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r r(t) r(t) x(t) y(t) z(t) t = x(t) + y(t) + z(t) r(t) ex ey ez.

x = cos, 0 <, y = sin, 0 < 2, z = z.

= cos + sin, e ex ey = - sin + cos, e ex ey = ez ez.

= + z r(t) e ez.

x = cos sin, 0 <, y = sin sin, 0 < 2, z = cos, 0.

= cos sin + sin sin + cos, e ex ey ez = cos cos + sin cos - sin, e ex ey ez = - sin + cos.

e ex ey = r(t) e.

= - r12(t) r2(t) r1(t).

t t s(t) = ()d.

t d r (t) = = dt = H1q1 + H2q + H3q e1 e2 e3.

Hj 2 2 dx dy dz Hj = + +.

dqj dqj dqj = [ ].

r, = i sin(t) + cos(t) r j x = sin(t) y = cos(t) dx dy x = = cos(t), y = = - sin(t).

dt dt dx dy ax = = -2 sin(t), ay = = -2 cos(t).

dt dt || = 2 + 2 = sin2(t) + cos2(t) =.

x y d a = = 0, dt a = a2 + a2 = x y R = a = = t = 0 = 0 d = =.

dt d = dt.

= C exp(t), C d = 0 exp(t), = = 0 exp(t).

dt = 0 exp(t) = 0 exp(t) x y x = y = 0 = a +bx a b i j i j x y y(x) y- x dt dy = ydt, dx = xdt, y = bx x = a dy = (b/a)xdx.

x bxy = (b/a)xdx =, 2a y = (b/2a)x x = r cos y = r sin x = ln k0t - 1 k k 0 e E = A sin(kt) A k eA sin(kt) x = t - m mk k M y = ex/ + e-x/.

x = 0t cos, y = 0t sin - t2/2, 0 0 x H L T x = cos(kt), y = sin(kt), z = t.

x = 2 cos2(kt/2), y = sin(kt), k x = R cos2(kt/2), y = (R/2) sin(kt), z = R sin(kt/2), x = a sin(kt) a k x = x = 1 x = x2 = Ox = x t = 0 x = x0 s y(x) m x = (1 - cos(t)), y = (1 - cos(t), 0 < < x = (1 - cos(t)), y = (1 - sin(t), 0 < < dr m = F, dt F m k 0 d m = -k2.

dt d k = - dt, 2 m 1 k - = - t + C, m C t = 0 1 k C = -1/0 - = - t - m = 0/2 m t = km t = k m t = 0 F F = kt k d m = F cos.

dt m = F sin.

kt cos a =.

m m t0 t0 = k sin t0 t0 tk cos m cos = adt = tdt =, m 2k sin0 t0 tk cos m2 cos s = dt = t2dt =, 2m 6k2 sin0 2 m cos m2 cos = s = 2k sin2 6k2 sin m F = F0 sin(t) t = 0 = r = 0 r d F= sin(t).

dt m d dt 0 t F0 t (t) - (0) = sin(t)dt.

m (0) = 0 F (t) = 1 - cos(t).

m d r dt r d = (t)dt.

r 0 t F0 t - = 1 - cos(t) dt.

r(t) r(0) m = r(0) F = t - sin(t).

r(t) m F = t - sin(t) r(t) m 12 12 20 90 300 5 10-4 Fc l k 5 m t = 0 F = F0 cos(t) F0 0 k m t = 0 0 F = k k 0 h 0 > m x = a cos(kt) y = b sin(kt) F P P -a x = 0 1 - exp t, a P 0 a m H 0 a0 = 0 m Rc = k f 0 m F = Q sin(kt) t m1 t1 m U(r) r = | r| dU r F = - U(r) = r dr r K = F, r, M = p =.

r, M (r, ) m L = 2 + r22 - U(r).

L M = = mr2 =.

m 2 + r22 + U(r) = E =.

= M/(mr2) 2 M = E - U(r) -.

m m2r > 0 < 0 r E Ueff, M Ueff(r) = U(r) + 2mr M F = -k2m r r m k2 M0 a, - 0 k O M x y t = 0 x = a, y = -, = 0, = 0.

k P = m F = -k2m r O m = m - k2m r r.

x y + k2x = 0, + k2y = -. 2 + k2 = 0 1,2 = ki x = C1 cos(kt) + C2 sin(kt), C1 C2 C1 C = -C1k sin(kt) + C2k cos(kt) t = 0 x = a t = 0 = 0 C1 = a C2 = 0 C1 C x = a cos(kt). y = y1 + y2, y2 y1 + k2y = 0. y = C3 cos(kt) + C4 sin(kt). y2 = A A = y = A y2 = -. k y = C3 cos(kt) + C4 sin(kt) -. k C3 C4 = -C3k sin(kt) + C4k cos(kt). t = 0 y = - k t = 0 = 0 C3 = 0 C4 = k y = sin(kt) -. k k x k cos(kt) =, sin(kt) = y +, a 0 k y + x2 k+ = 1.

a2 0 k A 0, - a k k m e E 0 m e E 0 m e H 0 m e E = E0 cos(t) 0 m e E = E0 cos(t) E0 t = 0 = (0) = 0 (t) r(0) r0 r(t) m O F = -k r t = 0 r 0 O m Fc = -k (t) r(t) = 0, 0, r0 0 = 0 cos, 0, 0 sin r U(x) = -Ax4 m z = 0 x = a (kt) y = b (kt) a E H = 0, 0, H eH cos(y/a) = 0 (0) = 0, a, 0 = r(0) mc md dm = Fex + u.

dt dt m Fex u dm dt M u a dm = dt u u md = Fex - u.

dt Fex - u = Fex = m, dm m = =.

dt u ma = Fex - u.

dm Fex - ma = =.

dt u Fex - ma = m /u = u m0 = 100 mf = 50 u = 800 md dm = Fex - u, dt dt m dm dt d dt Fex = 0 T m0-mf 1 dm d = -, u m 0 m m.

T = u ln m0 - mf m T = u ln = 555 m0 - mf u (t) m d/dt = 0 dm m + u = 0, dt dm = - dt. m u m ln = - t. m0 u u m0 u t = ln = ln(1 + ), m = (m0 - m)/m dm = - = m, dt u m m = m0e- t/u = m0e- t/u.

u u t = ln(1 + ) = m0e- t/u u m0 0 u u m dt md dm = Fex + u dt dt dt Fex = 0 dm d = u.

m dm < 0 u = 0 d dt dm |d| u.

d = = 0 0 m u = ln(m0/m).

u = ln(m0/m) u m = m0f(t) R(x, ) m0 n u a u T u = 2.4 9 a u k u m0 m0 u m t t = 0 F (t) t = 0 m m0 t = 0 m0 dm/dt = -c m u t M u dm/dt = k = m qi d L L - = 0, dt q q L(q, q, t) = T (q, q) - U(q, t) T U = 1, 2,..., s s q p = 1, 2,..., s s L p =. q q p H s H(q, p, t) = pq - L(q, q, t), = q p q t p(t) q(t) dp H dq H = -, =, = 1,..., s. dt q dt p H(q, p, t) = T (p, q) + U(q, t), T U (p, q, t) (p, q, t) s {, } = -.

p q q p, = -,.

c =, c 0.

c1 + c2, = c1, + c2,, c1, c2 , =, +,.

, =, +,.

t t t = p, p =.

q = q, q =.

p p, q =,, p, p = q, q = 0.

,, +,, +,, = 0.

f(q, p, t) df f = + H, f.

dt t p q P = P(q, p, t), Q = Q(q, p, t) P Q dP H dQ H = -, =, = 1,..., s.

dt Q dt P H = H P Q q, Q; q, P; p, Q; p, P.

F = F(q, Q, t) q, Q F F F p =, P = -, H = H +.

q Q t m L = 2 + 2 + 2.

d L L d L L d L L - = 0, - = 0, - = 0, dt x dt y dt z m = 0, m = 0, m = 0.

z = r(t 0) = (t = 0) = 0 r = + 0t.

r(t) r m F = -kx k = F = -gradU(x), kxU(x) =.

mT =, m2 kxL = -.

2 + 2x = 0, 2 = k/m F = - e, U() =.

2 = 2 + 22 m L = 2 + 22 -.

d L L d L L - = 0, - = 0, dt dt m - m2 + = 0, m2 + 2m = 0, ma = -, ma = 0, a a q p m L(x, y, z,,, ) = 2 + 2 + 2 - mgz, L L L px = = m, py = = m, pz = = m.

p2 + p2 + px y z H(x, y, z, px, py, pz) = + mgz.

2m x = 0, y = 0, z = -mg, px py pz =, =, =.

m m m px(t) = p0x, py(t) = p0y, pz(t) = p0z - mgt, p0xt p0yt gt2 p0zt x(t) = + x0, y(t) = + y0, z(t) = - + + z0.

m m 2 m p0x p0y p0z x0 y0 z0 m L(,, z,,, ) = 2 + 22 + 2 - mgz, L L L p = = m, p = = m2, pz = = m.

p H(,, z, p, p, pz) = p2 + + p2 + mgz.

2m 2 z m L(,,,,, ) = 2 + 22 + 22 sin2 - mg cos, L L L p = = m, p = = m2, p = = m2 sin2.

p1 p H(,,, p, p, p) = p2 + + + mg cos.

2m 2 sin m l m l x = x(t) m l y = y(t) x = x(t) m l x = a cos t p2 H = + q1 + q2.

2 p H( = - ( a), ( = ) p, r) p, a 2m c| p| H( = p, r) n( r) 2 2xL(x, ) = - - x3 + x2, 2,, M = p r, N N M = pi ri, i= q1 q 2 L = 6q2 + q q2 + 2q1 + 4q L = 2q1 + q q2 + 3q1 + 4q1q L = 2q1 + 2q2 + 4q2 + 2q1q2 2 dE = dt d pi = dt i dM = dt H r = + p p = p, r r, H( p) = H(r, p ).

r, H = 0 r H P = = - = 0, P =.

r r = +,, p p = p +, p, r r r H( p) = H(r, p ), r, H H, +, p = r p r d = -, +, p = - p r r, dt M = p =.

r, P = M = p = r, m z = 0 x = R cos(t) y = R sin(t),, z,, m z = 0 x = (kt) y = (kt) U(r) = r r A =, M + r r e H = q req r A = M- c r e H e A = M, H + H M r, 2c K K x = (x + V t), y = y, z = z, V x t = t +, c = V 1 c K K -V x K K x = (x - V t), y = y, z = z, V x t = t -.

c K dx dy dz,, = u + u + u, u i j k x y z dt dt dt K dx dy dz,, = ux + uy + uz u i j k dt dt dt u + V x ux =, u V x 1 + c u V y uy = 1 -, u V cx 1 + c u V z uz = 1 -.

u V c x 1 + c K x l0 K x l K V l = l0 1 -.

c K 0 K K =.

V 1 c K x K K V x =, c x K mu p =, V 1 c m0 u K m0cE =.

V 1 c u E2 = m2c4 + p2c2.

p = 0 E = m0c2 T T = E - m0c2.

T p2 = T + 2m0.

c K K x u + V x ux =, u V x 1 + c ux = -c K u x K V = c K u x ux - V u =.

x uxV 1 c u = -c x -c u x K x K u K x y ux = u cos, uy = u sin.

x uy u = u2 + u2, =.

x y ux K ux - V u cos - V u = =, x uxV uV cos 1 - 1 c2 c 2 uy V u sin V u = 1 - = 1 -.

y uxV uV cos c2 c 1 - 1 c2 c K x u u sin V y = = 1 -, u u cos - V cx 2 u cos - V + 1 - V /c2 u2 sinu =.

uV cos 1 c (u cos - V )2 + (1 - V /c2)2u2 sin u = 1 - uV cos /c u sin V = 1 - u cos - V c 0 K K K K s = V V =, 1 - V /c 2 V = c 1 - 0 /2 s = c 2 - s = c 2 - p x E V x m0 p l0 x K K V x K M1 M2 T2 M V l0 = 5 V K V K l = 3 0.7c s = 1.35 = 5 0.8 0.9 p0 t = E p0E 1% 10% m0 0.6 0.8 = + r er e a = -k dx eA eA = = 1 - cos(kt) min = 1 - cos 0 = dt mk mk eA 2eA min = 1 - cos = mk mk x 22 x x = y = ax = - ay = y y3 ya = y x y = x - 22 cos2 2 0 H = sin2 L = sin 2 T = 2 sin 2 r = = kt z = t (x - )2 + y2 = s = kt x2 + y2 + z2 = R2 (x R/4)2 + y2 = R2/4 = R = kt/2 = kt/ 2x2 - 2x2 2 - 1 2 2 1 1 a = k = 2 - 2 x2 - x1 2 2 (t) = x0 exp(t) a(t) = 2x0 exp(t) (x0 + s) s =.

ln (x0 + s)/x 2 sin2 0 ymax = xmax = sin 2 sin Hmax = H + = 13.83 xmax = 0t cos = 23.8 0 sin 0 sin 2H t = + + m0 m Fc = - = -1.2 104 l = = 7.5 10-2 t 2Fc 2F s = m m k t = ln + k m kt m0 kt = 0 exp s = exp - m k m 20 sin( - ) t = t - t = (cos + cos ) 1- h 2 - 1 - / t = 2- 0 1 - 1 - / F = k2mr -a Rx = -ax x = 0 exp t P mb = m 2 + 2a0H / 2 + 2 H 0 m k0 + m f t = ln k k1 + m f Q = 1 - cos(kt) sin mk t2 = t1 1 + m2/m eE y = x2m2m T = eE e0H 0/H H = m eE z = 1 - cos x/0 Oz m eE = + 0t + 1 - cos(t) r(t) rm eE (t) = 0 + sin(t) m = cos(t) + sin(t) r(t) r (t) = 0 cos(t) - sin(t) r 0 r x r 2 x2 sin2 - xy sin(2) + y2 cos2 + r02/2 = r0 sin2 0 = k/m r m m m = + 0 1 - e-kt/m + t - 1 - ekt/m r(t) r0 g k k k m (t) = 0e-kt/m + 1 - e-kt/m g k 0 g mg m2g kx z = r0 + x + + ln 1 - k0 cos k2 m0 cos x(0) x(t) = 2A 1 x(0)t m F = mk2x, mk2y, F = 2E/a a = a (t) x = a ln( t) = y = a arcsin( t) t m = m0e- t/u m = m0e-(a+ )t/u f(t) R(x, ) = - - u - f(t) m0f(t) n + 1 n m = m0 exp - t m = m0 exp - t u u T = u ln a + 98 s = u2 ln k /(2a) m0u e - s = e (t) = u ln m0/m - t F m = ln = F t/(m0 + t) m0 - t m (t) = 0 - t + u ln m0 - ct m0 M = u ln - m0 - M k M M2 u mz = u - - m0 - M ln k 2k2 k m0 - M ml L = 2 + m l cos m x = x(t) + l sin y = l cos ml L = 2 + m l cos - ml sin ml L = 2 + m l cos - m(t)l cos - m(t)l sin m = -m r a2 + 0 = cos t 0 = /l = l p2 H = - q1 + q2 m p/m - a L = L = 0 p2 2x2 p H(p, x) = 2 + 2 + x3 = 1 + 2x 2 1 + 2x p = 2 - 2x - 3x1 + 2x = -pz My, pz = -px Mz, px = -py Mx, py 0 = 0 p Mx, px = My, py Mz, z = 0 Mx, p = i, p My, p = j, p Mz, p = k, p Mx, My = -Mz My, Mz = -Mx Mz, Mx = -My 2 H = -6p2 + p1p2 - 2q1 - 4q H = p1p2 - 2p2 - 3q1 - 4q1q H = (1/8)p2 + (1/8)p2 - 4q2 - 2q1q1 Mx = m - z sin - mz cos My = m z - cos - mz sin Mz = m M2 = m222 2 + z2 + m2 - z Mx = -m2 sin + sin cos cos My = m2 cos - sin cos sin Mz = m2 sin2 M2 = m24 2 + sin2 M = {0, 0, mk} mk E = 2 - z pz = x2 y - = 2 EV px 1 - V/c c2 E E - pxV p = = E = E x 2 1 + V/c 1 - V /c2 1 - V /c p2 p M = 2m0 1 + M = 2m0 1 + m2c2 2m2c0 V l = l 1 - cos c 2M1T2 c T2(2M2c2 + T2) M = (M1 + M2) 1 + V = (M1 + M2)2c2 (M1 + M2)c2 + T V = c 1 - (l/l0)2 = 4/5c m p = = 4.91 10-19 1 - 2/c T = mc2 - 1 = 6.02 10-11 1 - 2/c s = 1 - = 2.2 10-6 cu + u = = 2.97 108 1 + u/cmcE = = 1.173 T = E - E0 = 0.662 1 - 2/c = y/x = eEt/p /c = 2 - = 0.14 = 0.01 0.45 = 0. A = 0.42m0c2 A = 0.14m0c adu = au + C, a R, C =, u = u(x).

um+umdu = + C, m = -1.

m + du = 2 u + C, eudu = eu + C.

u du au = ln |u| + C, audu = + C.

u ln a cos udu = sin u + C, sin udu = - cos u + C.

du du = u + C, = - u + C.

cos2 u sin2 u du u du 1 u = arcsin + C, = + C.

a a a2 - u2 a2 + u2 a du u = ln + u2 a2 + C.

u2 a u - a du = ln u + C.

u2 - a2 2a + a cos( ) = cos cos sin sin.

sin( ) = sin cos cos sin.

1 + cos 2 1 - cos cos2 =, sin2 =.

2 1 1 x x = ex - e-x, x = ex + e-x, x =.

2 2 x











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