θ (t) ω cos θ(t) = ( : θ, θ. ( ) ( ) ( 5) l () θ (t) = ω sin θ(t). ω := g l.. () θ (t) θ (t)θ (t) + ω θ (t) sin θ(t) =. [ ] d dt θ (t) ω cos θ(t

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1 7 8, /3/, 5// l (, simple pendulum) m g mlθ (t) = mg sin θ(t) () θ (t) + ω sin θ(t) =, ω := ( m ) ( θ ) sin θ θ θ (t) + ω θ(t) = ( ) ( ) g l θ(t) = C cos ωt + C sin ωt (C, C ) T = π ω = π l g g l ( ) ( ) ( () ) ( ) ( simple-harmonic-motion/) ( pi.pdf)

2 θ (t) ω cos θ(t) = ( : θ, θ. ( ) ( ) ( 5) l () θ (t) = ω sin θ(t). ω := g l.. () θ (t) θ (t)θ (t) + ω θ (t) sin θ(t) =. [ ] d dt θ (t) ω cos θ(t) =. (3) θ (t) ω cos θ(t) = E (E ). (mle ) 3 9 ( < θ 9 θ θ θ ) 9 (9 < θ 8 ) ( )

3 E (θ ) θ 4 θ = θ θ = ω cos θ = E. E = ω cos θ. (3) (4) θ (t) ω (cos θ(t) cos θ ) =. θ = (θ ) = ω ( cos θ ) = 4ω sin θ θ = ±ω sin θ. θ < θ < π (5) k := sin θ (6) θ() =, θ () = kω θ θ(t) = sin (k sn (ωt, k)) (4) dθ dt = ω cos θ cos θ. (± θ () > t ) (7) kz = sin θ 4 θ 3

4 cos θ cos θ = ( sin θ ) ( sin θ ) ( = sin θ θ ) sin = (k k z ) = k ( z ). k dz = cos θ dθ k dz dθ = = cos θ k dz k z. ( (4) z(t) ) t = t = ω dt = z θ ω ds ( s )( k s ). dθ = z cos θ cos θ ω ωt = z ds ( s )( k s ). k s Jacobi sn (ωt = sn (z, k) ) k ds k s sn(ωt, k) = z = k sin θ (8) θ(t) = sin (k sn(ωt, k)). (), (6) 3 ( ) θ (t) = ω θ(t), θ() =, θ () = kω (9) θ(t) = k sin (ωt). θ =.,., t, θ 4

5 . harmonic theta= : θ =.: ( ).3 harmonic theta= : θ =.3:.8 harmonic theta= : θ = ( 57.3 ) 5

6 4 θ (s 4 ) T = 4 ds ω ( s )( k s ) = 4 l g K(k), k := sin θ. K(k) k : K(k) := π/ dt k sin t. l T h := π g T T h = π K(k) T (θ = ) T h = , T (θ = 45 ) T h = , T (θ = 9 ) T h = θ = 45 4% θ = 9 8% [ K(k) = π ( ) ( ) ( ) ] k + k 4 + k () T T h = π K(k) = + ( ) k + ( ) 3 k ( ) 3 5 k k θ = 45 k = sin π k = ( ) =

7 θ ( ) T/T h < θ < 8 < θ < 9 ( ) A Mathematica A. vs θ θ(t) θ harmonic (t) Runge-Kutta GSL (GNU Scientific Library) Jacobi /* * furiko.c --- * = arcsin (k sn( g/l t,k)) */ #include <stdio.h> #include <math.h> #include <gsl/gsl_specfunc.h> 7

8 4 "toujisei.txt" : : int main() { double g,l,omega,theta,k; int i,n; double t,dt,sn,cn,dn; g = 9.8; l =.; omega = sqrt(g / l); scanf("%lf", &theta); k = sin(theta / ); dt =.; for (i = ; i <= ; i++) { t = i * dt; gsl_sf_elljac_e(omega * t, k*k, &sn, &cn, &dn); printf("%f %f %f\n", t, * asin(k * sn), * k * sin(omega * t)); return ; MacPorts GSL gcc -I/opt/local/include furiko.c -L/opt/local/lib -lgsl Mathematica g=9.8; l=; omega=sqrt[g/l]; theta=; k=sin[theta/]; Plot[{k Sin[omega t], ArcSin[k JacobiSN[omega t,k^]], {t,,] : GSL 8

9 gsl sf elljac e(omega * t, k, &sn, &cn, &dn) ( ) Mathematica JacobiSN[omega t,k] ( ) GSL, Mathematica k m(= k ) ( /3/) A. 79 (AGM) ( GSL GSL () ) /* * toujisei.c --- * T=4 l/g K(k), k=sin / * Th= l/g * */ #include <stdio.h> #include <math.h> double pi; /* * K(k), E(k) */ int kanzen(double k, double *K, double *E) { int i; double a,b,an,bn,cn,anp,bnp,cnp; double power, s, AGM; if (k < k >= ) { fprintf(stderr, "k must be in [,)\n"); *K = *E =.; return ; a =.; b = sqrt(. - k * k); an = a; bn = b; cn = sqrt(an * an - bn * bn); /* s = ^(n-) * cn * cn (n=,,,...) */ power =.5; s = power * cn * cn; for (i = ; i <= ; i++) { anp = (an + bn) / ; bnp = sqrt(an * bn); cnp = (an - bn) / ; an = anp; bn = bnp; cn = cnp; /* printf("%.5f, %.5f, %.5f\n", an, bn, cn); */ power *= ; s += power * cn * cn; 9

10 if (fabs(an-bn) < e-6) break; if (fabs(an-bn)>e-5) { fprintf(stderr,"k=%f: does not converge. i=%d, an=%f, bn=%f, an-bn =%e\n", k, i, an, bn, fabs(an-bn)); return ; AGM = an; *K = pi / / AGM; *E = (a - s) * (*K); return ; int main() { double theta,k,k,e; int deg; pi = 4 * atan(.); for (deg = ; deg < 8; deg++) { theta = deg * pi / 8.; k = sin(theta / ); kanzen(k, &K, &E); printf("%d %.5f\n", deg, * K / pi); return ; Mathematica Plot[ EllipticK[Sin[t Degree/]^] / Pi, {t,,79, PlotRange->{,4] B Runge-Kutta ( 5 ) 6.63% ( ) C Y ( ) K(k) = π/ dt k sin t 5

11 Mathematica k = / K(k) f[t_,k_]:=/sqrt[-k^ Sin[t]^] k=/ g=plot[f[t,k],{t,-pi,pi] π Integrate[f[t,k],{t,,Pi/] EllipticK[/4] Mathematica K(/) N[%,5] Mathematica NIntegrate[] NIntegrate[f[t, k], {t,, Pi/, AccuracyGoal -> 5, WorkingPrecision -> 6] TR[N_] := Block[{i, h, S, h = (Pi/)/N; S = ; h (f[, k]/ + Sum[f[i*h, k], {i,, N - ] + f[pi/, k]/)] N[Table[TR[n], {n,, 8], 6] %-EllipticK[k^] {.4564,.49737*^-7, 4.685*^-,.65*^-,.*^-4,.*^-6,.*^-6

12 C

13 period.c #include <stdio.h> #include <math.h> double k =.5; double sqr(double x) { return x * x; double f(double x) { return / sqrt(- sqr(k * sin(x))); int main(void) { int i, n; double h,s,pi; pi = 4 * atan(.); for (n = ; n <= 8; n++) { h = pi / / n; s = (f() + f(pi/)) / ; for (i = ; i < n; i++) s += f(i*h); s *= h; printf("%d %.5f\n", n, s); $ gcc period.c $./a.out $ ( 7 ) AGM 3

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