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.

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2 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.

3 1.1 V 1 ev = J e = C 1.1 L M T Q 1. imensional analysis m k A T T = m α k β A γ F = kx [k] = MLT /L = M/T = T, = M α M T β L γ = M α+β T β L γ α + β = 0, β = 1, γ = 0 α = 1/, β = 1/, γ = 0 T = m k 1.3 T = π m/k A 1. g T l g m G = m 3 kg 1 s M = kg R = m M = kg R = m 3

4 1.3 rt = xt, yt, zt v a vt = rt, at = vt = rt 1.4 v = ṙ = ẋ, ẏ, ż ft + h ft ft = lim h 0 h 1.5 t n sin t f f + g = + g, f fg = g + f g cf = cf x fxt = x f 1 x = fx x y fy 1.9 y = f 1 x x = fy f = x f x, y x = 1 x y φ t φ = φt sin φ φ sin φ sin φ = φ = φ cos φ cos φ 4

5 1.3 1 ft f = f, n n ftgt = n r=0 gt ft = ġf g f f n f r tg n r t r f r t ft r n r n n! = r r!n r! e = e = lim n = n n n=0 1 n! 1.10 log t = t 1 u u log t = 1 t 1.11 logxy = log x + log y y = e x x = log y 1.1 e x+y = e x e y logxy = log X + log Y log1 + t = 1 + t, et = e t log1 + t = t t + t3 3 = 1 n 1 tn n n=1 e t = 1 + t + t! + t3 3! + = t n n! n=0 1 < t 1 sin t = cos t, cos t = sin t, tan t = sec t tan t sec t = 1/ cos t 5

6 e it = cos t + i sin t 1.13 i = 1 e ix+y = e ix e iy sin t = t t3 3! + t5 t, cos t = 1 5!! + t4 4! 1.14 e t = 1 + t + t! + t3 3! + e it = 1 + it + it! = cos t + i sin t + it3 3! + = 1 t! + t4 4! + i t t3 3! + t5 5! g m z = mg z = g 1.15 z0 = z 0, ż0 = v 0 zt = z 0 + v 0 t 1 gt m g µ v 0 t vt T L 1.6 m v m v = kv + mg v0 = v 0 lim t vt 6

7 1.7 T a T a 3 T a 10 8 km ε T a f = G Mm r M = m = a 7

8 8

9 .1 1. f = 0 a = 0. ma = f 3. f 1 = f 1 4. f = G m 1m r f a f 1 = f 1 = 0 k f = kx ω = k/m mẍ = kx ẍ = ω x.1 xt = A cosωt + B sinωt. x0 = x 0, ẋ0 = v 0 A = x 0, B = v 0 /ω..1 + iω iω xt = 0 + iω xt = 0, iω xt = 0 e at / = ae at e ±iωt sinωt cosωt..1 1 mẋ 1 kx = m ω x E = m ẋ + ω x E 9

10 V x E = m ẋ + V x ẋ mẋẍ = V xẋ mẍ = V x.3 1 mẋ + V x = 0 E = 1 mẋ + V x =.4.1 F mẍ = mω x + F x0 = 0, ẋ0 = 0 Et = m ẋ + ω x. = V x x ẋ Lx, ẋ Lagrangian Lx, ẋ = 1 mẋ V x.5.3 L L ẋ x = 0.6 mẋ V x = 0 mẍ + V x = 0 mẍ = V x.3 10

11 r = x 1, x, x 3 Lr, ṙ L L ṙ r = 0 L L = 0 j = 1,, 3.7 x j x j L = m ẋ + ẏ mω x + y L L ẋ x = 0 mẍ + mω x = 0 L ẏ L y = 0 mÿ + mω y = 0 x, y. l m θ L = m l θ mgl1 cos θ θ ml θ = mg sin θ.3 r = r f f = r V r = V rˆr ˆr = r r = r.8 L = r p, p = mv = mṙ.9 r p L x = yp z zp y, L y = zp x xp z, L z = xp y yp x.10 11

12 L = 0 L = ṙ p + r ṗ = 0.11 ṙ = p/m ṗ = f r 1 r ṙ L = r p.4 M, m G V r = GMm/r L = m ṙ + GMm r.1 xy x = r cos φ, y = r sin φ ẋ = ṙ cos φ r φ sin φ, ẏ = ṙ sin φ + r φ cos φ.13 ṙ = ẋ + ẏ = ṙ cos φ r φ sin φ + ṙ sin φ + r φ cos φ = ṙ + r φ.14 L = m ṙ + r φ + GMm r.15 L L ṙ r = 0, L φ L φ = L m r mr φ + GMm r = 0, r φ = = h mr φ = 0.17 r = h r 3 GM r.18 1

13 m r = rφ t φ = h r = h r φ.19 u = 1/r.18 GM u = u + φ h.0 φ t.1 ω = 1 u = GM l h h + A cos φ r =, l = 1 + ε cos φ GM, ε = Ah GM.1 l ε ε 0 ε < 1 ε = 1 ε > 1.3 E E = m ṙ + r φ GMm r.1 0 ε < 1 E.4 T E = GMm 1 ε l.18 ṙ t ṙ = C h r + GM r, C = = r min = r max r 1 = h 1 1 r min r r 1 r max r min = l/1 + ε, r max = l/1 ε T T = r max r min h rmax r min rr rmax rr r min r = r min cos θ + r max sin θ T = πl h1 ε 3/ = 4π a 3 a = r min + r max / = l/1 ε GM 1/ 13

14 3 3.1 Lx, ẋ I = t1 t 0 Lx, ẋ 3.1 xt, ẋt I functional xt ẋt xt xt + δxt, ẋt ẋt + δẋt I I I + δi δi δi = t1 t 0 Lx + δx, ẋ + δẋ Lx, ẋ = t1 t 0 L L δx + x ẋ δẋ 3. δẋ = δx δxt 0 = δxt 1 = 0 δi = t1 t 0 L x L δxt 3.3 ẋ = 0 I 3.3 δi δxt = L x L ẋ 3.4 fx f x = 0 I 3.4=0 principle of the least action 3.1 I = t1 t 0 m ẋ ω x xt 0 = x 0, xt 1 = x 1 I 14

15 3. m 1, m r 1, r V r 1 r L = 1 R r r 1, r m1 ṙ 1 + m ṙ V r1 r 3.5 R = m 1r 1 + m r m 1 + m, r = r 1 r 3.6 r 1 = R + m m 1 + m r, r = R m 1 m 1 + m r 3.7 R, r reuce mass L = m 1 + m Ṙ m 1 m + ṙ V r 3.8 m 1 + m µ = m 1m m 1 + m 1 µ = 1 m m L = L + L M, m µ = Mm/M + m V r = GMm/r L = µ ṙ + GMm r m µ m/m = µ = m µ = m/ m 1 + m R = 0, µ r = r V r 3.3. ω = g/l θ = ω sin θ

16 θ sin θ θ θ = ω θ 3.11 θ 1 θ ω cos θ = = ω cos θ θ 0 θ/ θ = ω cos θ cos θ 0 = 4ω sin θ 0 θ sin t = 0 θ0 = θ 0, θ0 = 0 0 < θ 0 < π θ θ 0 θ = ω sin θ0 sin θ t = ωt 3.14 T T 3.14 T = ω = 4 ω θ0 0 π 0 θ sin θ0 sin θ φ 1 k sin φ k = sin θ 0 sin φ = sinθ/ sinθ 0 / 4 Kk 3.15 ω Kk 0 k < 1 π φ Kk = 1 k sin φ = π k + k K0 = π/, lim k 1 Kk = T = π/ω = π l/g 60 θ 0 = π/3 k = sinπ/6 = 1/ =1/16 6 % x = x +, 1 xα = 1 αx + x < 1 16

17 4 4.1 canonical formalism x x, ẋ Lx, ẋ x x p Hx, p x, p canonical variables Lx, ẋ 1 Lx, ẋ p = L ẋ x p = mẋ L = m ẋ V x 4.1 L = m p = L = mẋ 4. ẋ ṙ + r φ V r p r = L ṙ = mṙ, p φ = L φ = mr φ 4.3 p φ Hx, p = ẋp Lx, ẋ 4.4 Hamiltonian ẋ p = L/ ẋ ẋ x, p 4. H = ẋp Lx, ẋ = p m 4.3 H = ṙp r + φp φ L = p r m + p φ mr m m pr m p V x = m p + V x 4.5 m + r p φ mr = 1 p r + p φ m r + V r

18 3 x, p ẋ = H p, ṗ = H x 4.7 ẋ ẋ = ẋx, p Hx, p = ẋx, pp Lx, ẋx, p H p H x = ẋ p p + ẋ L ẋ ẋ p = ẋ = ẋ L x p x + L ẋ ẋ x = ṗ = L x = L ẋ L/ ẋ = p x, p p/ x = H = p m + V x ẋ = H p = p m, ṗ = H x = V x H = 1 p r + p φ m r + V r ṙ = H = p r p r m, φ = H = p φ p φ mr, p r = H r = p φ mr 3 V r, 4.9 p φ = H φ = p r, p φ r, φ ṗ φ = 0 p φ = mr φ = 4. H = p m + mω x, ẋ = p m, ṗ = mω x

19 p ẍ = ω x ξ = p + imωx i = 1 ξ = ṗ + imωẋ = mω x + iωp = iω p + imωx = iωξ ξt = ξ0 e iωt 4.1 ξt = pt + imωxt, ξ0 = p0 + imωx0 e iωt = cosωt + i sinωt pt = p0 cosωt mωx0 sinωt, xt = x0 cosωt + p0 sinωt 4.13 mω x, p Hx 1, x, p 1, p = 1 p 1 + p + ω x 1 + x + γx 1 x 4.14 m = 1 ẋ 1 = H p 1 = p 1, ẋ = H p = p, ṗ 1 = H x 1 = ω x 1 γx, ṗ = H x = ω x γx 1 p 1, p ẍ 1 = ω x 1 γx, ẍ = ω x γx x = x 1 + x, y = x 1 x ẍ = ω + γx, ÿ = ω γy ω x = ω + γ, ω y = ω γ xt = x0 cosω x t + ẋ0 ω x sinω x t, yt = y0 cosω y t + ẏ0 ω y sinω y t, 4.16 x 1, x 4.3 m q E B m r = q E + ṙ B

20 4.1 E = E 0, 0, 0 B = 0, 0, B 0 m v = q E + v B v0 = φ A L = A ṙ mṙ + qa = m r + q t L r E = graφ A, B = rota 4.18 t Lr, ṙ = m ṙ + qṙ A qφ ṙ A, = q φ r + q φ ṙ A = q + q ṙ A + ṙ rota r r m r = q A t graφ + qṙ rota = q E + ṙ B A Ar, t = + ṙ A, graa b = a b + b a + a rotb + b rota 4.0 t 4. gauge transformation φ φ χ t, A A + graχ 4.18 E B p = L ṙ = mṙ + qa 4.1 0

21 p mṙ m Hr, p = ṙ p L = ṙ mṙ + qa ṙ + qṙ A qφ = m ṙ + qφ = 1 m p qa + qφ D t roth = j, ive = ρ, B + rote = 0, ivb = t j ρ D = ε 0 E, H = B/µ 0 ε 0 µ 0 c c = 1/ε 0 µ 0 E, B, φ, A L = 1 ε 0 E 1µ0 B ε 0 E φ + Ȧ 1 B rota ρφ + j A 4.4 µ 0 t1 I = mc 1 ẋ c 4.5 t 0 ẋ ṙ 1 ẋ /c 1 ẋ /c I = t1 t 0 m ẋ mc 4.5 mẋ = ẋ /c p = mẋ/ 1 ẋ /c ṗ = 0 1

22 5 5.1 L = ẋp H t1 t1 I = L = t 0 x x + δx, p p + δp t 0 ẋp Hx, p 5.1 δi = = = t1 t 0 t1 t 0 t1 t 0 ẋ + δẋp + δp Hx + δx, p + δp ẋp Hx, p δp ẋ H p p δp ṗ + H x δẋp + ẋδp H H δx x δx 5. δxt 0 = δxt 1 = 0 δi = 0 ẋ = H p, ṗ = H x x, p X, P Hx, p H X, P Ẋ = H P, P = H X canonical transformation generating function 5.4 L = ẋp Hx, p = ẊP H X, P + W 5.5 W W x, P, t 5.5 W = W t + W x ẋ + W P P ẋp H = ẊP H + W t + W x ẋ + W P P

23 ẊP = XP X P p = W x, X = W P, H = H + W t 5.6 XP x, y Ω X, Y x = X cosωt + Y sinωt, y = X sinωt + Y cosωt 5.7 W x, y, P X, P Y, t W = P X x cosωt y sinωt + P Y x sinωt + y cosωt X = W P X = x cosωt y sinωt, p x = W x = P X cosωt + P Y sinωt, 5.6 W/ t Y = W P Y = x sinωt + y cosωt, 5.9 p y = W y = P X sinωt + P Y cosωt 5.10 H = H + Ω XP Y Y P X 5.11 H Z L Z ΩL Z 5.1 Ω Hx, y, p x, p y = 1 p m x + p mω y + x + y X, Y 5.3 x, p X, P W x, P, t W = xp X = W P = x, W = xp ɛ p = W x = P 5.1 W = xp + ɛgx, P 5.13 G generator G ɛ Gx, p P 3

24 5.6 X = x + ɛ G P, G X = x + ɛ G p, p = P + ɛ G x P = p ɛ G x 5.14 G P p x X = x + ɛ W = xp + ɛp X = W P = x + ɛ, p = W x = P 5.15 Ω ɛ 5.9 x X = x ɛy, G = xp Y yp X W = xp X + yp Y + ɛxp Y yp X X = W P X = x ɛy, p x = W x = P X + ɛp Y, y Y = y + ɛx Y = W P Y = y + ɛx, 5.16 p y = W y = P Y ɛp X 5.17 ɛ G = xp y yp x = L z t T = t + ɛ xt + ɛ xt = ɛẋ = ɛ H p, pt + ɛ pt = ɛṗ = ɛ H x 5.18 G G

25 6 6.1 x, p x, p Ax, p, Bx, p {A, B} = A B x Poisson bracket p A p x 1, x,, x N, p 1, p,, p N {A, B} = N j=1 B x A B A B x j p j p j x j {x, x} = 0, {x, p} = 1, {p, x} = 1, {p, p} = {x j, x k } = 0, {x j, p k } = δ jk, {p j, x k } = δ jk, {p j, p k } = G 5.14 X = x + ɛ G p, P = p ɛ G x 6.5 x = X x p = X p X + P x X + P p P = 1 + ɛ G x p P = G ɛ p X + X G ɛ x P, 1 ɛ G x p P A B x p A p B x = 1 + ɛ G x p = A X ɛ G p A X + B P A P A X G A ɛ x P 1 ɛ G x p A P ɛ X + ɛ G B p 1 + ɛ G x p B P 1 ɛ G x p B X G B ɛ x P B X + Oɛ {A, B} = {B, A}. {AB, C} = {A, C}B + A{B, C} 5

26 3. {A, {B, C}} + {B, {C, A}} + {C, {A, B}} = 0 {A, B} = A B x p B A B x p = A x p A x B = {B, A} p {x, p} = {x, p}x + x{x, p} = x, {x n, p} = nx n 1 {x, p } = {x, p}p + p{x, p} = p, {x, p n } = np n n 6.1 {x, p } 6. x, p Ax, p, t A Ax, p, t = t + A x ẋ + A p ṗ = A t + A x H p A p H x = A + {A, H} 6.7 t A explicitly A/ t = 0 A = {A, H} x, p X, P G 6.5 X = x + ɛ G p, P = p ɛ G x F X, P = F x, p + ɛ{f, G} 6. F F = H 6.8 HX, P = Hx, p + ɛ{h, G} = Hx, p ɛ G G G 6

27 6.3 m = 1 H = 1 p + ω x 6.9 a = 1 ω ωx + ip, a = 1 ω ωx ip 6.10 {a, a } = 1 1 {ωx + ip, ωx ip} = ω ω iω{x, p} + iω{p, x} = i, {a, a} = 0, {a, a } = 0 H = ωa a 6.11 a, a a = {a, H} = ω{a, a a} = iωa, a = {a, H} = ω{a, a a} = iωa 6.1 at = a0e iωt, a t = a 0e iωt 6.13 L = L x, L y, L z L x = yp z zp y, L y = zp x xp z, L z = xp y yp x {L x, L y } = {yp z zp y, zp x xp z } = y{p z, z}p x + x{z, p z }p y = xp y yp x = L z, {L y, L z } = {zp x xp z, xp y yp x } = z{p x, x}p y + y{x, p x }p z = yp z zp y = L x, 6.15 {L z, L x } = {xp y yp x, yp z zp y } = x{p y, y}p z + z{y, p y }p x = zp x xp z = L y, 6.3 L ± = L x ± il y, L = L x + L y + L z {L ±, L z } = ±il ±, {L +, L } = il z, {L, L x } = {L, L y } = {L, L z } = 0 7

28 7 7.1 x 1, x,, x N p 1, p,, p N x 1,, x N, p 1,, p N N phase space N N H = p m + mω x 7.1 p m + mω x = E 7. x, p N N N 1 7. x, p x, p X, P XP = xp X, P x, p = G 6.5 X, P x, p X x = ɛ G p, = X x P x = 1 + ɛ P p = ɛ G x X p P = p G x p G p x 1 + ɛ G x p ɛ G x + Oɛ ɛ G p 1 ɛ G p x 7.4 = 1 + Oɛ 7.5 x, p G = H T xt + T pt + T = xtpt 7.6 [ ] 8

29 7.3 action variable J = px 7.7 = Hx, p = E 7.8 p p x E 7.7 p m + mω x = E J = π me E/mω = πe/ω J h +1/ E n = n + 1 hω, h h/π g m l A ω = g/l l l + δl ω A δe 0 δω/ω δa/a δe/e δl δe/ω = 0 E/ω aiabatic invariant J = πe/ω 9

30 1.1 1 T 1 Hz s 1 L 1 MT Pa Nm 3 L MT 1 Q Ω VA 1 4 L 3 M 1 T Q F/m / CV 1 m g = GM/R g = 9.8m/s T = l α g β m γ α = 1/, β = 1/, γ = 0 T = l/g 3.15 = 3 g = GM/R G g /g = M /M R /R = 0.17 T /T = g /g = gt = 1/ft ftgt = 1 fg + fġ = 0 g = f 1 f 1 / = f/f g/f = g f 1 n [ n ftgt = + n ftgt ] = t=t n r=0 t = t [ n r f n r ] g r r n r = t=t n r=0 n f r g n r r 1.4 u = 1 + t u log1 + t = u log u = 1 1 u = t u = e t t = log u /u = log u/u = 1/u u/ = u e t / = e t m v = µmg vt = v 0 µgt 3 T = v 0 /µg, L = v 0 T/ = v 0/µg v = k m v mg k v mg k = k m v mg k vt mg/k = v 0 mg/ke kt/m v = mg/k x F mω = ω x F mω xt = F 1 cosωt/mω E/ = mẍ + ω xẋ = F ẋ Et = F xt 30

31 . v = l θ K = ml θ / V = mgl1 cos θ L = K V L θ L θ = 0 ml θ = mg sin θ.3 φ = 0 r min = l/1 + ε ṙ = 0, φ = h/r min E = m h 1 + ε l h = GMl GMm 1 + ε l = GMm 1 ε l ṙ ṙ r = h r 3 ṙ GM r ṙ ṙ = ṙ r, 1 r = ṙ r 3, 1 r = ṙ r r min, r max ṙ = 0 r = r min cos θ + r max sin θ r = r max r min sin θ cos θθ, r r min = r max r min sin θ, r max r = r max r min cos θ T = 4 r max r min h π/ = πr max + r min r max r min h 0 rmin cos θ + r max sin θ θ = πl h1 ε 3/ a 3.1 ẍ = ω x xt 0 = x 0, xt 1 = x 1 xt = A sinωt t 0 +B sinωt 1 t A = x 1 / sinωt, B = x 0 / sinωt T t 1 t 0 I = = t1 mω x sin 0 cosωt 1 t + x 1 cosωt t 0 x 0 x 1 cosωt 1 + t 0 t ωt t 0 mω x sinωt 0 + x 1 cosωt x 0 x 1 31

32 L L Ṙ R = 0 m 1 + m R = 0 L L = 0 µ r = ṙ r r V r 3.3 1/ 1 x = 1 + x/ + π/ Kk = φ k sin φ + = π 1 + π/ 0 φ sin φ = π/4 0 1 k m v x = qe 0 + v y B 0, m v y = qv x B 0, m v z = 0 v0 = 0 ω = qb 0 /m v x t = E 0 B 0 sinωt, v y t = E 0 B 0 1 cosωt, v z t = 0 4. E = graφ χ t A A + graχ = graφ t t = E, B = rota + graχ = rota = B rotgraχ a b = a x b x + a y b y + a z b z a = a x x + a y y + a z z 4.4 ṙ = H p = 1 p qa, m ṗ = H r = q graφ + 1m p qa A + p qa rota 4.0 p = mṙ + qa m r = q graφ A t ṙ rota = qe + ṙ B H = 1 P m X + PY mω + X + Y + ΩXP Y Y P X 3

33 Ẋ = P X m ΩY, P X, P Y P X = mω X ΩP Y, Ẏ = P Y m + ΩX, P Y = mω Y + ΩP X Ẍ = ω Ω X ΩẎ, Ÿ = ω Ω Y + ΩẊ ξt = Xt + iy t ξ = ω Ω ξ + iωξ ξt = ξ0e iωt cosωt Xt, Y t ω Ω 6.1 {AB, C} = AB x C p AB C p x = = {A, C}B + A{B, C} A C A C x B + A B x p p B + A B p x {x, p } = {x, p }x + x{x, p } = p x + x p = 4xp 6. X = x + ɛ G/ p, Y = y ɛ G/ x F X, Y = F x + ɛ G p, p ɛ G = F x, y + ɛ x F G x p F p G = F x, p + ɛ{f, G} x ɛ 6.3 {L +, L z } = {L x + il y, L z } = L y + il x = il x + il y = il +, {L, L z } = {L x il y, L z } = L y il x = il x il y = il, {L +, L } = {L x + il y, L x il y } = il z, {L, L x } = {L x + L y + L z, L x } = L y {L y, L x } + L z {L z, L x } = L y L z + L z L y = ml θ = mg sin θ mgθ 33

34 θt = A sinωt ω = g/l E = 1 mgla l l + δl δe δe = W + W δl < 0 W = mgδl W = = ml θ +mg cos θ T = π/ω W = 1 T T 0 ml θ + mg 1 θ = mg mga cos θ 1 θ / θt = A cosωt δe = mg mga δl + mgδl = 1 4 mga δl E = 1 mgla δe E = 1 δl l, δω ω = 1 δl l ω = g/l δ E ωδe Eδω = ω ω = E 1 δl ω l + 1 δa E = 1 mgla δl = 0 l δe E = δl l + δa A δa A = 3 δl 4 l 34

35 [ ] mω x mγẋ mẍ + γmẋ + mω x = 0 i 0 < γ < ω ii ω < γ x0 = x 0, ẋ0 = 0 xt [ ] Ω mẍ + mω x = F cosωt = Re F e iωt Re xt = ReAe iωt A = 0 mẍ + mγẋ + mω x = F cosωt = Re F e iωt A A = A e iδ A δ Ω [ ] V x m 1 E = m ẋ + V x T T = x x 1 m E V x x x 1 < x V x = E a V x = K x b V x = V 0 1 e ax [ ] L = r p ṗ = α r 3 r A = ṙ L α r r A/ = 0 A 35

36 [ ] NMR M = M x, M y, M z M x M y M z = γm B x M x T = γm B y M y T = γm B z M z M eq T 1 γ T 1, T M eq M z M eq = M z M B B = 0, 0, B 0 M z t M z 0 Mt = M x t + im y t Mt Mt M0 = M x 0 + im y 0 M x t, M y t [ ] 4.1 vt rt r0 = 0 E B y E B [ ] m = 1 H = 1 p α x x 0 α > 0, x 0 > 0 xt = = x 0, xt = + = +x 0 E = 0 [ ] r, p Hr, p rp ρr, p, t ρ = {H, ρ} t ρ 0 = Cexp H/k B T C ρ 0 rp = 1 ρ 0 0 Ar, p < A >= Ar, pρ 0 r, prp H = p /m C < H > 36

37 [ ] m mg kv m v = mg kv v0 = 0 [ ] m Ze µ z x = r cos φ, y = r sin φ cgs-gauss L = m ṙ + r φ + Zeµ φ c r r, φ p r, p φ p φ φ/ p φ H = p r ṙ + p φ φ L H = 1 p r + 1r p φ Zeµ m cr m r + 1 mr p φ Zeµ = E cr r/ p φ = 0 t r φ = ± c Zeµ r3 m E Zeµ mc r 4 φ < 0 φ > 0 + r φ = ±Ar r 4 r0 4 A, r 0 φ = 0 r = r 0 r 0/r = u r = r 0 φ x 0 u 1 u = sin 1 x 37

38 [ ] x = e αt α + γα + ω = 0 i 0 < γ < ω α = γ ± γ ω xt = e γt A cos ω γ t + B sin ω γ t ii ω < γ xt = e γt Ae γ ω t + Be γ ω t x0 = x 0, ẋ0 = 0 i 0 < γ < ω A = x 0, B = γ ω γ x 0 ii ω < γ A = x γ, B = x 0 γ 1 γ ω γ ω [ ] xt = ReAe iωt Re m Ω + ω Ae iωt = Re F e iωt A = F mω Ω A xt = a cosωt + b sinωt + F mω Ω cosωt xt = ReAe iωt Re m Ω + iγω + ω Ae iωt = Re F e iωt A = A A A δ F A = m γω, tan δ = ω Ω + 4γ Ω Ω ω Ω ω ω Ω + 4γ Ω 4ω Ω ω + γ A F 1 mω Ω ω + γ F mω Ω + iγω 38

39 Ω Lorentzian [ ] m T x + V x = E x = ± E V x m x T = x 1 x = m E V x x x 1 < x E = V x x 1 m x E V x x 1 x a V x = K x x 0 = E/K x0 m m T = x x 0 E K x = K m = K x 0 = 4 me K b V x = V 0 1 e ax y T = x x 1 x0 m x E V 0 1 e ax y = e ax y = ayx y1 y m T = ay E V 0 1 y = 1 m a x 1 < x y < y 1 V 0 y1 y 0 x x0 x y y y y y 1 y y 1 = 1 + E V 0, y = 1 E V 0 y = y cos θ + y 1 sin θ y1 y y π/ y y y y 1 y = 0 θ y cos θ + y 1 sin θ = π y1 y T T = π m a V 0 E 39

40 [ ] A = ṙ L α r r t Ȧ = r L + ṙ L α r ṙ + α r ṙr L = 0 r = α mr 3 r L = r p = mr ṙ Ȧ = α r 3 r r ṙ α r ṙ + α r ṙr r r ṙ = r ṙr r rṙ, r r = r, r ṙ = 1 r r = 1 r = rṙ r r [ ] B = 0, 0, B 0 M z M x, M y M B x = M y B z M z B y = B 0 M y, M B y = M z B x M x B z = B 0 M x, M B z = M x B y M y B x = 0, M z = M z M eq T 1 M z t M eq = M z 0 M eq e t/t1 M x = γb 0 M y M x T, M y = γb 0 M x M y T M = M x + im y M = M iγb 0 + 1T M Mt = M0exp iγb t T M x t = ReMt = e t/t M x 0 cosω 0 t + M y 0 sinω 0 t M y t = ImMt = e t/t M x 0 sinω 0 t + M y 0 cosω 0 t ω 0 = γb 0 [ ] - ẋ v x = E 0 B 0 sinωt, ẏ v y = E 0 B 0 1 cosωt t x0 = y0 = 0 xt = E 0 ωb 0 1 cosωt, yt = E 0 ωb 0 ωt sinωt 40

41 [ ] 1 + x α x x 0 = 0 x = ± αx 0 x x 0 x x 0 x = α 1 x 0 x 0 t 0 xt = x 0 tanh αx 0 t 1 x 0 x + 1 = 1 x0 + x log x 0 + x x 0 x 0 x xt t t t 0 xt = x 0 tanh αx 0 t [ ] 0 = ρ = ρ t + ρ r ṙ + ρ pṗ ρ t = H ρ r p ρ r H p = {H, ρ} ρ 0 = C exp H/k B T H r, p H {H, ρ 0 } = 0 ρ 0 t = 0 H = p /m ρ 0 rp = 1 C r e p /mk B T p = CV πmk B T 3/ = 1 C = 1 V πmk B T 3/ C p H = Hρ 0 rp = C p r m e p /mk B T p = 1 m mk BT 3 = 3 k BT [ ] v = g k m v = k v m v mg, v k 41

42 v v v v = k m [ ] 1 v + v log = k v v v m t vt = v kv t tanh = m 0 t 0 mg kg k tanh m t [ ] p r = L ṙ = mṙ, p φ = L φ = Zeµ mr φ + cr L ṙ L φ L r = 0 p r L φ = 0 p φ = 0 mr φ Zeµ φ cr = 0 r = r φ Zeµ φ mcr, p φ = m H = p r ṙ + p φ φ L = ṙ + r φ 1 = p r + 1r p φ Zeµ m cr φ = 1 mr p φ Zeµ cr m r + 1 mr p φ Zeµ = E cr r = E 1 m mr p φ Zeµ cr p φ = 0 r = ± E Zeµ m mc r 4 p φ = 0 φ = Zeµ mcr 3 4

43 t r φ = r/ φ/ = ± cr3 Zeµ m E Zeµ mc r 4 ±Ar r 4 r 4 0 r r 0 r 4 0 = Zeµ mc E r ± c me Zeµ r3 = ±Ar 3 A = c me Zeµ = 1 r0 + φ = 0 r r 0 r r r 4 r 4 0 = A φ 0 φ Aφ u = r0/r u = ur/r = 1 u u r0 = u r0 sin 1 u π u = r 0 π r = sin r 0Aφ = cosφ r = r 0 cosφ A = 1/r0 φ = π/4 φ = 0 r = r 0 φ = π/4 φ = π/ 43

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