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" dt = "r(t "T s dt = "r(t "T s T T s dt T "T s = "r ln(t "T s = "rt + rt 0 T = T s + Ae "rt T(0 = T 0 T(0 = T s + A A = T 0 "T s T(t = T s + (T 0 "T s e "rt

dy dx = f (x, y (Euler dy dx = f (x, y y y(x n+ = y(x n + "y n "y n f (x n, y n "x y n+ y n x n x n+ x x s x e N x 0 = x s x n = x 0 + "x n "x = x e x s N N "

x = kx x(0 = t = x(t = exp(kt x i+ = x i + "t kx i = x i (+ k"t "t = n x e = x( = + k n( lim x = lim + k n*+ e n*+ n ( n n = e k p [(-p,(+p]

RC cross over N n Z p A =N+Z H Z=, N=0, A= C Z=6, N=6, A= D Z=, N=, A= 4 C Z=6, N=8, A=4 T Z=, N=, A=3 35 35 U Z=9, N=43, A=35 38 38 U Z=9, N=46, A=38 Z=3, N=65, A=78

" " 4 He n p+e + EC p n+e + + p+e n e (Z (A " - -4 + 0 0 0 + - 0 EC - 0 4 7N+ 0 n" 4 6 C+ H 4 6C " 4 7N + e 5730 years 9 6C 6.5 ms 0 6C 9.55 s 6C 0.59 m 6C 5730 y.449 s 98.89 3 6C. 4 6C 0-0 5 6C " ""

C 4 C 4 5730 C 4 460 " dn = "N dn N = " lnn = "t + t 0 N(t = N 0 e "t (N(0 = N 0

dn dn = " N = N " N N = N 0 e " t dn + N = N 0 e " t dn + " N = 0 N = Ae " t " " N = Be " t " Be " t + Be " t ( " B = N 0 B = " N = " " N 0 = N 0 e " t " N 0 e " t + Ae " t

N (0 = 0 0 = " N = " N 0 + A, A = N 0 ( e " t " e " t " " N 0 " = " N = Ae " t N = Cte " t dn = Ce " t " Cte " t Ce " t " Cte " t + Cte " t = N 0 e " t C = N 0 N = " N 0 te " t + Ae " t = (" N 0 t + Ae " t

T / N(t = N 0 e "t N(T / = N 0 = N 0 e"t / e "T / = T / = ln positron + ZN A 76 Kr " 36 4.8 hr E.C. 76 Br " 35 6.hr + 76 34 Se

N Z A 0.9hr 8 Mg " 8 3 Al ".4 min 8 4 Si " " Z=86 6 Rn 85 At 84 07 Po Po 83 07 Bi 8 07 Pb 5.8 hr 3.6 hr A=07 08 09 0 0.56 s 5 hr 74 7. hr Rn At Po Bi Pb

Z 9 U U 9 Pa 90 Th Th 89 88 Ra 87 86 Rn 85 At 84 Po Po Po 83 Bi Bi 8 Pb Pb Pb 8 Tl Tl 06 0 4 8 6 30 34 38 A RC R V V RI C Q I = dq V " RI = Q C Q RI = V " Q C R dq = V " Q C I = dq

dn = "N N(t + "t = N(t + dn(t "t + d N(t ("t + d N(t = " dn(t " N(t + t = N(t N(tt + N(t(t + N E N(t N(tt ( N(t + t = N(t N(tt + t N(t N E = N(t [ N(t + N E ]t (Heun dn = "N N(t + "t = N(t + dn(t "t + d N(t ("t + d N(t = " dn(t " N(t + t = N(t N(tt + N(t(t + N E N(t N(tt ( N(t + t = N(t N(tt + t N(t N E = N(t [ N(t + N E ]t (Heun

Kinematics Kinetics y(t v(t a(t v(t = dy a(t = dv F(y, v,t a(t = m F(y, v,t m d y(t = F(y,v,t m

g = F " 9.8 N kg m y(0 = y 0, v(0 = v 0 v(t = v 0 " gt y(t = y 0 + v 0 t " gt F = GMm (R + y = GMm " R + y = mg " ( y R + R R g " GM R G y v mg F d F d F = "mg + F d F d (v

y = y(t v = dy a = dv F m = a(t v = v(t F = F(t = F(v = "mg + F d F d (v y(t y(t = y meas (t v " v " v t F d (v = k v F d (v = k v F d " k,k F d = mg a = 0 k v t = mg " v t = mg k k v t = mg " v t = mg k a = 0 " v F d, = k v t v t = mg " v v t " v F d, = k v t v t " = mg v v t

v(t = dy(t a(t = dv(t Euler Euler-Cromer v n+ = v n + a n "t y n+ = y n + v n "t n n+ v n+ = v n + a n "t y n+ = y n + v n+ "t n+ a n a n+ a n a n+ v n v n+ v n v n+ y n y n+ y n y n+ Euler-Richardson Euler t mid = t + "t y mid, v mid dv = a = m F(y,v,t dy = v a n = m F(y n, v n,t v mid = v n + a n"t y mid = y n + v n"t a mid = m F(y mid, v mid,t n + "t "t v n+ = v n + a mid "t y n+ = y n + v mid "t y(t + "t y(t = " <

Euler-Richardson y (t + "t y(t "t ("t y(t + "t y (t + "t y(t "t / y ER (t + "t ("t y(t + "t y(t + "t Euler-Richardson y(t + "t Taylor y(t + "t = y(t+ "t dy + d y t ("t + (("t 3 v(t = dy t, a(t = d y t t y(t + "t = y(t+ v(t"t + a(t("t + (("t 3 y (t + "t = y(t+ v(t"t + a(t("t a(t = a( y(t,v(t,t y (t + "t = y(t + + v(t + "t"t + a(t + "t("t y(t + "t "t = y(t+ v(t + a(t("t y (t + "t = y(t+ v(t+ v(t + "t ( "t + a(t+ a(t + "t ( ("t a(t + "t = a(t+ da "t + a(t t y (t + "t = y(t+ "t v(t+ v(t + ( + "t, (a(t +. * - = y(t+ v(t+ v(t + "t ( "t + 4 ("t y (t + "t y (t + "t y(t + "t ("t

y (t + "t y (t + "t y(t + "t ("t y ER (t + "t = (y (t + "t+ (y (t + "t = y (t + "t y (t + "t = y(t+ v(t + "t"t v ER (t + "t = v(t+ a(t + "t"t precise exact accurate double precision double accuracy precise accurate

precise

dy = f (y(t Euler y(t + "t = y(t+ dy "t + d y ("t + = y(t+ dy "t + (("t [("p,(+p] dx = kx k= t " [0,], t = n [(,(+] x i+ = x i + x i t + "t + " +,+ n n ( [ ",+ ] " x i+ = x i + n [(,(+] ( *[(,(+] + = + ( *(, + + (. - *(+ 0 x, n n i / + + n + ( n *,+ n + + (. - n * 0 x, i + ( /

+ n " ( n >>, n << n < x < + n + ( e(" n < x < e(+ n n n dx = f (x,t, x(t s = x 0, t " [t s,t e ] Euler f (x,t for t " [t s,t e ], x x s b f (x,t " M t e " t s < b M x, x, t L f (x,t " f (x,t L x " x

Euler x i (n " x(t i (n Ch x i (n x(t i (n Euler [t s,t e ] nh = t e " t s (n n Euler (n x (n i+ = x (n i + hf (x (n i,t i x(t i+ ( n t i+ (n = x(t (n (n i + " f (x(t,t = x(t (n i + hf (x(t (n i,t i + (h t i ( n (n x (n i+ x(t i+ = x (n (n i x(t (n i + h[ f (x i,t (n (n i f (x(t (n i,t i ] (h x i+ (n (n " x(t (n i+ x (n i " x(t (n i + h[ f (x (n i,t i " f (x(t (n i,t (n i ] + Ah e i (n " x i (n x(t i (n (n e i+ e n (n " (+ hle i (n + Ah n " (+ hl n (n e 0 + Ah (+ hl k = Ah (+ hl k n k= 0 k= 0 = Ah (+ hl n + hl = Ah L (+ hln { } " Ah [ ] L e(t e t s L

(a-(b y(t + "t = y(t+ dy "t + d y ("t + d 3 y 6 3 ("t3 + = y(t+ f (y(t"t + y(t = y(t + "t f (y(t + "t"t + d y = d y + d3 y "t dy dy dy 3 t+"t t t d y t("t + d 3 y ("t 3 + 6 3 t d y t+"t("t d 3 y ("t 3 + 6 3 t+"t (a (b ( y(t + "t y(t = y(t y(t + "t+{ f (y(t+ f (y(t + "t}"t + ("t 3 y(t+ f (y(t"t = y E y(t + "t = y(t+ { f (y+ f (y E }"t + (("t 3 dy = f (y, f (y = "y y(t + t = y(t " ( y(t+ y Et y E y(t " y(tt

Euler d y = a dy = v, dv = a v n+ = v n + a n "t y n+ = y n + v n "t Taylor ( ( ( y(t + "t = y(t+ y(t"t + ("t = y(t+ v(t"t + ("t y n+ = y n + v n "t + ("t Euler Euler-Cromer v n+ = v n + a n "t y n+ = y n + v n+ "t = y n + v n "t + a n ("t Taylor y(t + "t = y(t+ y(t"t + y(t("t + ("t 3 ( = y(t+ v(t"t + a(t("t + (("t 3 = y(t+ v(t"t + a(t("t a(t("t + (("t 3 Euler-Cromer

Euler-Richardson v n+ = v n + a "t n+ y n+ = y n + v "t n+ v = v n+ n + a n "t " y n+ = y n + v n t + a n (t Taylor y(t + "t = y(t+ y(t"t + y(t("t + (("t 3 = y(t+ v(t"t + a(t("t + (("t 3 Euler-Richardson Kepler T /a 3 T a

F M + F m = 0 M m m r = F m = " GmM r " R ( r " R = " GmM 3 3 M R = F M = GmM r " R ( r " R = GmM 3 3 R r r " R = r " R, F m m " F M M F m = " F M = " GmM 3 " = G mm " 3 m + mm M µ * mm m+ M + µ " = GmM " " 3 " = GmM ( " 3 m + " = GmM " M( " 3 µ

m=5.99 x 0 4 kg M=.99 x 0 30 kg M>>m µ " m = 5.99 0 4 kg G = 6.67 "0 m 3 kg s d x = " GM x r 3 d y = " GM r 3 dx dy y " d v = GM r 3 d dr r = r d x = d y = d " d " dx dy d ( x + y = GM r 3 = ( GM r 3 = ( GM r 3 d r = GM r dr r = d ( * r x dx = ( GM d r 3 x y dy = ( GM d r 3 y dr

d v = " GM dr r = GM d ( r " d v GM r = 0 ( E * mv GMm r = const E = mv " GMm r mv > GMm r v " large r " large mv < GMm r v " large r " small v " small r " larg e E v " GMm r

L = r " p, L = (0,0,L z p = m v L z = ( r " m v z = m(xv y yv x m d r = " GMm r r 3 d x d y = " GM r 3 = " GM r 3 x y d L z = m d (xv y " yv x = m d (x y " yx = m( x y + x y " y x " y x = m(x y " y x = m " GM r 3 xy + GM r 3 yx( = 0 Euler r = (x, y, v = (v x, v y, a = (a x, a y r L z = m(xv y " yv x, F = "F(r r v n+ x = v n x + a n x "t v y n+ = v y n + a y n "t x n+ = x n + v x n "t y n+ = y n + v y n "t v x n+ = v x n + a x n "t v n+ y = v n y + a n y "t x n+ v n+ y = (x n + v n x "t(v n y + a n y "t y n+ v n+ x = (y n + v n y "t(v n x + a n x "t L n+ z = x n v n y y n v n x + v n x v n y "t v n y v n x "t + x n a y n "t y n a x n "t + v x n a y n ("t v y n a x n ("t = L n z F(r mr L n z("t = F(r mr ("t n L z ( x n+ v y n+ = (x n + v x n "t(v y n + a y n "t y n+ v x n+ = (y n + v y n "t(v x n + a x n "t x n a y n "t y n a x n "t = F(r mr (xn y n y n x n = 0

x = r cos"t y = rsin"t x = "r cost x + y = r y = "r sint a = x + y = r x = "rsint y = r cost v = x + y = r a = v r " m d r = m a = m v r m v r = GMm, v = GM r r r r = GMm r ( r r b P F F a : e : ae a F P + F P = a e = " b a

T = "r v T = 4" r T r 3 v = 4" r r GM = 4" GM r 3 AU AU=.496 x 0 m year=3.5 x 0 7 s T= year a= AU GM r 3 GM = 4" T = 4" = 4" (AU 3 / year G, MMKS GM = 6.67 "0 ".99 "0 30 =.33"0 0 (m 3 /s GM = 4" r 3 T = 4".496 3 3.5 = 39.48 3.348 033 /(9.930 4 =.33 0 0 (m 3 /s

T log T a AU log a 0.4-0.68 0.387-0.4 0.65-0. 0.73-0.4.0 0.0 0.88 0.74.53 0.83.86.074 5.0 0.76 9.5.470 9.539 0.980 84.0.94 9.8.83 65.7 30.06.478 48.394 39.44.596 T = Ar 3 " logt = 3 log r + log A 930006 Titius-Bode Law Johann Titius Johann Bode r i = 0.4 + 0.5 " i (AU (i =,,,3,... i r i a 0.4 0.387 0.7 0.73 3.6.54 " 4.8 (.77 5 5. 5.03 6 0.0 9.535 7 9.6 9.8 8 [38.8] <30.06> 9 [77.] 39.44

dx d d x +" x dx = 0 d x +" x = 0 x(t = Acos("t + d dx ( + d " x = 0 mv +, * m" x - +. = 0 v(t = "Asin("t + E = mv + m" x = const = " A Euler v n+ = v n " x n t x n+ = x n + v n t E n = mv n + m" x n = (v n +" x n E n+ = m ( v n " x n t + ( m" x n + v n t = v n m" x n v n t + m" 4 x n (t + m" x n + m" x n v n t + m" v n (t E = E n+ E n = m" (" x n + v n (t m" 4 A (t E n+ E n ( +" (t Euler "(t

Euler-Cromer v n+ = v n " x n t x n+ = x n + v n+ t = x n (" t+ v n t E n = mv n + m" x n = (v n +" x n ( + [ m" x n ( " (t + v n t] E n+ = m v n " x n t = v n m" x n v n t + m" 4 x n (t + m" x n + m" x n " 4 (t 4 + m" v n (t m" x n " (t m" x n " (t v n t + m" x n v n t E = E n+ E n = m" (t { v n " x n " x n v n t + x n " 4 (t } v n " Asin(t + x n " Acos(t + v n " A sin (t + x n " A cos (t + v n x n " A cos (t + "E m ("t x n v n " A sin(t +cos(t + = A sin(t + x n " A cos (t + = A { A cos(t + [ + cos(t + ] [ ] ( + 4 A "tsin(t + + A 4 ("t + cos(t + "E m6 A ("t 4 4 ( " (t 3 Euler-Cromer

Euler-Richardson dv = " x dx = v t v mid = v n " x n t x mid = x n " v n v n+ = v n " x mid t = v n " t x n + v n t ( * x x n+ = x n + v mid t = x n + t v n " n t ( * v n+ = v n " x n t " v n (t x n+ = x n + v n t " x n (t "E = E n+ E n E n = mv n + m x n E n+ = mv n+ + m x n+ = mv n + m 4 x n ("t + 8 m 4 v n ("t 4 m v n x n "t m v n ("t + m 4 v n x n ("t 3 + m x n + m v n "t + 8 m6 x n ("t 4 + m x n v n "t m 4 x n ("t m 4 x n v n ("t 3 " E n+ E n = 8 m 4 (t 4 (v n + x n ( " (t 3 Euler-Richardson

(a,0 (0,v b a x a + y b = x x a + y y b = 0 x + x x y + + y y = 0 a b y = v, x = 0, x = a, y = 0 a x a + v b = 0 " x = a b v m d x = " GMm r r = a " a b v = " GM a v = b GM a a v = b a GM a

Configuration Space, Velocity Space, Phase Space d x = a = F m d v (v x,v y ( x, v y m r m r r M d r m = " GMm 3 r d r m = " GMm 3 r r + Gm m r 3 r " Gm m r 3 r r x

d r d r = " GM r 3 = " GM r 3 r + Gm r 3 r " Gm r 3 r r m M " 306, m M " 9.04 Kirkwood gaps http://en.wikipedia.org/wiki/kirkwood_gap

d r ( m = " GMm 3 r ( d r m = " GMm 3 r r + Gm m r 3 r " Gm m r 3 r r d r = " d r = " r r GM 3 r r 3 GM " r r GM m 3 M ( Kirkwood gaps J: Jupter A: Asteroid Kepler s Law: T /a 3 =const T A /T J /3 /5 3/7 / /3 a A /a J 0.48 0.543 0.568 0.63 0.763 a J...... 5.03 (AU...... a A.50.8.95 3.8 3.97

d r = F = a = " m e r 3 d r = m e F = a = " r 3 r + r " r " r r 3 r " r 3 r f (t t + "t f (t "t / t + "t / f (t + "t / f (t + "t "E " U =0 5 " L =0 7 t + "t f (t + "t f (t + "t "E f (t + "t "t "E > U : "t "t / "E < L : "t "t L < "E < U : "t "t

d r m = " GMm 3 r d r m = " GMm 3 r r r d r " r, d r r + Gm m r 3 r r " Gm m r 3 r d ( m d r " + * - =. GMm 3, r d ( m d r " + * - =. GMm 3, r d (r d (r + Gm m r 3. Gm m r 3 d r ( r. r d r ( r. r " GMm r 3 d (r = " GMm dr r = GMm d ( r d ( m " * dr + m " dr " Gm m r 3 +, = d - GM " m + m. Gm m 3 r r r d ( r " r = Gm m d r " r d ( r. r d m " dr * + + m " dr " (GM m + m ( Gm, m r r r ( r - = 0. E = m v + m v "GM m + m r r (" Gm m r " r

m = m = GMm = GMm = Gm m = " E = v + v " + r r ( + r " r

Action Action Integral Action Variable " L " L = 0 p j p j p j q j q j J j = p j dq j or p j dq j " t = t 0, t r = r (t, t 0 " t " t I = ( t t 0 m r = r 0, " d r r = r (t+" r (t (" r (t 0 = " r (t = 0 r "I

t m d "I = t 0 ( r +" r t * ( t 0 t d r + d t0 m " r = m" r d r t d = * m d r " r t 0 ( m t0 t * d r d m d " r = 0 ( r = r 0 + v (t t 0 ( d m d t r " r t 0 ( t d "I = m d r ( *" r t 0 d m d " r = m d r = f = (U t " I = U r t t = U = U t 0 I J J " t 0 m d t r ( U( * - t t 0 + r. = 0 (T U = L t 0, t / 0 0t 0 t 0

f U f = "U m d r = f = "U r = r (t, t 0 " t " t J = t t " L = " (T U = t 0 t 0 t = t 0, t L = L( r, r = T "U " t t 0 * +, m d r ( U( r -. / r = r 0, r "J = " m d t r ( "U( * - 0 + r. = t 0, / 0 t t 0 d m d * r ( + U( - + r., / " r m d r + "U( r = 0 t "J = " L = 0 : t 0

r N i r i = r i (q,q,"q 3N" M,t L = L( q, q L = T( q, q "U( q d L q " L j ( q = 0 q =" q = q 3 = r = a = const ( r L = ma ( " + sin " + mgacos"

"L = "L " q " = ma "L = "L "q " = ma sin cos mgasin " d (ma = mgasin + ma sin cos "L = "L " q " = ma sin " d (ma sin = 0 ma sin " = const = M z m a " + M z ma " = mgasin" + M z cos" ma sin 3 " mgacos" = const = E ma sin " Cyclic Variable q c L(q, q q c "L = 0 d "L = 0 "q c " q c ( * "L = const " q c d "L ( "L = 0 " "q i q i

q i I 0 i= I 0 i= *, + q i I i= i d "L "L - ( q i / = 0 " q i "q i. * d "L "L "L -, q i ( q i q i / = 0 + " q i " q i "q i. "L "L I d q q i + q i ( = i "L + dq i "L ( = dl " q i "q i i= " q i "q i " d ( I i= L q i L+ = 0 q i * I "L q i L = const " q i i= q Euler q I I "L "T q i = q i = T " " i= q i i= " T (T U = T +U = const q i

t " L( r, r,t = 0 t 0 r = d r r r r r = d r r t " L( r, r,t ( r d r ( + * = 0 t 0 r, t t 0 "L " r r r * ( + = 0 " = L r p * t t 0 L( r, r,t " p r + p d r ( = 0 H( r, p,t " p r L( r, r,t

t " p d r H( r, p *,t( = 0 t 0 p t " p d r { H} = 0 t 0 t " p d r H( r, p,t( t * = d r +H t 0 + p * ( " p = 0 t 0 d r = "H " p r t " p d r H( r, p,t( t * = p d t 0 " r +H + r d * r ( = 0 t 0 t p d " r = p " t r d t p t t0 0 " d t p r = t 0 " r t 0 " r " (t 0 = " r " (t = 0 t " p d r H( r, p,t( d * t p = t 0 + +H + r * ( " r = 0 t 0 d p = "H r Canonical Equations of Motion

Euler Method d x dx = " 0 dx = v dv = " 0x x v n+ = v n " 0 x n t x n+ = x n + v n t Euler-Cromer Method v n+ = v n " 0 x n t x n+ = x n + v n+ t Euler-Richardson Method v mid = v n " t 0 x n t x mid = x n + v n a mid = F(x mid, v mid,t n + t = x mi = " t ( 0 x n + v n * v n+ = v n + a mid "t = v n 0 x n + "t v ( n *"t "t x n+ = x n + v mid "t = x n + v n 0 x ( n* "t

m d x = "kx m =, v = p H = p + " 0x d x = " 0x " 0 k m H = p m + kx = p m + m" 0x dx = "H "p dp = "H "x " x V(x H = p +V(x V(x = " cos x dx = p dp = " sinx d x = " sinx

x t = x 0 cos"t + " p 0 sin"t p t = "x 0 sin"t + p 0 cos"t "x t "x 0 = cost, "x t "p 0 = sint "p t = sint, "p t = cost "x 0 "p 0 "x t "p t = (x t, p t (x 0, p 0 "x 0 "p 0 J (x t, p t (x 0, p 0 = x t x t x 0 p 0 p t p t x 0 p 0 J = H = p + " x dx = H p, dp = H x x(t = x(0cos"t + " p(0sin"t p(t = "x(0sin"t + p(0cos"t

( H(t = p(t + " x(t = { " x(0 sin "t "x(0p(0sin"tcos"t + p(0cos "t} + { " x(0 cos "t + "x(0p(0cos"tsin"t + p(0sin "t} = p(0 + " x(0 = H(0 h = H = p + " x p h + x h " ( = 0 Euler Euler-Cromer Euler-Richardson

x n+ = x n + p n "t p n+ = p n x n+ "t x n+ = x n + p n "t p n+ = p n H ( * x x=x n+ "t x n+ = x n + p n "t p n+ = p n sinx n+ "t x n+ = x n + "t H, p n+ = p n * "t H p ( x ( x n,p n x n,p n x n+ = x n + "t H, p n+ = p n * "t H p ( x ( x n,p n x n+ = x n + "t H, p n+ = p n * "t H p ( x ( x n+,p n x n+,p n x n+,p n

H = p + " x = " p " +"x ( " d x = ( x* " x x " p y " H = (x + y dx = "H "y dx = y dy = "H "x dy = x " d x = ( x* (x, y = (x 0, y 0 x(t = x 0 cos"t + y 0 sin"t y(t = x 0 sin"t + y 0 cos"t " x(t " cos(t sin(t " = x(0 y(t sin(t cos(t y(0

Euler x n = x n" + y n" t " x n y n y n = y n" " x n" t " (t" = x n* *(t y n* " x n y n " = A cos( sin( sin( " x n cos( t sin" = + t cos" = + t y n y n y n" Acos" = Asin" = t " " n x n x n" A = + ("t sec" = + t t " = sin ( + + t * " " x n y n x n y n " " (t" = x n* = +( t *(t y n* * " cos+ sin+ " = sec+ x n* *sin+ cos+ " = sec n ( cos( sin( y n* sin( " cos( n x 0 y 0 (t +( t +( t " (t +( t +( t " = sec n ( cosn( sinn( " n n"t "t n = nsin ( + +" t * t " n = nsin ( + nt = + t * 3 n 3 t 3 +O( 5 t 5 H n = H(n"t = (+ "t H 0 sinn( " x 0 cosn( y 0 x n* y n*

x n = x n" + y n" t y n = y n" " x n t = y n" "t(x n" + y n" t = "x n" t + y n" (" t " x n *t ( = +*t + *t ( x n+ ( y n t* " sin ( + " x n y n, y n+ " " = sec ( cos n ( sinn( " " sinn( cos n + ( x 0 y 0 x n = x n" + y n" t y n = "x n" t + y n" (" t sin " ( *+t x n = x n" + sin y ( n" y n = "sin x n" + " 4sin y n" ( (( " 4sin * = 4 " sin - +. (, (/ " 3 = 4 cos " 3 ( * = cos 4 cos 3 ( " 3cos - +., (/ = ( cos cos 3 ( (

cos " x n = cos " x n + sin " cos " cos " y n cos " x n = cos " x n + sin" cos " y n y n = sin " cos " cos " cos 3" x n + cos " y n = sin" cos 3" cos " x n + cos " " x n y n " = sec ( cos ( sin( " sin( cos 3( x n y n " x n y n " " = sec ( cos n ( sinn( " " sinn( cos n + ( x 0 y 0 n= " x y " = sec ( cos ( sin( " sin( cos 3 ( x 0 y 0

" n n+ x n+ y n+ " = sec ( cos ( sin( " x n sin( cos 3( y n " = sec ( cos ( " " sin( cos n ( sinn( " sin( cos 3( " sinn( cos n + ( " " cos n = sec ( ( cos ( sinn(sin( sinn( cos ( + cos " n + (sin( " sinn( cos 3( cos " n (sin( cos " n + ( cos 3( sinn(sin( ( " A = sec A " x 0 A A y 0 x 0 y 0 x 0 y 0 A = cos n " ( cos " sinnsin = [ cosn + cos(n " + cos(n + " cos(n " ] [ ] = cosn + cos(n + = * cos n + ( cos " sin n + ( sin + cos n + ( cos + sin n + ( sin -, / +. = cos n + ( cos A = "sinn cos + cos n + sin ( = * sin(n + " sin(n " + sin(n + 3 " sin(n " +, -. / = * sin(n + + sin(n + 3 +, -. / = * sin n + ( cos +, + cos ( n + sin + sin n + ( cos " cos n + ( sin -. / = sin( n + cos

A = "cos n " (sin " sinn cos 3 = " * sin(n + " sin(n " 3 + sin(n + 3 + sin(n " 3 +, -. / = " * sin(n + + sin(n + 3 +, -. / = " * sin n + ( cos +, = "sin( n + cos + cos ( n + sin + sin n + ( cos " cos n + ( sin -. / A = "sinnsin + cos n + cos 3 ( = [ cos(n + " cos(n " + cos(n + + cos(n " ] [ ] = cos(n + + cos(n + = * cos n + 3 ( cos " sin n + 3 ( sin + cos n + 3 ( cos + sin n + 3 sin -, / + (. = cos n + 3 cos ( + " x n+ ( = sec cos (n + *., - / 0 sin(n + ( y n+ + *sin(n + cos (n ++ (., - / 0 ( x 0 y 0 ( n+ n=n

" n = n nt t * = nsin, nt ( + - = n t + 3 t * 0., + nt / 6 ( + = 4 n 3 t 3 +O( 5 t 5 " n = n 3 t 3 4 +O(5 t 5 Euler (/8 (x 0, y 0 = (r,0 H n = + "t,"t 4 " t sin ( n / 5 4 - + 07 H 3. * 0 6

Euler-Cromer x " y y n = y n" " x n" t x n = x n" + y n" t = x n" (" t + y n" t x n = x n" " y n" t y n = x n" t + y n" (" t "t "t = y n" "tx n = y n" "t(x n" + y n" t y n = "x n" t + y n" (" t x n = x n" + y n" t