ẍ = kx, (k > ) (.) x x(t) = A cos(ωt + α) (.). d/ = D. d dt x + k ( x = D + k ) ( ) ( ) k k x = D + i D i x =... ( ) k D + i x = or ( ) k D i x =.. k.

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1 K E N Z OU F = kx x 3 678

2 ẍ = kx, (k > ) (.) x x(t) = A cos(ωt + α) (.). d/ = D. d dt x + k ( x = D + k ) ( ) ( ) k k x = D + i D i x =... ( ) k D + i x = or ( ) k D i x =.. k. D = ±i dt = ±iωx, (ω = k/)... x = A exp(iωt), x = A exp( iωt) A, A.3 e iθ = cosθ + i sin θ x = A exp(iωt) + A exp( iωt) (.3) x = (A + A )cosωt + i(a A )sinωt (.4) x (A + A ), i(a A ) A, A A = Ā, A = Ā ξ, η,.4 A = (ξ + iη), A = (ξ iη) (.5) x = ξ cos ωt η sin ωt = ( ) ξ + η ξ η cos ωt ξ + η ξ + η sin ωt. = A(cos α cos ωt sin α sin ωt) = A cos(ωt + α) ( A = ξ + η, α = tan (η/ξ)) A ω [rad/sec] α [rad] ωt + α [rad] f [Hz] f = ω/π) T [sec] T = /f = π/ω) (.6) θ. θ = ω θ, (ω = g/l, g :, l ) (.7)

3 T A A T (T = π l/g) rad rad [rad] rad. E k E k = ẋ, U = k(x x) (.8) x = U = kx W. ẋ... W = E k + U = ẋ + kx (.9) W.9 ẋẍ kẋ = d [ ẋ + kx ] = dt ẋ + kx = W = const (.) W = ω A sin (ωt + α) + ka cos (ωt + α) = ω A (.) A < E k >, < U > < E k >= T < U >= T T T... < E k >=< U > ẋ dt = T kx dt = T T T ω A sin (ωt + α)dt = 4 ω A = 4 ka ka cos (ωt + α)dt = 4 ω A = 4 ka Galileo Galilei

4 .3 U(x) x = x x U(x) x U(x) = U(x ) + (x x )U (x ) + (x x ) U (x ) + (x x ) 3 U (x ) + (.)! 3! x = x U (x ) = U (x ) > (x x ) 3 U (x) U(x) U(x ) + U (x ) (x x ) (.3) U(x).3 ẍ = d U(x) (.4) ẍ = U (x )(x x ) (.5) ω = U (x )/, x x = x ẍ = ω x (.6) F F = U U = k cosh x Fig. F ig. y y = cosh x = (e x + e x )/ e x = + x + (/)x + e x = x + (/)x + x.4 ẍ = d d U(x) = k cosh x = k d (ex + e x ) = k (ex e x ) (.7) x e x e x x = e x = + x + (/)x +, e x = x + (/)x + U(x ), U (x ) 4

5 ẍ = k ( + x + + x ) = kx (.8) ω = k/ T T = π/ω T W W = ( ) + U(x) dt dt = {W U(x)}/.. x. t = + const (.) W U(x) 3. W U(x) x = A x = A V (x) x W U(x) = (x A )(A x)v (x), (A > A ) (.) x = A x = A T/ T = A A A W U(x) A (x A )(A x)v (x) (.3) x = (A + A ) (A A ) cos φ, ( φ π).3 = (/)(A A ) sin φdφ W U(x) = (A A )V (x) sin φ, (A > A ) T = A A = π dφ (.4) W U(x) V (x) 3 Huygens) 3 5

6 F ig. y ds = + dy = a cos(θ/)dθ a ds θ gy s = θ ds = 4a sin(θ/) = ay y = a ( s ) πa πa x T. ṡ dt = / s ds = θ a cos(θ/)dθ / = θ a cos(θ/)dθ / W U(s) W U(s) ṡ (3.) W U ax x U(s) E k W = U ax = ga U(s) = gy = g a ( s ) E k = ( ) ds = W U(x) = ga g ( s ) dt a ṡ ( ṡ = 4ga g ( 4a s) / = 4ga sin θ ) = ga cos θ 3. θ π T/ π a cos(θ/) a π a T/ = ga cos(θ/) dθ = dθ = π g g.. a. T = 4π g (3.) (3.3) ω ω = π T = g a (3.4) 4 4. θ = g l sin θ = ω sin θ, (ω = g/l) (4.) 6

7 sin θ = θ θ3 3! + θ5 5! (4.) 3 θ = ω θ F ig.3 y E k = (/)(ẋ + ẏ ) U = gl cos θ θ l T ẍ = g T cos θ ÿ = T sin θ = θ = (g/l) sin θ x = l cos θ y = l sin θ ẍ = l( φ sin φ + φ cos φ) ÿ = l( φ cos φ φ sin φ) x g W W = E k + U W = ( ) l θ gl cos θ = l θ ω cos θ (4.3) 4. θ θ θ = ω sin θ θ d dt ( ) θ ω cos θ = (4.4) E E = θ ω cos θ E l t = θ = θ, θ = θ = ω (cos θ cos θ )... θ = ω sin (θ /) sin (θ/) (4.5) cos φ = sin (φ/) θ = θ = θ θ t = T/4 t = T 4 = θ ω sin (θ /) sin (θ/) dθ = ω π/ dφ (4.6) a sin φ sin(θ /) = a, sin(θ/) = a sin φ dθ = (/ a sin φ )dφ φ θ φ θ = φ = θ = θ φ = π/ T T = 4 ω K(a) = π/ π/ a sin φ dφ = 4 K(a) (4.7) ω dφ (4.8) a sin φ 7

8 4.8 coplete elliptic integral 4 a sin φ = + a sin φ a4 sin 4 φ a6 sin 6 φ + = n= (n )!! a n sin n φ (n)!!!! n 6!! = 6 4 = 48 π/ T K(k) = π/ sin n θdθ = π T = 4 ω K(k) = π ω a sin φ dφ = π (n ) (n 3) 3 n (n ) 4 = π (n )!! n!! { + 4 a + 9 a } a6 + [ + 4 a + 9 a ] ( ) θ a6 +, a = sin T θ a a a T T h T h = π/ω, T = (4/ω)K(k) T/T h = (/π)k(k) θ 5 θ T/T h θ T/T h θ T/T h θ T/T h θ T/T h θ 79 θ (4.9) T Th Π T Th Π Φ, a sin Θ a sin Φ Θ 4 Karl Gustav Jakob Jacobi(

9 θ 5 3 x Matheatica For[ =Degree, <=9Degree, = +5Degree,a=Sin[ /]^;Ans=N[(/Pi)*ElliptickK[a]]; Print[{,Ans}]] Plot[EllipticK[Sin[ /]*(/Pi),{,,79Degree},PlotStyle->Thickness[.4], Ticks->{Table[ Degree,{ Degree,Degree,8Degree,Degree}],Autoatic}, AxesLabel->{"θ ",T/Th},PlotLavel->StyleFor["T/Th=(/ )K(a),a=sin(θ /), K(a)= (π/) a dφ, sin φ FontFaily->"Ties",FontSize->4,FontWeight->"Bold"]] 4. F = ẍ = (αx + βx 3 ), α >, β (4.) Duffing equation 5 β β > β < Fig.4 U(x, β) = (/)αx + (/4)βx 4 F = ẍ = d U = αx βx3 (4.) F ig.4 F β > β = F = ẍ = (αx + βx 3 ), (α > ) W = E k + U β < x U = (/)αx + (/4)βx 4 t = ± p / Z x p W U(x) x = A x = A A T β 5 4. sinθ = θ θ 3 /3! + θ 5 /5! θ = ω (θ θ 3 /6) 9

10 F ig.5 U U = (/)αx + (/4)βx 4, (β > ) U ax = (/)αa + (/4)βA 4 (β < ) A v(x) A x x v(x) dt dt = /v x = A x = A t T / t = T/ = A A v(x) (4.) v(x) W x = A U ax (A, β) = W = αa + 4 βa4 (4.3) x E k (x, β) W U(x, β) E k (x, β) = W U(x, β) = αa + 4 βa4 αx + 4 βx4 E k (x, β) = (/)v 4. = α(a x ) + 4 β(a4 x 4 ) (4.4) { = (A x ) α + } 4 β(a + x ) (4.5) T = A T β β > β = T / A = T A v(x) = E k (x, β)/ (4.6) A v(x) = / (4.7) A Ek (x, β) β < β = T / A = T T W = ( ) + U(x) dt dt = ± {W U(x)}/... x dt = ± {W U(x)}/ (4.8) T

11 4.8 dt = ± x {W U(x)} = ± φ dφ V (x)/ (4.9) 3 Hailton x = a, a (a > ) W U(x) = (a x )V (x) (4.) x E U(x) = x = a, (a > ) a W αa 4 βa4 =,... W = αa + 4 βa4 (4.) V (x) = Ax + Bx + C W U(x) = αa + 4 βa4 αx 4 βx4 = (a x )(Ax + Bx + C) (4.) x A, B, C A = 4 β, B =, C = α + 4 βa... V (x) = 4 {α + β(x + a )} (4.3) V (x) W U(x) W U(x) = W αx 4 βx4 = 4 (a α + a 4 β αx βx 4 ) = 4 (a x ){α + β(a + x )} x t = ± = ± x W U(x) (a x ){α + β(a + x )} = ± ϕ a sin ϕdϕ a sin ϕ (α + βa k sin ϕ ϕ = ± α + βa dϕ k sin ϕ

12 4. T π/ dϕ T = 4 α + βa k sin ϕ = 4 K(k) (4.4) α + βa k k = βa ( + a ) β T T = 4 K() = π α α = π, ω ( ω = α/ ) (4.5) α/ t t β/α x x α s = t t y = β/α x x /ds = x, d x/ds = x ( ) x(t) = x /α s, ẋ = ( ) /α x /α s, ẍ(t) = ( ) α x /α s ẍ(t) = α d y ds = d ( dy ds α β (y + y3 ) 4. ) = ds β d x β α ds = β α α ẍ = α 4.6 (/dt) t dt dt = (x + x3 ) dt d dt d α α α β (y + y3 ) = (y + y 3 ) ẍ = (x + x 3 ) (4.6) ( ) = d ( dt dt x + ) 4 x4 ( ) ( = W dt x + ) 4 x4 = W U(x) (4.7) W U(x) = x + 4 x4 (4.8) 4.7 dt = ± (W U(x)) t = ± x (4.9) W U(x) E U(x) x E U(x) = x = a, (a > ) E a 4 a4 = a = + + 4E (4.3)

13 E U(x) = E x 4 x4 = (a x )V (x) (4.3) V (x) x V (x) = Ax + Bx + C x E x 4 x4 = (a x )(Ax + Bx + C) A = 4, B =, C = a x = a cos ϕ = a sin ϕdϕ t = ± x (a x )( + a + x ) = ± + a ϕ dϕ k sin ϕ (4.3) (4.33) k = a ( + a ) k T T = 4t = 4 π/ + a T W U(x) = W αx 4 βx4 = 4 (a α + a 4 β αx βx 4 ) dϕ k sin ϕ = 4 kk(k) (4.34) a = 4 (a x ){α + β(a + x )}... t = ± = ± ϕ x = ± α + βa = ± x W U(x) a sin ϕdϕ a sin ϕ (α + βa k sin ϕ ϕ T T = 4 α + βa k dϕ k sin ϕ π/ k = (a x ){α + β(a + x )} dφ k sin ϕ = 4 K(k) (4.35) α + βa βa ( + a ) ẍ = (αx + βx 3 ) = 6 βx 3 ẍ + ω x + βx 3 =, ω = α (4.36) ω x βx 3 6 α/ α, /β β t = x = a, ẋ = 3

14 3 ẍ + ω x = x = a cos ωt 4.36 ω p x = a cos pt (4.37) ω (ω p ) (4.38) ω 4.36 ω = p + (ω p ), (4.39) ẍ + p x = (ω p )x βx 3 (4.4) ẍ + p x = (ω p )a cos ωt βa 3 cos 3 pt = {a(ω p ) + 34 } βa3 cos pt 4 βa3 cos 3pt (4.4) ẍ + p x = f cos pt (4.4) ẍ + ω = F cos ωt x = a cos(ω t + α) + F ω cos ωt ω ω = ω 4.4 x = a cos(pt + α) + f p cos ωt (4.43) p 4.4 cos pt p p a(ω p ) βa3 = p = ω a β (4.44) 4.4 ẍ + p x = 4 βa3 cos 3pt (4.45) 7 cos3θ = 4cos 3 θ 3 cos θ 4

15 8 A x = Acos3pt (4.46) ( 9p A + p A) cos 3pt = 4 βa3 cos 3pt A = βa3 3p (4.47) ẍ + p x = c c c cos pt + c sin pt x = c cos pt + c sin pt + Acos3pt = c cos pt + c sin pt + βa3 cos 3pt 3p c, c t ẋ = pc sin pt + pc cos pt 3βa3 3p c, c t = x = a a = c + βa3 3p c = a βa3 3p t = ẋ = = pc c = sin 3pt (4.48) x = ) (a βa3 3p cos pt + βa3 cos 3pt (4.49) 3p T = π/p p p = ω + (3/4)a β (4.5) β > β < secular ter β cos 3pt } ω = p + C β, C : (4.5) x = ϕ (t) + βϕ (t) β x = ϕ + βϕ + β ϕ + β 3 ϕ 4 ω = p + C β + C β + x = ϕ + βϕ + β ϕ + } (4.5) 8 5

16 β 4.4 C, C,, p ϕ, ϕ, ϕ x β 3 ( 4 ϕ + β ϕ + β ϕ + β 3 ϕ 3 + p (ϕ + βϕ + β ϕ + β 3 ϕ 3 ) = ϕ + p ϕ + β( ϕ + ϕ ) + β ( ϕ + ϕ ) + β 3 ( ϕ 3 + ϕ 3 ) (4.53) = {βc ϕ + β (C ϕ + C ϕ ) + β 3 (C 3 ϕ + C ϕ + C ϕ )} (4.54) {βϕ 3 + β (3ϕ ϕ ) + β 3 (3ϕ ϕ + 3ϕ ϕ )} (4.55) = {β(c ϕ + ϕ 3 ) + β (C ϕ + C ϕ + 3ϕ ϕ ) (4.56) +β 3 (C 3 ϕ + C ϕ + C ϕ + 3ϕ ϕ + 3ϕ ϕ )} (4.57) β β ϕ + p ϕ = ϕ + p ϕ = C ϕ ϕ 3 t = x = a, ẋ = ϕ + p ϕ = C ϕ C ϕ 3ϕ ϕ ϕ 3 + p ϕ 3 = C 3 ϕ C ϕ C ϕ 3ϕ ϕ 3ϕ ϕ ϕ () + βϕ () + β ϕ () + β ϕ 3 () = a ( ϕ ) t= + β( ϕ ) t= + β ( ϕ ) t= + β 3 ( ϕ 3 ) t= = (4.58) (4.59) β ϕ () = a, ( ϕ ) t= ϕ () =, ( ϕ ) t= ϕ () =, ( ϕ ) t= ϕ 3 () =, ( ϕ 3 ) t= (4.6) 4.58 ϕ + p ϕ = ϕ (t) = a cos pt + const 4.6 const = ϕ (t) = a cos pt (4.6) 4.58 ϕ + p ϕ = C a cos pt cos 3 pt = (C a + 34 a3 ) cos pt 4 a3 cos 3pt cos pt C C a a = C = 3 4 a (4.6) ϕ ϕ + p ϕ = 4 a3 cos 3pt 6

17 ϕ = ϕ 4.5 a3 (cos 3pt cospt) (4.63) 3p x = ϕ + βϕ = a cos pt + βa3 (cos 3pt cospt) 3p (4.64) ) = (a βa3 3p cos pt + βa3 cos 3pt 3p (4.65) p = ω a β (4.66) 3 ϕ = a cos pt, ϕ = a3 3p (cos 3pt cospt), C = 3 4 a (4.67) ) ϕ + p ϕ = C a cos pt ( 3a a 3 4 3p (cos 3pt cospt) 3a cos pt a3 (cos 3pt cospt) 3p = C a cos pt + a5 8p (cos3 pt cos pt) a5 3p cos pt(cos 3 pt cos pt) ) = (C a + a5 8p cos pt + 6a5 8p cos3 pt 48a5 8p cos5 pt ) = (C a + a5 8p cos pt + 3a5 8p ( cos3 pt 6 cos 5 pt) = (C a + a5 8p ) cos pt + 3a5 (5 cos pt cos 5pt) 8p ) = (C a 3a5 8p cos pt 3a5 cos 5pt (4.68) 8p cos pt C = 5 9 cos 5θ = 6 cos 5 θ cos 3 θ + 5 cos θ 4.68 x = A cos 5pt 4.7 A 3a4 8p (4.69) ϕ + p ϕ = 3a5 cos 5pt (4.7) 8p ϕ + p ϕ = 5Ap cos 5pt + Ap cos 5pt = 3a5 cos 5pt 8p... A = a5 4p (4.7) cos 4θ = 4 sin θ cos θ 8 sin 3 θ cos θ, sin 4θ = 4 sin θ cos θ 8 sin 3θ cos θ, sin θ 5 sin 5θ = 6 sin 5 θ sin 3 θ + 5 sin θ 7

18 t = ϕ () =, ϕ (t) = C cospt + C sin pt + a5 cos 5pt (4.7) 4p4 ϕ () = C, C a5 C = 4p 4, C = ϕ (t) = a5 cos pt + a5 cos 5pt (4.73) 4p4 4p4 x = ϕ + βϕ + β ϕ = (a βa3 3p β a 5 ) 4p 4 cos pt + βa3 3p cos 3pt + β a 5 cos 5pt 4p4 p = ω C β C β = ω a β 3a4 8p β 4 p p p p = A ± B p 3 Fig.5 3 F ig.5 x ω =, a =, β =. x = a βa3 x = 3p β a 5 4p 4 a βa3 cos pt + βa3 cos 3pt 3p 3p cos pt + βa3 3p cos 3pt + β a 5 4p 4 cos5pt x = a cos ωt t T ( = ) a =, α =, β =. 4 /(α + βa ) K(k),.88 π/ α, (α ω ) π/ α a β π / α a β + (α + 3 a αβ + 5a4 3 β ) / (!!) 8

19 4.36 L ẍ + ω x + βx 3 = (4.74) L = (ẋ ω x ) β 4 x4 (4.75) L x ( d l ) ẋ = 4.74 β x = A cos ω t β ω I I = t t Ldt δi = δ t t Ldt = (4.76) x x = A cos ωt δi t, t t t A x I = π/ω Ldt = π A ω (ω ω) 3 π 6 ω A4 β (4.77) F ig.6 A x = A cos ωt t t ωt ω = ω + (3/4)A β δi = I A I A = ( π ω ω 3 ) ω 4 A β = ω = ω A β A ω 4.5 9

20 ( ) G OOD L U C K! S E E Y OU A G A I N!

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