) d [a(t)e(t)] =ti +9t j +4k, = d a(t) d[a(t)e(t)] e(t)+ da(t) d f (t) =i +18tj

More information

64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () m/s : : a) b) kg/m kg/m k

64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () m/s : : a) b) kg/m kg/m k 63 3 Section 3.1 g 3.1 3.1: : 64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () 3 9.8 m/s 2 3.2 3.2: : a) b) 5 15 4 1 1. 1 3 14. 1 3 kg/m 3 2 3.3 1 3 5.8 1 3 kg/m 3 3 2.65 1 3 kg/m 3 4 6 m 3.1. 65 5

More information

1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0

1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx

More information

pdf

pdf http://www.ns.kogakuin.ac.jp/~ft13389/lecture/physics1a2b/ pdf I 1 1 1.1 ( ) 1. 30 m µm 2. 20 cm km 3. 10 m 2 cm 2 4. 5 cm 3 km 3 5. 1 6. 1 7. 1 1.2 ( ) 1. 1 m + 10 cm 2. 1 hr + 6400 sec 3. 3.0 10 5 kg

More information

08-Note2-web

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)

More information

( ) ( )

( ) ( ) 20 21 2 8 1 2 2 3 21 3 22 3 23 4 24 5 25 5 26 6 27 8 28 ( ) 9 3 10 31 10 32 ( ) 12 4 13 41 0 13 42 14 43 0 15 44 17 5 18 6 18 1 1 2 2 1 2 1 0 2 0 3 0 4 0 2 2 21 t (x(t) y(t)) 2 x(t) y(t) γ(t) (x(t) y(t))

More information

Note.tex 2008/09/19( )

Note.tex 2008/09/19( ) 1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................

More information

W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)

W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2) 3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)

More information

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

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

More information

Gmech08.dvi

Gmech08.dvi 51 5 5.1 5.1.1 P r P z θ P P P z e r e, z ) r, θ, ) 5.1 z r e θ,, z r, θ, = r sin θ cos = r sin θ sin 5.1) e θ e z = r cos θ r, θ, 5.1: 0 r

More information

2004 2 1 3 1.1..................... 3 1.1.1................... 3 1.1.2.................... 4 1.2................... 6 1.3........................ 8 1.4................... 9 1.4.1..................... 9

More information

δ ij δ ij ˆx ˆx ŷ ŷ ẑ ẑ 0, ˆx ŷ ŷ ˆx ẑ, ŷ ẑ ẑ ŷ ẑ, ẑ ˆx ˆx ẑ ŷ, a b a x ˆx + a y ŷ + a z ẑ b x ˆx + b

δ ij δ ij ˆx ˆx ŷ ŷ ẑ ẑ 0, ˆx ŷ ŷ ˆx ẑ, ŷ ẑ ẑ ŷ ẑ, ẑ ˆx ˆx ẑ ŷ, a b a x ˆx + a y ŷ + a z ẑ b x ˆx + b 23 2 2.1 n n r x, y, z ˆx ŷ ẑ 1 a a x ˆx + a y ŷ + a z ẑ 2.1.1 3 a iˆx i. 2.1.2 i1 i j k e x e y e z 3 a b a i b i i 1, 2, 3 x y z ˆx i ˆx j δ ij, 2.1.3 n a b a i b i a i b i a x b x + a y b y + a z b

More information

K E N Z OU

K E N Z OU K E N Z OU 11 1 1 1.1..................................... 1.1.1............................ 1.1..................................................................................... 4 1.........................................

More information

V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H

V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H 199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)

More information

006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................

More information

1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.

1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1. 1.1 1. 1.3.1..3.4 3.1 3. 3.3 4.1 4. 4.3 5.1 5. 5.3 6.1 6. 6.3 7.1 7. 7.3 1 1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N

More information

7-12.dvi

7-12.dvi 26 12 1 23. xyz ϕ f(x, y, z) Φ F (x, y, z) = F (x, y, z) G(x, y, z) rot(grad ϕ) rot(grad f) H(x, y, z) div(rot Φ) div(rot F ) (x, y, z) rot(grad f) = rot f x f y f z = (f z ) y (f y ) z (f x ) z (f z )

More information

18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb

18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb 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)

More information

2 Chapter 4 (f4a). 2. (f4cone) ( θ) () g M. 2. (f4b) T M L P a θ (f4eki) ρ H A a g. v ( ) 2. H(t) ( )

2 Chapter 4 (f4a). 2. (f4cone) ( θ) () g M. 2. (f4b) T M L P a θ (f4eki) ρ H A a g. v ( ) 2. H(t) ( ) http://astr-www.kj.yamagata-u.ac.jp/~shibata f4a f4b 2 f4cone f4eki f4end 4 f5meanfp f6coin () f6a f7a f7b f7d f8a f8b f9a f9b f9c f9kep f0a f0bt version feqmo fvec4 fvec fvec6 fvec2 fvec3 f3a (-D) f3b

More information

構造と連続体の力学基礎

構造と連続体の力学基礎 II 37 Wabash Avenue Bridge, Illinois 州 Winnipeg にある歩道橋 Esplanade Riel 橋6 6 斜張橋である必要は多分無いと思われる すぐ横に道路用桁橋有り しかも塔基部のレストランは 8 年には営業していなかった 9 9. 9.. () 97 [3] [5] k 9. m w(t) f (t) = f (t) + mg k w(t) Newton

More information

23 7 28 i i 1 1 1.1................................... 2 1.2............................... 3 1.2.1.................................... 3 1.2.2............................... 4 1.2.3 SI..............................

More information

6kg 1.1m 1.m.1m.1 l λ ϵ λ l + λ l l l dl dl + dλ ϵ dλ dl dl + dλ dl dl 3 1. JIS 1 6kg 1% 66kg 1 13 σ a1 σ m σ a1 σ m σ m σ a1 f f σ a1 σ a1 σ m f 4

6kg 1.1m 1.m.1m.1 l λ ϵ λ l + λ l l l dl dl + dλ ϵ dλ dl dl + dλ dl dl 3 1. JIS 1 6kg 1% 66kg 1 13 σ a1 σ m σ a1 σ m σ m σ a1 f f σ a1 σ a1 σ m f 4 35-8585 7 8 1 I I 1 1.1 6kg 1m P σ σ P 1 l l λ λ l 1.m 1 6kg 1.1m 1.m.1m.1 l λ ϵ λ l + λ l l l dl dl + dλ ϵ dλ dl dl + dλ dl dl 3 1. JIS 1 6kg 1% 66kg 1 13 σ a1 σ m σ a1 σ m σ m σ a1 f f σ a1 σ a1 σ m

More information

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 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

More information

1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2

1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2 2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6

More information

3 filename=quantum-3dim110705a.tex ,2 [1],[2],[3] [3] U(x, y, z; t), p x ˆp x = h i x, p y ˆp y = h i y, p z ˆp z = h

3 filename=quantum-3dim110705a.tex ,2 [1],[2],[3] [3] U(x, y, z; t), p x ˆp x = h i x, p y ˆp y = h i y, p z ˆp z = h filename=quantum-dim110705a.tex 1 1. 1, [1],[],[]. 1980 []..1 U(x, y, z; t), p x ˆp x = h i x, p y ˆp y = h i y, p z ˆp z = h i z (.1) Ĥ ( ) Ĥ = h m x + y + + U(x, y, z; t) (.) z (U(x, y, z; t)) (U(x,

More information

Z: Q: R: C: sin 6 5 ζ a, b

Z: Q: R: C: sin 6 5 ζ a, b Z: Q: R: C: 3 3 7 4 sin 6 5 ζ 9 6 6............................... 6............................... 6.3......................... 4 7 6 8 8 9 3 33 a, b a bc c b a a b 5 3 5 3 5 5 3 a a a a p > p p p, 3,

More information

A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π

A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π 4 4.1 4.1.1 A = f() = f() = a f (a) = f() (a, f(a)) = f() (a, f(a)) f(a) = f 0 (a)( a) 4.1 (4, ) = f() = f () = 1 = f (4) = 1 4 4 (4, ) = 1 ( 4) 4 = 1 4 + 1 17 18 4 4.1 A (1) = 4 A( 1, 4) 1 A 4 () = tan

More information

21 2 26 i 1 1 1.1............................ 1 1.2............................ 3 2 9 2.1................... 9 2.2.......... 9 2.3................... 11 2.4....................... 12 3 15 3.1..........

More information

振動と波動

振動と波動 Report JS0.5 J Simplicity February 4, 2012 1 J Simplicity HOME http://www.jsimplicity.com/ Preface 2 Report 2 Contents I 5 1 6 1.1..................................... 6 1.2 1 1:................ 7 1.3

More information

数学演習:微分方程式

数学演習:微分方程式 ( ) 1 / 21 1 2 3 4 ( ) 2 / 21 x(t)? ẋ + 5x = 0 ( ) 3 / 21 x(t)? ẋ + 5x = 0 x(t) = t 2? ẋ = 2t, ẋ + 5x = 2t + 5t 2 0 ( ) 3 / 21 x(t)? ẋ + 5x = 0 x(t) = t 2? ẋ = 2t, ẋ + 5x = 2t + 5t 2 0 x(t) = sin 5t? ẋ

More information

I ( ) 2019

I ( ) 2019 I ( ) 2019 i 1 I,, III,, 1,,,, III,,,, (1 ) (,,, ), :...,, : NHK... NHK, (YouTube ),!!, manaba http://pen.envr.tsukuba.ac.jp/lec/physics/,, Richard Feynman Lectures on Physics Addison-Wesley,,,, x χ,

More information

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) θ

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) θ 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)

More information

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=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

More information

77

77 O r r r, F F r,r r = r r F = F (. ) r = r r 76 77 d r = F d r = F (. ) F + F = 0 d ( ) r + r = 0 (. 3) M = + MR = r + r (. 4) P G P MX = + MY = + MZ = z + z PG / PG = / M d R = 0 (. 5) 78 79 d r = F d

More information

grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( )

grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( ) 2 9 2 5 2.2.3 grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = g () g () (3) grad φ(p ) p grad φ φ (P, φ(p )) y (, y) = (ξ(t), η(t)) ( ) ξ (t) (t) := η (t) grad f(ξ(t), η(t)) (t) g(t) := f(ξ(t), η(t))

More information

量子力学 問題

量子力学 問題 3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,

More information

ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 +

ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + 2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j

More information

(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0

(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0 1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45

More information

m(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120)

m(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120) 2.6 2.6.1 mẍ + γẋ + ω 0 x) = ee 2.118) e iωt Pω) = χω)e = ex = e2 Eω) m ω0 2 ω2 iωγ 2.119) Z N ϵω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j 2.120) Z ω ω j γ j f j f j f j sum j f j = Z 2.120 ω ω j, γ ϵω) ϵ

More information

I 1

I 1 I 1 1 1.1 1. 3 m = 3 1 7 µm. cm = 1 4 km 3. 1 m = 1 1 5 cm 4. 5 cm 3 = 5 1 15 km 3 5. 1 = 36 6. 1 = 8.64 1 4 7. 1 = 3.15 1 7 1 =3 1 7 1 3 π 1. 1. 1 m + 1 cm = 1.1 m. 1 hr + 64 sec = 1 4 sec 3. 3. 1 5 kg

More information

211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,

More information

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5. A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c

More information

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y [ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)

More information

Untitled

Untitled http://www.ike-dyn.ritsumei.ac.jp/ hyoo/dynamics.html 1 (i) (ii) 2 (i) (ii) (*) [1] 2 1 3 1.1................................ 3 1.2..................................... 3 2 4 2.1....................................

More information

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11

More information

Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e

Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e 7 -a 7 -a February 4, 2007 1. 2. 3. 4. 1. 2. 3. 1 Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e z

More information

5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1

5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1 4 1 1.1 ( ) 5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1 da n i n da n i n + 3 A ni n n=1 3 n=1

More information

29

29 9 .,,, 3 () C k k C k C + C + C + + C 8 + C 9 + C k C + C + C + C 3 + C 4 + C 5 + + 45 + + + 5 + + 9 + 4 + 4 + 5 4 C k k k ( + ) 4 C k k ( k) 3 n( ) n n n ( ) n ( ) n 3 ( ) 3 3 3 n 4 ( ) 4 4 4 ( ) n n

More information

Aharonov-Bohm(AB) S 0 1/ 2 1/ 2 S t = 1/ 2 1/2 1/2 1/, (12.1) 2 1/2 1/2 *1 AB ( ) 0 e iθ AB S AB = e iθ, AB 0 θ 2π ϕ = e ϕ (ϕ ) ϕ

Aharonov-Bohm(AB) S 0 1/ 2 1/ 2 S t = 1/ 2 1/2 1/2 1/, (12.1) 2 1/2 1/2 *1 AB ( ) 0 e iθ AB S AB = e iθ, AB 0 θ 2π ϕ = e ϕ (ϕ ) ϕ 1 13 6 8 3.6.3 - Aharonov-BohmAB) S 1/ 1/ S t = 1/ 1/ 1/ 1/, 1.1) 1/ 1/ *1 AB ) e iθ AB S AB = e iθ, AB θ π ϕ = e ϕ ϕ ) ϕ 1.) S S ) e iθ S w = e iθ 1.3) θ θ AB??) S t = 4 sin θ 1 + e iθ AB e iθ AB + e

More information

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345

More information

1 3 1.1.......................... 3 1............................... 3 1.3....................... 5 1.4.......................... 6 1.5........................ 7 8.1......................... 8..............................

More information

最新耐震構造解析 ( 第 3 版 ) サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 3 版 1 刷発行時のものです.

最新耐震構造解析 ( 第 3 版 ) サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます.   このサンプルページの内容は, 第 3 版 1 刷発行時のものです. 最新耐震構造解析 ( 第 3 版 ) サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/052093 このサンプルページの内容は, 第 3 版 1 刷発行時のものです. i 3 10 3 2000 2007 26 8 2 SI SI 20 1996 2000 SI 15 3 ii 1 56 6

More information

II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re

II ( ) (7/31) II (  [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier

More information

l µ l µ l 0 (1, x r, y r, z r ) 1 r (1, x r, y r, z r ) l µ g µν η µν 2ml µ l ν 1 2m r 2mx r 2 2my r 2 2mz r 2 2mx r 2 1 2mx2 2mxy 2mxz 2my r 2mz 2 r

l µ l µ l 0 (1, x r, y r, z r ) 1 r (1, x r, y r, z r ) l µ g µν η µν 2ml µ l ν 1 2m r 2mx r 2 2my r 2 2mz r 2 2mx r 2 1 2mx2 2mxy 2mxz 2my r 2mz 2 r 2 1 (7a)(7b) λ i( w w ) + [ w + w ] 1 + w w l 2 0 Re(γ) α (7a)(7b) 2 γ 0, ( w) 2 1, w 1 γ (1) l µ, λ j γ l 2 0 Re(γ) α, λ w + w i( w w ) 1 + w w γ γ 1 w 1 r [x2 + y 2 + z 2 ] 1/2 ( w) 2 x2 + y 2 + z 2

More information

S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt

S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt S I. x yx y y, y,. F x, y, y, y,, y n http://ayapin.film.s.dendai.ac.jp/~matuda n /TeX/lecture.html PDF PS yx.................................... 3.3.................... 9.4................5..............

More information

II 2 II

II 2 II II 2 II 2005 yugami@cc.utsunomiya-u.ac.jp 2005 4 1 1 2 5 2.1.................................... 5 2.2................................. 6 2.3............................. 6 2.4.................................

More information

K E N Z U 2012 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.2................................... 4 1.2.1..................................... 4 1.2.2.................................... 5................................

More information

C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0

C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0 1 2003 4 24 ( ) 1 1.1 q i (i 1,,N) N [ ] t t 0 q i (t 0 )q 0 i t 1 q i (t 1 )q 1 i t 0 t t 1 t t 0 q 0 i t 1 q 1 i S[q(t)] t1 t 0 L(q(t), q(t),t)dt (1) S[q(t)] L(q(t), q(t),t) q 1.,q N q 1,, q N t C :

More information

( ) ( 40 )+( 60 ) Schrödinger 3. (a) (b) (c) yoshioka/education-09.html pdf 1

( ) ( 40 )+( 60 ) Schrödinger 3. (a) (b) (c)   yoshioka/education-09.html pdf 1 2009 1 ( ) ( 40 )+( 60 ) 1 1. 2. Schrödinger 3. (a) (b) (c) http://goofy.phys.nara-wu.ac.jp/ yoshioka/education-09.html pdf 1 1. ( photon) ν λ = c ν (c = 3.0 108 /m : ) ɛ = hν (1) p = hν/c = h/λ (2) h

More information

液晶の物理1:連続体理論(弾性,粘性)

液晶の物理1:連続体理論(弾性,粘性) The Physics of Liquid Crystals P. G. de Gennes and J. Prost (Oxford University Press, 1993) Liquid crystals are beautiful and mysterious; I am fond of them for both reasons. My hope is that some readers

More information

Chebyshev Schrödinger Heisenberg H = 1 2m p2 + V (x), m = 1, h = 1 1/36 1 V (x) = { 0 (0 < x < L) (otherwise) ψ n (x) = 2 L sin (n + 1)π x L, n = 0, 1, 2,... Feynman K (a, b; T ) = e i EnT/ h ψ n (a)ψ

More information

z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy

z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy z fz fz x, y, u, v, r, θ r > z = x + iy, f = u + iv γ D fz fz D fz fz z, Rm z, z. z = x + iy = re iθ = r cos θ + i sin θ z = x iy = re iθ = r cos θ i sin θ x = z + z = Re z, y = z z = Im z i r = z = z

More information

(MRI) 10. (MRI) (MRI) : (NMR) ( 1 H) MRI ρ H (x,y,z) NMR (Nuclear Magnetic Resonance) spectrometry: NMR NMR s( B ) m m = µ 0 IA = γ J (1) γ: :Planck c

(MRI) 10. (MRI) (MRI) : (NMR) ( 1 H) MRI ρ H (x,y,z) NMR (Nuclear Magnetic Resonance) spectrometry: NMR NMR s( B ) m m = µ 0 IA = γ J (1) γ: :Planck c 10. : (NMR) ( 1 H) MRI ρ H (x,y,z) NMR (Nuclear Magnetic Resonance) spectrometry: NMR NMR s( B ) m m = µ 0 IA = γ J (1) γ: :Planck constant J: Ĵ 2 = J(J +1),Ĵz = J J: (J = 1 2 for 1 H) I m A 173/197 10.1

More information

ver F = i f i m r = F r = 0 F = 0 X = Y = Z = 0 (1) δr = (δx, δy, δz) F δw δw = F δr = Xδx + Y δy + Zδz = 0 (2) δr (2) 1 (1) (2 n (X i δx

ver F = i f i m r = F r = 0 F = 0 X = Y = Z = 0 (1) δr = (δx, δy, δz) F δw δw = F δr = Xδx + Y δy + Zδz = 0 (2) δr (2) 1 (1) (2 n (X i δx ver. 1.0 18 6 20 F = f m r = F r = 0 F = 0 X = Y = Z = 0 (1 δr = (δx, δy, δz F δw δw = F δr = Xδx + Y δy + Zδz = 0 (2 δr (2 1 (1 (2 n (X δx + Y δy + Z δz = 0 (3 1 F F = (X, Y, Z δr = (δx, δy, δz S δr δw

More information

2.2 h h l L h L = l cot h (1) (1) L l L l l = L tan h (2) (2) L l 2 l 3 h 2.3 a h a h (a, h)

2.2 h h l L h L = l cot h (1) (1) L l L l l = L tan h (2) (2) L l 2 l 3 h 2.3 a h a h (a, h) 1 16 10 5 1 2 2.1 a a a 1 1 1 2.2 h h l L h L = l cot h (1) (1) L l L l l = L tan h (2) (2) L l 2 l 3 h 2.3 a h a h (a, h) 4 2 3 4 2 5 2.4 x y (x,y) l a x = l cot h cos a, (3) y = l cot h sin a (4) h a

More information

webkaitou.dvi

webkaitou.dvi ( c Akir KANEKO) ).. m. l s = lθ m d s dt = mg sin θ d θ dt = g l sinθ θ l θ mg. d s dt xy t ( d x dt, d y dt ) t ( mg sin θ cos θ, sin θ sin θ). (.) m t ( d x dt, d y dt ) = t ( mg sin θ cos θ, mg sin

More information

M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 -

M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 - M3............................................................................................ 3.3................................................... 3 6........................................... 6..........................................

More information

housoku.dvi

housoku.dvi : 1 :, 2002 07 14 1 3 11 3 12 : 3 13 : 5 14 6 141 6 142 8 143 8 144 8 145 9 2 10 21 10 : 2 22 11 23 11 24 11 A : 12 B 14 C 15 ( ) ( ) ( ),, (,, ) 2,, : 1 3 1 11 (t),, ρv 0 (1) t (t) ( A ) ( ) ρ (t) t +ρ

More information

S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d

S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5....

More information

微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.

微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます.   このサンプルページの内容は, 初版 1 刷発行時のものです. 微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)

More information

2000年度『数学展望 I』講義録

2000年度『数学展望 I』講義録 2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53

More information

2011de.dvi

2011de.dvi 211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37

More information

A 99% MS-Free Presentation

A 99% MS-Free Presentation A 99% MS-Free Presentation 2 Galactic Dynamics (Binney & Tremaine 1987, 2008) Dynamics of Galaxies (Bertin 2000) Dynamical Evolution of Globular Clusters (Spitzer 1987) The Gravitational Million-Body Problem

More information

1

1 Chapter Fgure.: x x s T = 2 mv2 mgx = 0 (.) s = X 0 x 0 x x v = 2gx + + ( ) 2 2 y x 2 ( ) 2 2 y x 2 /2gx (.2) y(x) 2 . S = L(t, r, ṙ) r(t) ṙ = r L(t, r, ṙ) t, r, ṙ L x t r, ṙ r + δr, ṙ + δṙ δs δs = δr

More information

50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq

50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq 49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r

More information

phs.dvi

phs.dvi 483F 3 6.........3... 6.4... 7 7.... 7.... 9.5 N (... 3.6 N (... 5.7... 5 3 6 3.... 6 3.... 7 3.3... 9 3.4... 3 4 7 4.... 7 4.... 9 4.3... 3 4.4... 34 4.4.... 34 4.4.... 35 4.5... 38 4.6... 39 5 4 5....

More information

数値計算:常微分方程式

数値計算:常微分方程式 ( ) 1 / 82 1 2 3 4 5 6 ( ) 2 / 82 ( ) 3 / 82 C θ l y m O x mg λ ( ) 4 / 82 θ t C J = ml 2 C mgl sin θ θ C J θ = mgl sin θ = θ ( ) 5 / 82 ω = θ J ω = mgl sin θ ω J = ml 2 θ = ω, ω = g l sin θ = θ ω ( )

More information

85 4

85 4 85 4 86 Copright c 005 Kumanekosha 4.1 ( ) ( t ) t, t 4.1.1 t Step! (Step 1) (, 0) (Step ) ±V t (, t) I Check! P P V t π 54 t = 0 + V (, t) π θ : = θ : π ) θ = π ± sin ± cos t = 0 (, 0) = sin π V + t +V

More information

R = Ar l B r l. A, B A, B.. r 2 R r = r2 [lar r l B r l2 ]=larl l B r l.2 r 2 R = [lar l l Br ] r r r = ll Ar l ll B = ll R rl.3 sin θ Θ = ll.4 Θsinθ

R = Ar l B r l. A, B A, B.. r 2 R r = r2 [lar r l B r l2 ]=larl l B r l.2 r 2 R = [lar l l Br ] r r r = ll Ar l ll B = ll R rl.3 sin θ Θ = ll.4 Θsinθ .3.2 3.3.2 Spherical Coorinates.5: Laplace 2 V = r 2 r 2 x = r cos φ sin θ, y = r sin φ sin θ, z = r cos θ.93 r 2 sin θ sin θ θ θ r 2 sin 2 θ 2 V =.94 2.94 z V φ Laplace r 2 r 2 r 2 sin θ.96.95 V r 2 R

More information

[1.1] r 1 =10e j(ωt+π/4), r 2 =5e j(ωt+π/3), r 3 =3e j(ωt+π/6) ~r = ~r 1 + ~r 2 + ~r 3 = re j(ωt+φ) =(10e π 4 j +5e π 3 j +3e π 6 j )e jωt

[1.1] r 1 =10e j(ωt+π/4), r 2 =5e j(ωt+π/3), r 3 =3e j(ωt+π/6) ~r = ~r 1 + ~r 2 + ~r 3 = re j(ωt+φ) =(10e π 4 j +5e π 3 j +3e π 6 j )e jωt 3.4.7 [.] =e j(t+/4), =5e j(t+/3), 3 =3e j(t+/6) ~ = ~ + ~ + ~ 3 = e j(t+φ) =(e 4 j +5e 3 j +3e 6 j )e jt = e jφ e jt cos φ =cos 4 +5cos 3 +3cos 6 =.69 sin φ =sin 4 +5sin 3 +3sin 6 =.9 =.69 +.9 =7.74 [.]

More information

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc2.com/ 1 6 3 6.1................................ 3 6.2.............................. 4 6.3................................ 5 6.4.......................... 6 6.5......................

More information

#A A A F, F d F P + F P = d P F, F y P F F x A.1 ( α, 0), (α, 0) α > 0) (x, y) (x + α) 2 + y 2, (x α) 2 + y 2 d (x + α)2 + y 2 + (x α) 2 + y 2 =

#A A A F, F d F P + F P = d P F, F y P F F x A.1 ( α, 0), (α, 0) α > 0) (x, y) (x + α) 2 + y 2, (x α) 2 + y 2 d (x + α)2 + y 2 + (x α) 2 + y 2 = #A A A. F, F d F P + F P = d P F, F P F F A. α, 0, α, 0 α > 0, + α +, α + d + α + + α + = d d F, F 0 < α < d + α + = d α + + α + = d d α + + α + d α + = d 4 4d α + = d 4 8d + 6 http://mth.cs.kitmi-it.c.jp/

More information

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,. 9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,

More information

I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )

I A A441 : April 15, 2013 Version : 1.1 I   Kawahira, Tomoki TA (Shigehiro, Yoshida ) I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17

More information