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1 * *1

2

3 iii 1. Newton Newton Lagrange Lagrange Hamilton Noether

4 iv Kepler Euler Legendre Poisson Poisson

5 1 1. Newton 1.1 Newton Newton 10 7 m t P 3 (x(t, y(t, z(t r(t (x(t, y(t, z(t (1.1 O P P r(t u(t dr(t dt lim t 0 r(t + t r(t t (1.2 u(t ( v x (t, v y (t, v z (t ( dx(t, dy(t, dz(t dt dt dt (1.3 u P P P a(t d2 r(t dt 2 du(t dt lim t 0 u(t + t u(t t (1.4 a(t ( a x (t, a y (t, a z (t ( d 2 x(t dt 2 (, d2 y(t, d2 z(t dvx (t dt 2 dt 2 dt, dv y(t, dv z(t dt dt (1.5

6 2 1. Newton Newton (1 (2 m u F m du dt F (1.6 (3 A B F BA B A F AB F BA F AB (1.7 A B A B Newton Newton Mach A B m A m B a A, a B m A a A F AB, m B a B F BA (1.8 F BA F AB m A a A m B a B (1.9

7 1.1 Newton 3 Einstein (1 (2 Newton Yes Newton Maxwell (1 No. c t A r A V B r r Vt u u V Newton v/c 0 v *1 (86400s s *1 L. D. E. M.

8 4 1. Newton c A B B A B A A 1s 299,792, m p mu x p x p > h (1.10 h kg m 2 s 1 Planck 1.2 N i r (i ( x (i, y (i, z (i 3N x (x 1, x 2,, x 3N ( r (1, r (2,, r (N (1.11 i u (i (dx (i /dt, dy (i /dt, z (i /dt u (v 1, v 2,, v 3N ( u (1, u (2,, u (N, (1.12 i F (i ( F (i x, F y (i, F z (i F (F 1, F 2,, F 3N ( F (1, F (2,, F (N (1.13 t t 0 (x(t 0, u(t 0 Newton (x, u F u du/dt

9 1.3 5 t du/dt t (x, u m µ dv µ dt F µ (x, u, t (µ 1, 2,, 3N (1.14 i m (i (m 1, m 2,, m 3N ( m (1, m (1, m (1, m (2, m (2, m (2,, m (N, m (N, m (N (1.15 Newton t t 0 t τ t 0 τ N t (τ t 0 /N t n t 0 + n t, (i 0, 1,, N t n t n+1 t x 1 (t n+1 x 1 (t n v 1 (t n x 3N (t n+1 x 3N (t n v + 3N (t n v 1 (t n+1 v 1 (t n F 1 (x(t n, u(t n, t n /m 1 v 3N (t n+1 v 3N (t n F 3N (x(t n, u(t n, t n /m 3N t (1.16 t 2 t 0 t t 1 t t 2, t t N N t t N N t 2 t N Euler 1.3 MKS l [m] l 0.05

10 6 1. Newton l cm 5cm m 0.05m t l t + l A [A] x [x] [L] t [t] [T] [L] [T] [M] [LT 1 ] [LT 2 ] [MLT 2 ] Taylor e x 1 + x + x 2 /2 + x *2 t exp( t/τ τ τ A(t A 0 exp( t/τ A(t t τ 1/e A(t τ x mẍ mg γẋ (1.17 g γ [MT 1 ] m d 2 x dt 2 γ m dx dt g (1.18 x(t 0 0, v(t 0 0 γ m γ/m m g γ *2 x n

11 1.3 7 g γ/m τ m γ, l gτ2 g ( 2 m (1.19 γ τ l x x/l t t/τ d d(t/τ d/dt d(t/τ/dt τ d dt (1.20 d 2 x d t 2 x d d t 1 (1.21 m g γ s m τ l mg γv 0 γ m v g v mg γ l τ (1.22 τ l

12 8 1. Newton Hooke mẍ kx 2n+1 (n 0, 1, 2 (1.23 k/m [L 2n T 2 ] x(t 0 x 0 v(t 0 0 x(t x 0 x 0 k/m T x0 2n m/k (1.24 T n Hooke n 0

13 Newton 1 2 m (1, m (2 r (1, r (2 2 1 F (12 Gm (1 m (2 r (1 r (2 r (1 r (2 (2.1 3 Newton G kg 1 m 3 s 2 Newton *1 Maxwell E(r, t B(r, t *2 Lorentz F(r, u, t q [E(r, t + u B(r, t] (2.2 q, r, u Maxwell 1 2 q (1, q (2 *1 Einstein *2

14 10 2. r (1, r (2 i Maxwell E(r, t q(i r r (i 4πϵ 0 r r (i 3 (2.3 ϵ F (12 q(1 q (2 r (1 r (2 4πϵ 0 r (1 r (2 3 (2.4 Coulomb *3 Coulomb Coulomb Coulomb 2.2 Coulomb N 1,2,, N i i i F (i j F F (i j (2.5 j( i R m *3 u Coulomb (v/c 2 Newton

15 O ρ(r P O O P x OP r a a + a a x x 0 x x 0 + x 0 2π a 2 x0 2 x 0 a/ a 2 x0 2 a ρ(a 2πa x 0 a x x A AP AP (r x (a 2 x0 2 r 2 + a 2 2rx 0 cos OPA (r x 0 /AP x f x Gm ρ(a2πa x 0 a r 2 + a 2 2rx 0 r x 0 Gm ρ(a2πa(r x 0 x 0 a ( r2 + a 2 2rx 0 a2 + r 2 2ax 3/2 (2.6 0 x F x Gm Gm Gm Gm r 2 R 0 R 0 R 0 R 0 daρ(a2πa daρ(a2πa a a a a dx 0 r x 0 ( a2 + r 2 2ax 0 3/2 dx 0 2r (a 2 + r 2 2rx 0 + (r 2 a 2 ( a2 + r 2 2ax 0 3/2 da ρ(a2πa 2r a2 + r 2 2rx 0 r 2 a 2 + r r a 2 + r 2 2rx 0 daρ(aπa { a r + (a + r + r2 a 2 r2 a 2 } a r a + r x 0 a x 0 a (2.7 r R F x Gm r 2 Gm r 2 R 0 R 0 ρ(aπa {a r + a + r + (r + a (r a} da ρ(a4πa 2 da GmM r 2 (2.8 M *4 r < R *4 a a + a 4πa 2 a

16 12 2. F x Gm r 2r 2 Gm 2r 2 Gm r 2 0 R r r 0 2πρ(aa {a r + a + r + (r + a (r a} da 2πρ(aa {r a + a + r (r + a (r a} da ρ(a4πa 2 da (2.9 r r *5 m mg m GM R 2 (2.10 R M g GM/R 2 9.8m s F ex ϵ ϵ 0 a > 0 F ex aϵ + o(ϵ (2.11 *5

17 ϵ x x *6 L x/l x F ex a x ( x L + o L (2.12 x/l 1 F ex x x F a x L kx (2.13 Hooke k a/l > 0 u γ η F(u (γ u + η u 2 + u u (2.14 u 0 u *7 γ [MT 1 ] η [ML 1 ] *6 x > 0 *7 u S S v v ρ ρs v 2

18 i r (i ( x (i, y (i, z (i N 3N x (x 1, x 2,, x 3N ( r (1, r (2,, r (N (2.15 i F (i ( F (i x, F (i y, z (i F (F 1, F 2,, F 3N ( F (1, F (2,, F (N (2.16 F x F µ (x U x µ (µ 1, 2,, 3N (2.17 U(x U(x F(x U(x F µ x ν 2 U x ν x µ 2 U x µ x ν F ν x µ, ( ν µ (2.18 U 0 x1 U(x U 0 0 x3n 0 x2 F 1 (x 1, 0,, 0dx 1 F 2 (x 1, x 2, 0,, 0dx 2 0 F 3N (x 1, x 2,, x 3N 1, x 3N dx 3N (2.19

19 U F 1 (x 1, 0,, 0 + x 1 x3n F 3N x2 0 F 2 x 1 (x 1, x 2, 0,, 0dx (x 1, x 2,, x 3N 1, x 3N 0 x dx 3N 1 x2 F 1 F 1 (x 1, 0,, 0 + (x 1, x 2 0 x, 0,, 0dx 2 2 x3n F (x 1, x 2,, x 3N 1, x 3N 0 x dx 3N 3N F 1 (x 1, 0,, 0 + [F 1 (x 1, x 2, 0,, 0 F 1 (x 1, 0, 0,, 0] + + [F 1 (x 1, x 2,, x 3N 1, x 3N F 1 (x 1, x 2,, x 3N 1, 0] F 1 (x 1, x 2,, x 3N (2.20 F µ U/ x µ (µ 1, 2,, 3N F U 1, U 2 U 2 U 1 d(u 1 U 2 du 1 du 2 3N µ1 F µ dx µ + 3N µ1 F µ dx µ 0 (2.21 x U U 1 U 2 x x 0 x x 0 *8 (0, 0,, 0 (x 1, 0,, 0 (x 1, x 2, x 3N U(x c (1 F(r F 0 U(r F 0 r + c (2.22 (2 (3 F(r F(r r r U(r r F(rdr + c (2.23 F(r (X(x, Y(y, Z(z U(r x X(xdx+ y Y(ydy+ z Z(zdz+c (2.24 *8

20 16 2. Coulomb 1,2 α (12 F (12 α (12 r (1 r (2 r (1 r (2 3 (2.25 m (1, m (2 α (12 Gm (1 m (2 Coulomb q (1, q (2 q (1 q (2 /4πϵ 0 r (1 r (2 + α (12 U (12 (r (1, r (2 r (1 r (2 (2.26 N U i> j U (i j (r (i, r ( j i> j (i j α r (i r ( j (2.27

21 17 3. Lagrange 3.1 F S 3.1 m d2 r dt 2 F + S (3.1 d 2 r/dt 2 S 6 *1 u u, u u 0 ( (3.2 *1

22 18 3. Lagrange S N F m du dt F + N (3.3 u (t 0 N F R u F m du dt F + R (3.4 R F R µ N (3.5 µ R µ N u u (3.6 *2 µ 0 < µ < µ ( Newton a (x, y, z (x, y, z (a sin θ cos ϕ, a sin θ sin ϕ, a cos θ (3.8 (θ, ϕ *2 u u + u u

23 *3 N Newton m µ ẍ µ F µ + S µ (µ 1, 2,, 3N (3.9 Newton Ȧ da/dt, Ä d 2 A/dt 2 i (i 1, 2,, N r (i x (x 1, x 2,, x 3N ( r (1, r (2,, r (N (3.10 i F (i S (i F (F 1, F 2,, F 3N ( F (1, F (2,, F (N (3.11 S (S 1, S 2,, S 3N ( S (1, S (2,, S (N, (3.12 i m (i (m 1, m 2,, m 3N ( m (1, m (1, m (1, m (2, m (2, m (2,, m (N, m (N, m (N (3.13 f q (q 1, q 2,, q f f 3N (x, y, z (θ, ϕ x x(q, t (3.14 q x 3N f f *4 t t + dt x µ x µ + dx µ q ν q ν + dq ν dx µ dx µ x µ dq ν + x µ dt (3.15 t *3 d *4

24 20 3. Lagrange dt ẋ µ dx µ /dt q ν q ν dq ν /dt t ẋ µ x µ q ν + x µ t (3.16 *5 ẋ µ x µ / q ν ẋ µ q ν ẋ µ q ν x µ (3.17 ẋ µ 2 x µ q ν + 2 x µ t ( xµ q ν + ( xµ t ( xµ ν 1 ν 1 d dt ( x µ m µ ẋ µ K 1 2 N i1 m (i ( ṙ (i N µ1 m µ (ẋµ 2 (3.19 m µ ẋ µ K ẋ µ, (3.20 q ν π ν K q ν (3.21 *5 x µ / t ẋ µ dx µ /dt x µ q ν, q ν, t q ν q ν t x µ / t q ν q ν t x µ x µ (q 1 (t, q 2 (t,, q f (t, q 1 (t, q 2 (t,, q f (t, t t t ẋ µ dx µ /dt

25 kinetic momentum ẋ µ dẋ µ ẋ µ q ν, q ν, t (3.17, (3.18 dk 3N µ1 3N µ1 K ẋ µ dẋ µ m µ ẋ µ 3N µ1 m µ ẋ µ d dt ẋ µ dq ν + ( xµ ẋ µ d q ν + ẋ µ q ν t dt dq ν + 3N µ1 m µ ẋ µ x µ d q ν + q ν, q ν t 3N µ1 ẋ µ m µ ẋ µ dt (3.22 t dk K dq ν + K d q ν + K dt (3.23 q ν t d q ν m µ ẋ µ π ν π ν K q ν 3N µ1 m µ ẋ µ x µ (3.24 dq ν K 3N µ1 m µ ẋ µ d dt ( xµ ( (3.15

26 22 3. Lagrange dx d x (vir + d x (def (3.26 d x (vir x dq ν (3.27 d x (def x dt (3.28 t dx (dx 1, dx 2,, dx 3N d x (vir t x x(q, t f d x (def f dx (vir dx (def d x (vir d x (def x (vir x (def ( dx d W (F + S dx F d x (vir + F d x (def + S d x (vir + S d x (def (3.29 *6 S f d x (vir f S d x (vir 3N µ1 S µ x µ dq ν 0 (3.30 dq ν 3N µ1 S µ x µ 0 (3.31 S d x (def f d x (vir d W (vir (F + S d x (vir F d x (vir (3.32 *6 d W F d W U d W 3N µ1 ( U/ x µ dx µ F F µ U/ x µ

27 3.5 Lagrange 23 d W (vir Q ν dq ν (3.33 Q ν δq ν (3.27 (3.32 ( x Q ν F 3N µ1 ( xµ F µ ( Lagrange Newton π ν d ( K d dt q ν dt 3N µ1 3N µ1 x µ m µ ẋ µ m µ ẍ µ x µ + 3N µ1 m µ ẋ µ d dt ( xµ Newton (3.31 3N µ1 m µ ẍ µ x µ 3N µ1 (F µ + S µ x µ 3N µ1 (3.35 F µ x µ Q ν (3.36 (3.25 K/ q ( d K K Q ν (ν 1, 2,, f (3.37 dt q ν Lagrange q (q 1, q 2,, q f q ( q 1, q 2,, q f t U(q, q, t Lagrangian Q ν d dt ( U q ν U (3.38 L(q, q, t K(q, q, t U(q, q, t (3.39 ( d L L 0 (3.40 dt q ν

28 24 3. Lagrange U(q, q, t U U(q, t Q ν 3N µ1 F µ U x µ (3.41 U x µ ( xµ U (3.42 *7 Lagrangian L(q, q, t q ν π ν K/ q ν p ν L q ν (3.43 U q π p U q 3.6 q(t ( d X X 0 (3.44 dt q ν X(q, q, t U U + X Lagrangian L L X gauge X *7 U (x 1, x 2,.x 3N U 3N µ1 U x µ x µ + ( x µ (q 1, q 2,, q ν 1, q ν+1,, q f q ν q ν U 3N U x µ 3N U q ν x µ1 µ q ν x µ1 µ U/ ( xµ

29 (3.44 ν 1 2 X q ν q ν + ν 1 2 X q ν + 2 X X 0 (3.45 q ν q ν t q ν q ν ν, ν 2 X/ q ν q ν 0 X X A 0 (q, t + (3.45 ( Aν ν 1 A ν (q, t q ν (3.46 A ( ν Aν q ν + A 0 0 (3.47 t q ν t q 0 A ν 2.4 A ν (ν 0, 1, 2,, f (3.48 Λ(q, t A ν (q, t (ν 0, 1,, f (3.49 Λ(q, t A ν X(q, q, t X(q, q, t Λ t Λ q ν d Λ(q, t (3.50 dt gauge Lagrangian U(q, q, t U(q, q, t d Λ(q, t, (3.51 dt L(q, q, t L(q, q, t + d Λ(q, t (3.52 dt 3.7 Lagrange l m

30 26 3. Lagrange ϕ 1 ϕ 2 u 1 u 2 1 (l ϕ 1 2 (3.53 u 2 u 1 (u 2 u 1 2 (l ϕ 2 2 (3.54 φ m v φ u 1 u 2 u 1 ϕ 2 ϕ 1 u m v u 2 2 (u 1 + u 2 u 1 2 u (u 2 u u 1 u 2 u 1 cos(ϕ 2 ϕ 1 (l ϕ (l ϕ l 2 cos(ϕ 2 ϕ 1 ϕ 1 ϕ 2 (3.55 ϕ 1 ϕ 2 K m 2 u2 1 + m 2 u2 2 m(l ϕ m 2 (l ϕ ml 2 cos(ϕ 2 ϕ 1 ϕ 1 ϕ 2 m(l ϕ m 2 (l ϕ ml 2 ϕ 1 ϕ 2 (3.56 L K U U mgl cos ϕ 1 mg (l cos ϕ 1 + l cos ϕ 2 2mgl 1 ϕ2 1 mgl 1 ϕ2 2 ( L 2ml ϕ 2 ϕ 1 + ml 2 ϕ 2 1 L ml ϕ 2 ϕ 2 + ml 2 ϕ 1 2 L 2mglϕ 1 ϕ 1 L (3.58 mglϕ 2 ϕ 2

31 Lagrange 2ml 2 ϕ 1 + ml 2 ϕ 2 + 2mglϕ 1 0 (3.59 ml 2 ϕ 2 + ml 2 ϕ 1 + mglϕ 2 0 ( m q r (x, y, z E(r, t B(r, t Lorentz F q [E(r, t + ṙ B(r, t] (3.61 E(r, t ϕ A t (3.62 B(r, t A (3.63 ϕ(r, t A(r, t Lagrangian d dt U(r, ṙ, t qϕ(r, t qṙ A(r, t (3.64 L K U 1 2 mṙ2 qϕ + qṙ A (3.65 ( U U ẋ x qda x q ϕ A + qṙ dt x x q A x t qẋ A x x qẏ A x y qż A x z q ϕ x + qẋ A x x + qẏ A y x + qż A z x [ ẏ ( Ay ( Ax ż (3.66 q ϕ x q A x + q t x A x y z A z x qe x + q(ẏb z żb y F x (3.67 ]

32 28 3. Lagrange χ(r, t ϕ ϕ t (3.68 A A + χ(r, t (3.69 E(r, t B(r, t Maxwell U U q χ t q(ṙ χ U q d χ(r, t (3.70 dt 3.9 Hamilton Lagrangian S [q(t] t2 t 1 L (q(t, q(t, t dt (3.71 t t 1 q q 1 t t 2 q q 2 { q(t1 q 1 q(t 2 q 2 (3.72 q q(t q(t q(t q(t + δq(t δs δq(t δs S [q(t + δq(t] S [q(t] t2 t 1 t2 t 1 (L (q(t + δq(t, q(t + δ q(t, t L (q(t, q(t, t dt L δq ν (t + L δ q ν (t q ν dt (3.73 δ q d(δq/dt δq(t 1 δq(t 2 0 δs t2 t 1 t2 t 1 L δq ν (t dt + L δq ν (t q ν ( L d dt ( L q ν tt 2 tt 1 t2 t 1 d dt ( L δq ν (t q ν dt δq ν (tdt (3.74

33 3.9 Hamilton 29 q(t q(t S [q(t] δq(t δs 0 q(t Lagrange (3.40 Hamilton S [q(t] * 8 q(t q(t Lagrange t (q, q t + dt (q, q q(t q(t S [q(t] q(t Lagrangian gauge Lagrangian L(q, q, t L (q, q, t L(q, q, t + d Λ(q, t (3.75 dt L (q, q, t S S t2 t 1 t2 L (q, q, tdt t 1 L(q, q, tdt + Λ(q 2, t 2 Λ(q 1, t 1 (3.76 S S... δs 0 δs Lagrange Newton Lagrange Lagrangian Newton *8 0 s 1 q 1 (t q 2 (t ss [q 1 (t] + (1 ss [q 1 (t] S [sq 1 (t + (1 sq 2 (t]

34 30 3. Lagrange Newton Lagrangian Newton

35 f Lagrange ( d L L 0 (ν 1, 2,, f (4.1 dt q ν f 2 f 2 f C (C 1, C 2,, C 2 f q ν q ν (t, C (ν 1, 2,, f (4.2 q ν q ν (t, C (ν 1, 2,, f (4.3 (4.2 (4.3 C 1, C 2,, C 2 f C ν C ν (q, q, t (ν 1, 2,, 2 f (4.4 C ν (q, q, t Lagrange q q(t d dt C ν(q(t, q(t, t 0 (4.5 C ν (q(t, q(t, t 2 f q, q, t q q Lagrange

36 Lagrangian q λ q λ q λ q λ q 1 Lagrangian q (q 2, q 2,, q f L( q, q 1, q, t (4.6 q 1 Lagrange q 1 p 1 dp 1 dt p 1 p 1 L/ q 1 q 1 d ( L 0 (4.7 dt q 1 q 1 q 1 ( q, q, t (4.8 q 1 q 1 q 1 ( q, q, t d L L 0 (ν 2, 3,, f (4.9 dt q ν L L L, L (ν 2, 3,, f (4.10 q ν q ν q 1 ( q, q, t q ν q ν, q ν ν 2, 3,, f Lagrangian Routhian L ( q, q, t L p 1 q 1 (4.11 Lagrangian p 1 q 1 q 1 L L d L d L p 1 d q 1 dq ν + L d q 1 + q 1 ν2 L dq ν + ν2 L ν2 q ν ν2 L d q ν + L q ν t dt p 1d q 1 d q ν + L dt (4.12 t

37 L q ν L q ν, L L (ν 2, 3,, f (4.13 L Lagrange d dt ( L q ν L 0 (ν 2, 3,, f (4.14 f f 1 L L Legendre 4.3 N ( r (1, r (2,, r (N U ( r (1, r (2,, r (N m (i (i 1, 2,, N Lagrangian L ({ r (i}, { ṙ (i} N i1 m (i 2 (ṙ(i 2 U ({ r (i } (4.15 n s n r (i r (i + sn (i 1, 2,, N (4.16 U ({ r (i } U ({ r (i} (4.17 Coulomb U u ( (i j r (i r ( j (4.18 i> j n ṙ (i ṙ (i (4.19 Lagrangian L ({ r (i }, { ṙ (i } L ({ r (i}, { ṙ (i} (4.20 s s s 0

38 d ds ( L ({ r (i }, { ṙ (i } s0 N i1 N i1 L r (i ( dr (i ds s0 + N i1 L ṙ (i ( dṙ (i ds s0 L n (4.21 r (i L r (i ( L x, L L, (i y (i z (i, ( L L ṙ (i ẋ, L L, (i ẏ (i ż (i (4.22 r (i (t Lagrange 0 d ds ( L ({ r (i }, { ṙ (i } s0 N i1 L N ( r n d L n d (P n (4.23 (i dt ṙ (i dt i1 P N L N ṙ m (i ṙ (i (4.24 (i i1 i1 n n M (k k m (i, (k 1, 2,, N (4.25 i1 R (k (k 1, 2,, N R (1 r (1, (4.26 R (k M(k 1 R (k 1 + m (k r (k M (k (4.27 r (k R (k r (k+1, (k 1, 2,, N 1 (4.28 R R N 1 M N m (i r (i, i1 M M(N N m (i (4.29 i1 K K N i1 1 ( 2 m(i ṙ (i M ( Ṙ 2 N 1 1 ( r + (k 2 2 µ(k k1 (4.30 µ (k M(k m (k+1 M (k+1, (k 1, 2,, N 1 (4.31

39 (4.16 R R + sn, r (k r (k (4.32 Lagrangian (4.16 R n n R L (n Ṙ n MṘ n P (4.33 Ṙ P/M n 4.4 Lagrangian (4.15 n ϕ φ n r ϕ r r r (i R n (ϕr (i, (i 1, 2,, N (4.34 θ n ( Coulomb U u ( (i j r (i r ( j (4.35 i> j (4.17 ṙ (i R n (ϕṙ (i (4.36 (ṙ(i 2 (ṙ(i 2 (4.37 Lagrangian (n (4.20 ϕ 0 r (i r (i r (i n r (i r (i ( r (i sin θ ϕ n r (i ϕ θ n r (i r (i n r (i ϕ ( ( r (i ṙ n r (i (i, ϕ ϕ n ṙ (i (4.38 ϕ0 ϕ0

40 36 4. (4.20 ϕ ϕ 0 0 d ( ({ } L r (i, { ṙ (i } N ( dϕ L dr (i ϕ0 r + (i dϕ i1 ϕ0 N L r (n (i r(i + r (i (t Lagrange i1 0 d ( ({ } L r (i, { ṙ (i } N dϕ ϕ0 i1 d dt N i1 N i1 ( L (n r (i + ṙ (i d dt ( L ṙ (n r (i (i L ṙ (i ( dṙ (i dϕ ϕ0 L ṙ (i (n ṙ(i (4.39 N i1 L ṙ (i (n ṙ (i d (n L (4.40 dt L N r (i L N ṙ r (i ( m (i ṙ (i (4.41 (i i1 i1 a (b c b (c a c (a b n n 4.5 t Lagrangian L(q, q q s Lagrangian t t t + s (4.42 Lagrangian Lagrangian q dq/dt t t dq dt dq dt ( dt dt 1 dq dt Lagrangian t L/ t 0 (4.43 dl dt L q ν + L q ν + L q ν t L q ν + L q ν q ν (4.44

41 4.6 Noether 37 q(t Lagrange dl ( d L L q ν + q ν d L dt dt q ν q ν dt q ν q ν (4.45 E L q ν q ν L (4.46 de/dt 0 x q x x(q U K 1 2 3N M νν (q µ1 3N ẋ µ m µ (ẋ µ µ1 Lagrangian m µ x µ x µ x µ q ν (4.47 ν 1 M νν (q q ν q ν (4.48 (4.49 L(q, q K(q, q U(q (4.50 t E L q ν q ν K q ν q ν ν 1 M νν (q q ν q ν 2K (4.51 E 2K (K U K + U (4.52 E 4.6 Noether Noether q t q t

42 38 4. q q (q, t, s t t (q, t, s (4.53 s s 0 q q (q, t, s 0 q t t (q, t, s 0 t (4.54 Lagrangian ( gauge Λ(q, t, s q(t t 2 t 1 L (q, dq t2 ( ( dt, t dt L q, dq t 1 dt, t + dλ(q(t, t, s dt (4.55 dt q(t q(t 1 q 1, q(t 2 q 2 q (t q (t 1 q 1 q(t 2 q 2 t i t (q i, t i, s q i q (q i, t i, s s 0 Λ 0 t (4.55 t2 ( ( R(s L (q, dq dt dt, t dt L q, dq dt, t t 1 dr t2 ds t 1 dλ dt R(s 0 0 ( d L (q, dq dt ds dt, t dt dλ dt dt 0 (4.56 dt 0 (4.57 (4.57 s 0 s s 0

43 4.6 Noether 39 dt 1 (4.58 ( dt dt d ( t (4.59 s dt dt s ( dq ν s dt ( s dq ν dt 1 dt dt [ ( dq ( ( ] ( ν dt dq ν dt dt 2 s dt dt dt s dt dt d ( q ν dq ν d ( t (4.60 dt s dt dt s 0 ( dr t2 ds L s0 t 1 s t2 ( L d dt t 1 t 1 ( dt dt ( t s + L t t s d dt t2 ( d dt + t2 t 1 d dt dc dt dt + + ( Λ s ( L q ν q ν s t 1 dt L q ν s + L q ν s + ( d dt ( L L q ν s L d q ν dt q ν ( L L q ν + q ν + L q ν q ν t t2 ( ( d L dt q ν L ( dq ν dt + L t t s d dt ( q ν s q ν s t s + L d dt L ( q ν s q ν ( L q ( C ν L t q ν s q ν L q ν s dl dt s0 L q ν + s0 L q ν + L q ν t ( t s L d q ν q ν dt ( Λ s dt ( t s ( d L t q ν dt q ν s d dt ( Λ dt s t dt (4.61 s ( Λ s s0 (4.62 (4.63 C s 0 Lagrange ( d L L 0 (ν 1, 2,, f (4.64 dt q ν q q(t C(t 2 C(t 1 t2 t 1 dc dt 0 (4.65 dt

44 40 4. C Noether Lagrangian Lagrangian Noether Noether q 1 q (q 2, q 3,, q f s L(q 1 + s, q, q 1, q, t L(q 1, q, q 1, q, t (4.66 Lagrangian q 1 q 1 + s (4.67 ( q ( 1 q ( 1, 2 q 3 q f, s s0 s s0 s s0 s s0 ( t s s0 0, Λ 0 (4.68 ṗ 1 d ( L 0 (4.69 dt q 1

45 Lagrange Lagrange q Lagrangian t L(q, q E L q L (5.1 q q q q(q, E (5.2 t t 0 + dq q(q, E t 0 E (5.3 x m U(x Lagrangian L 1 2 mẋ2 U(x (5.4 E 1 2 mẋ2 + U(x (5.5

46 42 5. ẋ dx/dt dx 2 dt ± (E U(x (5.6 m x dx t t 0 ± 2 (E U(x m t 0 E x U(x E E U(x E U(x 1 2 mẋ2 0 ( (5.7 x x x min x x max x T(E 2 xmax x min dx 2 (E U(x m (5.9 x x min x x max

47 U(x du/dx 0 x x 0 F du/dx 0 x x 0 U(x x x 0 x > x 0 (< x 0 F du/dx < 0 (> 0 x x 0 x x 0 U(x x x 0 x > x 0 (< x 0 F(x du/dx > 0 (< 0 x x 0 x x 0 E U(x m (1, m (2 r (1 r (2 R r r (1 M m (1 + m (2 R m(1 r (1 + m (2 r (2, r r (1 r (2 (5.10 M Lagrangian L 1 2 MṘ µṙ2 U(r (5.11 Coulomb µ µ (1 m(1 m (2 M (5.12 R P L Ṙ MṘ ( Lagrangian

48 44 5. L rel 1 2 M ( P M µṙ2 U(r P 1 2 µṙ2 U(r + (5.14 r r Coulomb L rel 4.4 L r µṙ L 0 L r 0 r L L 0 r ṙ r r (r, ϕ dt r dr ϕ dϕ r r dr rdϕ r ds ds (dr 2 + r 2 (dϕ 2 (5.15 r ṙ ds dt Lagrangian (dr dt P M 2 ( 2 dϕ + r 2 (5.16 dt L rel 1 2 µ ( ṙ 2 + r 2 ϕ 2 U(r (5.17 ϕ L z L rel ϕ µr2 ϕ (5.18 L z z µr 2 ϕ µ(xẏ yẋ 4.2 Lagrangian r Lagrangian L rad 1 ṙ2 ( 2 2 µ + r 2 Lz µr U(r L L z 2 z µr µṙ2 U eff (r (5.19 U eff (r U(r + L2 z 2µr 2 (5.20

49 U(r 5.1 r(t E L rad ṙ L rad 1 ṙ 2 µṙ2 + U eff (r (5.21 (5.6 dr dt ± 2 µ (E U eff(r (5.22 t t 0 + ± dr 2 µ (E U eff(r (5.23 t 0 E r r(t (5.18 dϕ L z dt (5.24 µr2 t ϕ (r(t, ϕ(t dt r ϕ L z dr ϕ c + ± r 2 2µ(E U eff (r (5.25 c E z L z (5.18 ϕ t ϕ r r min r max r r min r max r min ϕ ϕ ϕ 2π ϕ 2π r min r r max u(r r 2 u(r r 1 r Bertrand *1 *1 H. Goldstein, C. Poole and J. Safko Classical Mechanics (3rd ed. (Addison Wesley 3-6

50 Kepler r r 2 Kepler Coulomb U(r α r (5.26 α Gm (1 m (2 Coulomb q (1, q (2 α q (1 q (2 /4πϵ 0 α > 0 U eff (r r 0 r + r L 2 z /αm α 2 m/2l 2 z < 0 E < 0 E 0 α < 0 U eff (r r 0 r + E > 0 U eff (r α r + L2 z 2µr 2 (5.27 (5.25 r L z dr/r 2 ϕ c ± ( µαr L 2 z c ± arcsin eµαr 2µ(E + α/r Lz 2 /2µr 2 ( /2µr 2 U eff (r r c E z e α/r e 1 + 2EL2 z µα 2 (5.29 r L 2 z /µα 1 ± e sin(ϕ c 5.2 r ϕ { c ± π/2 (α > 0 ϕ min c π/2 (α < 0 (α > 0 (5.30 (5.31 r min l 1 + e l e 1 (α > 0 (α < 0 (5.32

51 5.3 Kepler 47 l L2 z µ α (5.33 ϕ min r l (α > 0 r 1 + e cos(ϕ ϕ min l (α < e cos(ϕ ϕ min (5.34 (x, y (r cos(ϕ ϕ min, r sin(ϕ ϕ min l a 1 e 2 α 2E (5.35 (x + ae 2 y 2 + a 2 ( 2 a 1 e 2 (x ae 2 y 2 a 2 ( a e (α > 0, 0 e < 1 y 2 + 2lx l 2 (α > 0, e 1 1 (α > 0, α < 0, e > 1 (5.36 e α > 0 E < 0 α > 0 E 0 α > 0 α < 0 E > 0 Kepler a b a 1 e α > 0 Kepler (5.18 Kepler 1 2 r2 ϕ L z (5.37 2µ

52 48 5. T πa2 1 e 2 L z 2µ πa2 l/a µαl 2µ 2π µ α a3/2 (5.38 Kepler µ T 2πa 3/2 α 2πα ( µ 1 3/2 (5.39 2E E 5.4 m (1 m (2 R m(1 r (1 + m (2 r (2 M 0, (M m (1 + m (2 (5.40 r (G1 r (G2 5.2 r r (G1 r (G2 Lagrangian L rel 1 2 µṙ2 U(r (5.41 U(r µ m (1 m (2 /(m (1 + m (2 Lagrangian 1 2 µṙ2 + U(r E( (5.42 µr ṙ L( (5.43 r(t r (G1 (t, r (G2 (t r (G1 m(2 M r, r(g2 m(1 M r (5.44

53 ṙ (G1 m(2 M ṙ, ṙ(g2 m(1 M ṙ (5.45 t t + r + U 0 i 1, 2 u (Gi i u i u (G1 i u (G2 i u f u (G1 f u (G2 f (5.45 u (Gi f E 1 2 µu2 i 1 2 µu2 f (5.46 v i v f, v (G1 i v (G1 f, v (G2 i v (G2 f (5.47 (5.43 r (5.44 r (G1 r (G2 xy z L t u (G1 i u (G2 i ρ r (1 (t v i ρ 5.4 ρ L z µrv i sin δ µrv i r µρv i, (r + (5.48 δ r u i u (G1 i u (G1 f u (G2 i u (G2 f u i u f θ Lagrangian (5.41 r t t + ρ θ ϕ 0 v f θ φ 0 r(t r (2 (t θ π 2ϕ 0 (5.49 ϕ 0 (5.25 r r r min r + v i ρ

54 50 5. ϕ 0 + r min + r min L z dr r 2 2µ(E U(r L 2 z r 2 ρdr r 2 1 2U(r µv 2 i ρ2 r 2 (5.50 r min q (1 q (2 Coulomb 5.3 α q (1 q (2 /4πϵ 0 < 0 ϕ 0 ( l 1 + e cos ϕ 0 (5.51 e e 1 + 2EL2 z µα µρu2 2 i α (5.52 θ θ 2arctan α (5.53 µρu 2 i θ v i ρ v i ρ ρ ρ + dρ u i dσ 2πρdρ (5.54 θ ρ ρ ρ(θ θ θ + dθ r dσ 2πρ(θ dρ dθ dθ (5.55 dρ/dθ < 0 dθ > 0 dθ dω 1

55 θ θ + dθ dω 2π sin θdθ dσ dω ρ(θ sin θ dρ dθ (5.56 Coulomb ρ α µu 2 i cot θ 2 (5.57 dσ dω α 2 4µ 2 u 4 i sin4 (θ/2 (5.58 Rutherford dσ σ dσ dω 2π dω π 0 dθ sin θ dσ (θ (5.59 dω r < R ρ > R dρ/dθ 0 ρ R dρ/dθ < 0 σ dσ R 0 2πρdρ πr 2 (5.60 i 1, 2 u (Li i u (L1 f u (Li f u (L1 i u i θ u (G2 i 5.5 v (G2 i θ' v (G2 f v (L1 f θ v (G1 i v (L1 i v (G1 f u (G2 i θ θ tan θ v (G1 f v (G1 f sin θ cos θ + v (G2 i sin θ cos θ + (m (1 /m (2 (5.61

56 52 5. (5.45 (5.47 v (G2 i v (G1 f v(g2 i v (G1 i m(1 m (2 (5.62 dω 2π sin θ dθ dσ/dω dσ dω (θ dσ dω (θ [1 + 2(m (1 /m (2 cos θ + ( m (1 /m (2 2 ] 3/2 1 + ( m (1 /m (2 cos θ (5.63

57 x m U(x Lagranigian xx 0 xx 0 xx 0 L 1 2 mẋ2 U(x ( x x 0 x x 0 U(x x x 0 k d 2 U/dx 2 > 0 η x x 0 η U(η U(x kη2 + (η (6.2 k d2 U dx 2 > 0 (6.3 xx0 η 0 U(η η ẋ Lagrangian η L 1 2 m η2 1 2 kη2 (6.4 Lagrangian Lagrange ( d L L m η + kη 0 (6.5 dt η η η(t Ce λt λ mλ 2 + k 0 (6.6

58 54 6. λ ±iω, ω k m (6.7 ω η(t C 1 e iωt + C 2 e iωt (6.8 C 1 C 2 C 1 C 2 C 2C 2 c 1 ReC, c 2 ImC η(t Re [ Ce iωt] (6.9 η(t c 1 cos ωt + c 2 sin ωt (6.10 c 1 c 2 η(t A cos(ωt + ϕ (6.11 A ϕ cos(ωt + ϕ cos ωt cos ϕ sin ωt sin ϕ (6.12 c 1, c 2 A, ϕ A c c2 2, tan ϕ c 2 (6.13 c 1 A ωt + α 6.2 q f Lagrangian L(q, q L i (q i, q i (6.14 i1 L i (q i, q i, t

59 q (q 1, q 2,, q f x (x 1, x 2,, x 3N t x x(q U(q Lagrangian L 1 2 M νν (q q ν q ν U(q (6.15 ν 1 M νν (q (4.49 U 0, (ν 1, 2,, f (6.16 q q 0 q q 0 U(q η q q 0 η η η q L 1 2 M νν η ν η ν U(q ν 1 k νν η ν η ν (6.17 ν 1 M νν (q 0 M νν k νν 2 U (6.18 qq0 ˆM M νν f f ˆk k νν f f M νν M ν ν k νν k ν ν ˆM T ˆM ˆk T ˆk U(q 0 Lagrangian L 1 2 ηt ˆM η 1 2 ηt ˆkη (6.19 η f η T η η 1 η 2.. η f, η T (η 1, η 2,, η f (6.20 η 0 ˆk η 0 η T ˆkη > 0 (6.21

60 56 6. η T ˆkη 0 η 0 η T ˆM η > 0 (6.22 f f  f u u T Âu > 0 (6.23  ˆk ˆM Lagrange d dt ( L L η η ˆM η + ˆkη 0 ( ω η(t ue iωt, (u (6.25 Lagrange λ ω 2 ( λ ˆM + ˆku 0 (6.26 ( λ ˆM + ˆk u 0 u 0 λ f det( λ ˆM + ˆk 0 (6.27 f λ (ν (ν 1, 2,, f λ λ (ν (6.26 u u (ν 0 ω (ν λ (ν u (ν λ ω 2 > 0 (6.27 λ (ν ω 2 λ (ν ω Imω < 0 (6.25 Lagrange λ (ν u (6.26 λ (ν u(νt ˆku (ν > 0 (6.28 u (νt ˆMu (ν ˆk ˆM

61 ( λ (ν ˆM + ˆku ν 0 u (ν T u (ν T ( λ (ν ˆM + ˆku (ν 0 (6.29 ( λ (ν ˆM + ˆku (ν 0 u (ν ˆM ˆk u (ν T ( λ (ν ˆM + ˆku (ν 0 (6.30 (6.29 (6.30 ( λ (ν λ (ν u (ν T ˆMu (ν 0 (6.31 λ (ν λ (ν u (ν T ˆMu (ν 0 (6.32 (6.27 (6.26 u (6.26 u u (νt ˆMu (ν 1 (6.33 u f f Û ( u (1, u (2,, u ( f (6.34 Û T ˆMÛ ˆ1 (6.35 ˆ1 f f Û T ˆkÛ u (1T u (2T. u ( f T ( λ (1 ˆMu (1, λ (2 ˆMu (2,, λ ( f ˆMu ( f diag ( λ (1, λ (2,, λ ( f (6.36 diag ( λ (1, λ (2,, λ ( f λ (1, λ (2,, λ ( f f f η ξ Û 1 η Û T ˆMη (6.37

62 58 6. η Ûξ (6.19 Lagrangian L 1 2 (( ξ (ν 2 ( λ (ν ξ (ν 2 (6.38 Lagrange ξ (ν + λ (ν ξ (ν 0 (ν 1, 2,, f (6.39 f C (ν (ν 1, 2,, f ω (ν λ (ν ξ ν (t Re [ C (ν exp ( iω (ν t ] (6.40 η(t Ûξ(t u ν Re [ C (ν exp ( iω (ν t ] (6.41 u (ν u (ν u (ν ϕ 1 ϕ 2 Lagrangian ϕ 1 ϕ 2 L m(l ϕ m 2 (l ϕ ml 2 cos(ϕ 2 ϕ 1 ϕ 1 ϕ 2 + mgl cos ϕ 1 + mg (l cos ϕ 1 + l cos ϕ 2 m(l ϕ m 2 (l ϕ ml 2 ϕ 1 ϕ 2 mglϕ 2 1 mgl 2 ϕ ηt ˆM η 1 2 ηt ˆkη (6.42 η ( ϕ1 ϕ 2, ˆM ( ( 2ml 2 ml 2 2mgl 0 ml 2 ml 2, ˆk 0 mgl (6.43 Lagrange ˆM η + ˆkη 0 η ue i λt ( λ ˆM + ˆk u 0 (6.44

63 det ( λ ˆM + ˆk ( ml 2 λ + mgl ( ml2 2 λ + mgl ( ml λ 0 (6.45 λ (1 ( 2 2 g l, λ(2 ( g l (6.46 u (ν ( ( u (1 1 (, u (2 1 2 ( ( C (ν 2 η(t Re C (ν u (ν exp ( iω (ν t (6.48 ω (1 (2 2g, ω (2 l (2 + 2g l (6.49 ϕ 1 (t Re [ C (1 e iω(1t + C (2 e ] iω(2 t (6.50 ϕ 2 (t Re [ 2C (1 e iω(1t 2C (2 e ] iω(2 t ( f +1 a k f m x 0 x ( f + 1a j x j ja η j Lagrangian 6.2 L j1 m f 1 2 η2 j k ( 2 k η j+1 η j 2 2 η2 1 k 2 η2 f (6.52 j1 1 2 ηt ˆM η 1 2 ηt ˆkη (6.53

64 60 6. ˆM mˆ1 ˆk k (6.54 Lagrange ˆM η + ˆkη 0 η ue i λt ( λ ˆM + ˆku 0 (6.55 λ u det( λ ˆM + ˆk 0 λ (µ (2 m k λ u j u j+1 u j 1 0 ( j 1, 2, f (6.56 u 0 u f +1 0 u j j u j ζ n ζ (2 m k λ ζ ζ (6.57 u (1 j u (2 j c (1, c (2 c (1 u (1 j + c (2 u (2 j α, β u j c (1 α j + c (2 β j (6.58 u 0 u 1 u j c (1 c (2 β 1/α u 0 0 c u j cα j ( α 2 j 1 (6.59 u f +1 0 α 2( f +1 1 α f ( α (ν πν exp i, (ν 1, 2,, f (6.60 f + 1

65 m k λ α(ν + 1 α (ν (6.61 λ f λ (ν 2k ( ( πν 1 cos, (ν 1, 2,, f (6.62 m f + 1 λ (ν u (ν u (ν T ˆMu (ν mu (ν T u (ν 0 (ν ν (6.63 u (νt u (ν 1 u (ν u (ν u (ν T u (ν δ νν (6.64 u (ν ( u (ν 2 π jν j f + 1 sin f + 1 (6.65 ( ( ( ω (ν 2k πν k 1 cos 2 m f + 1 m sin πν (6.66 2( f + 1 C (ν η(t Re C (ν u (ν exp ( iω (ν t (6.67 ( η j (t Re C (ν 2 π jν f + 1 sin exp ( iω (ν t f + 1 (6.68 η(t 0 η(t 0 u (ν u (νt η(t 0 Re [ C (ν] (6.69 u (νt η(t 0 Re [ iω (ν C (ν] ω (ν Im [ C (ν] (6.70

66 62 6. ( C (ν u (νt η(t 0 + i ω η(t 0 (ν 2 ( π jν sin f + 1 f + 1 j1 ( η j (t 0 + i ω η j(t 0 (ν ( l ( f + 1a f + a 0 ψ(x j, t η j (t x ψ(x, t ρ m m ρa (6.72 1/κ 1 k a κ (6.73 Lagrange ˆM η + ˆkη 0 η j (t ψ(x j, t ρa 2 ψ(x j, t t 2 κ a ( 2ψ(x j, t ψ(x j 1, t ψ(x j+1, t, ( j 1, 2,, f (6.74 ψ(0, t ψ(x 0, t ψ(l, t ψ(x f +1, t ψ(x, t a ψ(x j±1, t ψ(x j ± a, t ψ(x j, t ± ψ x a ψ xx j 2 x 2 a 2 (6.75 xx j Lagrange x j x ψ(0, t ψ(l, t ψ c 2 t 2 ψ ( x 2 c κ ρ (6.77

67 (6.68 a 0 x j x ψ(x, t Re Re + + 2a ( πxν ( C (ν l sin exp i πν l ( ic (ν a (6.66 l ct 1 ( (exp i πν ( 2l l (x ct exp i πν l (x + ct (6.78 ω (ν 2 k ( πν κ m sin 2l a πν 2 ρa 2 2l a cπν l (a 0 (6.79 *1 (6.71 ic (ν 2 a a l 2 l j1 l 0 ( πν ( sin l x j iψ(x j, t 0 + ( πν ( sin l x iψ(x, t c(πν/l 1 c(πν/l ψ t ψ t xx j,t0 dx (a 0 (6.80 t0 ( x 1 ( c t x + 1 ψ(x, t 0 (6.81 c t y ± x ± ct x (y + + y /2, t (y + y /2c x y ± y ± x + t y ± t 1 ( 2 x ± 1 c t ( ψ y + y 0 (6.83 y + ψ y (y (6.84 f (x g(x ψ f (y + g(y + f (x ct + g(x + ct (6.85 *1 ν f πνa/2l 1 ω (ν ψ(x, t x ψ(x, t a ν C (ν ν

68 64 6. d Alembert f (x x c g(x x c f (x g(x c *2 ψ(x 0, t ψ(x l, t 0 ψ(x 0, t 0 g(x f ( x ψ(x l, t 0 ψ(x, t f (x ct f ( x ct (6.86 f (x + 2l f (x (6.87 f (x + f (x + Re ( ic (ν a 1 ( exp i 2πν L L x L 2l f (x (6.80 (6.85 ic (ν a 2 L L/2 0 2 L L/2 L/2 2 L L/2 L/2 + 2 L L/2 ( 2πν ( sin L x ( 2πν ( sin L x ( 2πν i sin L/2 2 L/2 exp L L/2 ( 2πν cos i( f (x f ( x L 2πν ( f (x f ( x dx i f (x L 2πν f (x dx L x f (xdx 2 ( ] xl/2 2πν [cos L L x f (x x L/2 L x f (xdx f (xdx (ν 1, 2, (6.88 ( i 2πν L x L/2 L/2 L 1 L/2 f (xdx L/2 *2

69 f (x 1 L f (0 + 1 L f (0 + + ν f (ν f (ν 1 L L/2 L/2 + + f (ν 1 L exp [ ( Re 2 f (ν 1 L exp i 2πν ] L x ( 1 L exp i 2πν L x ( i 2πν L x ( exp i 2πν L x + + f (ν 1 ( exp i 2πν L L x (6.89 f (xdx (6.90 f ( ν f (ν f (x Fourier

70

71 Euler O x, y, z e x, e y, e z O O x, y, z e x, e y, e z P O P r r x e x + y e y + z e z (7.1 x, y, z e x, e y, e z e x, e y, e z e x, e y, e z O R f Euler (ϕ, θ, ψ e 0 (e z e z / e z e z e y e 0 ϕ e z e z θ e 0 e y ψ

72 68 7. (1 e z ϕ (2 e 0 θ (3 e z ψ (1 e y e 0 (2 e 0 e z e z (3 e z e 0 e y e y e y e z e z e x e x e x, e y, e z e x, e y, e z *1 n ζ 3 3 R n (ζ (1 R ez (ϕ (2 *2 _ e z. ψ θ e z. ϕ R e0 (θ R ez (ϕr ey (θr ez ( ϕ (7.2 (3 R ez (ψ R e0 (θr ez (ψr e0 ( θ (7.3 e x e y _ e x ϕ ψ _ e y θ. e 0 (e z e z / e z e z (1 (3 7.1 Euler R ez (ψr e0 (θr ez (ϕ R e0 (θr ez (ψr e0 ( θr e0 (θr ez (ϕ R ez (ϕr ey (θr ez ( ϕr ez (ψr ez (ϕ R ez (ϕr ey (θr ez (ψ (7.4 n n n n R n (ξr n (ξ R n (ξ R n (ξ (7.5 u *1 Euler Landau Euler *2 e 0 θ e z ϕ e 0 e y e y θ e z ϕ e y e 0

73 7.1 Euler 69 R n (dξr n (dξ u R n (dξ ( u + dξ n u u + dξ n u + dξn u (4.38 u + (dξn + dξ n u (7.6 R n (dξr n (dξ u R n (dξ R n (dξu (7.7 dξ n dξn + dξ n (7.8 n dξ R n (dξr n (dξ u u + dξ n u R n (dξ u (7.9 (7.8 P r OP R OO r OP r R + r r x e x + y e y + z e z (7.10 t t + dt r r + dr e x, e y, e z n ωdt ṙ dr dt dt(ωn r ω r (7.11 dt ω ωn t P ṙ Ṙ + ṙ Ṙ + ω r Euler Euler Euler (ωdtn ( ϕdte z + ( θdte 0 + ( ψdte z (7.12 ω ϕe z + θe 0 + ψe z (7.13

74 70 7. e z αx,y,z(e z e α e α e 0 αx,y,z(e 0 e α e α ω ω x e x + ω y e y + ω z e z (7.14 e 0 e x sin ψ (7.15 e 0 e y cos ψ (7.16 e 0 e z 0 (7.17 e z e x sin θ cos(π ψ sin θ cos ψ (7.18 e z e y sin θ cos(π/2 ψ sin θ sin ψ (7.19 e z e z cos θ (7.20 e z e x, e y sin θ e x π ψ, e y π/2 ψ ω x ϕ sin θ cos ψ + θ sin ψ (7.21 ω y ϕ sin θ sin ψ + θ cos ψ (7.22 ω z ϕ cos θ + ψ (7.23 *3 7.2 i m (i O r (i r ρ(r i x (i y (i z (i m (i ρ ( r (i x (i y (i z (i *3 R a (Rb (R 1 a (R 1 Rb (R 1 a b α x, y, z e z e α e z (R ez (ϕr ey (θr ez (ψe α ( R ez ( ϕe z ( Rey (θr ez (ψe α e z (R ey (θr ez (ψe α ( R ey ( θe z ( Rez (ψe α R ey ( θe z ( sin θ, 0, cos θ, R ez (ψe x (cos ψ, sin ψ, 0, R ez (ψe y ( sin ψ, cos ψ, 0, R ez (ψe z (0, 0, 1 e z e x sin θ cos ψ, e z e y sin θ sin ψ, e z e z cos θ

75 i O r (cm i m (i r (i i m (i 1 ρ(r (i x (i y (i z (i 1 M M i rρ(rd 3 r (7.24 M i m (i ρ(rd 3 r R Euler U(X, Y, Z, ϕ, θ, ψ R Xe x + Ye y + Ze z g z Z Euler U(Z, ϕ, θ, ψ ρ(rg ( e z R + xe z e x + ye z e y + ze z e z d 3 r ρ(rg (Z x sin θ cos ψ + y sin θ sin ψ + z cos θ d 3 r MgZ + Mg ( x (cm sin θ cos ψ + y (cm sin θ sin ψ + z (cm cos θ (7.25 (7.18-(7.20 K R Euler Lagrangian K U Euler K 1 m (Ṙ (i + ω ṙ (i 2 2 i M 2 Ṙ2 + 1 ( m (i ω 2 ṙ(i 2 + m (i Ṙ ṙ(i i K (cm + K (r + K (mix (7.26 i K (cm Ṙ K (r Euler K (mix K (r O l i ( r (i m (i ṙ (i (7.27 a (b c (c ab (a bc

76 72 7. l m (i r (i ( ω r (i i m ( (i r (i 2 ω m ( (i r (i ω r (i i Îω ( l α I αβ ω β,, l l α e α β i α (7.28 Î 3 3 e α I αβ α, β x, y, z I αβ e T α Î e β m ( (i δ αβ r 2 r α r β i ρ(r ( δ αβ r 2 r α r β d 3 r (7.29 r x x r y y r z z δ αβ Kronecker δ αβ { 1 (α β 0 (α β (7.30 O K (r K (r 1 2 i 1 2 i m (i ( ω ṙ(i 2 m (i ( ω 2 ( r (i 2 ( ω r (i ω l 1 2 ωt Îω 1 I α,β ω α ω β 2 (7.31 (6.19 K (mix R Euler O O Euler K (r α,β K K (r 1 I α,β ω α ω β ( O i m (i r (i Mr (cm 0 K (mix 0 α,β

77 K (cm K (r K K (cm + K (r M 2 Ṙ2 + 1 I α,β ω α ω β ( e α e α I αβ I αβ e T α Î e β I α δ αβ (7.34 I α 0 I α K (r K (r 1 I α ω 2 α ( I αβ 2 K (r / ω α ω β Î ω 2K (r ω T Îω 0 K (r Ṙ α α,β Îu λu λ (, ( λˆ1 + Îu 0 (7.36 det ( λˆ1 + Î 0 (7.37 λ 1, λ 2, λ 3 u u 1, u 2, u 3 1 u i u j 0 u 3 u 1 u 2 u 3 e x, e y, e z u 1, u 2, u 3 I x λ 1, I y λ 2, I z λ 3 (7.34 Î (cm Î (cm 2π/n n Î (cm n 3

78 74 7. *4 I (cm I αβ O r (cm I αβ ρ(r ( δ αβ r 2 r α r β d 3 r ( ( ρ(r δ αβ r r (cm + r (cm 2 ( rα r (cm α ( ρ(r ( I (cm + M ( δ αβ r r (cm 2 ( rα r (cm α + ( ρ(r δ αβ ( r (cm 2 r (cm α ( + r (cm α rβ r (cm β ( rβ r (cm β d 3 r ( δ αβ r (cm 2 r (cm α r (cm β r (cm β d 3 r + r (cm β d 3 r (7.38 ρ(r ( r r (cm d 3 r Mr (cm Mr (cm 0 (7.39 n n ψ ω ψn K K (r 1 2 I ψ 2 (7.40 I n T Î n n α I αβ n β α,β ρ(r ( r 2 ( n r 2 d 3 r ρ(rr 2 d 3 r (7.41 n n α n α n α e α r r 7.3 I 1 I x I y I 2 I z O *4

79 O z Lagrangian Euler L K 1 2 I ( 1 ω 2 x + ωy I 2ω 2 z 1 2 I ( 1 ϕ 2 sin 2 θ + θ I 2( ϕ cos θ + ψ 2 (7.42 ϕ ψ l e z L ϕ ( I 1 sin 2 θ + I 2 cos 2 θ ϕ + I 2 ψ cos θ (7.43 l e z L ψ I ( 2 ψ + ϕ cos θ (7.44 Noether l e z e z e z l e z l l l e z ( I 1 sin 2 θ + I 2 cos 2 θ ϕ + I 2 ψ cos θ (7.45 l e z l(e z e z l cos θ I 2 ( ψ + ϕ cos θ (7.46 θ 0 (7.47 ϕ l I 1 (7.48 ψ (I 1 I 2 I 1 I 2 l cos θ (7.49 l ϕ l/i 1 ω z (l e z /I 2 l cos θ/i I 1 I x I y I 2 I z O O z e z Lagrangian Euler

80 76 7. L K U 1 2 I ( 1 ω 2 x + ωy I 2ω 2 z U(Z, ϕ, θ, ψ 1 2 I ( 1 ϕ 2 sin 2 θ + θ I 2( ϕ cos θ + ψ 2 Mgd cos θ (7.50 (7.25 U(Z, ϕ, θ, ψ Z x (cm y (cm 0 d z (cm ϕ ψ l e z L ϕ ( I 1 sin 2 θ + I 2 cos 2 θ ϕ + I 2 ψ cos θ βi 1 (7.51 l e z L ψ I ( 2 ψ + ϕ cos θ αi 1 (7.52 ϕ, ψ ϕ α β cos θ sin 2 θ ψ I 1β I 2 cos α β cos θ sin 2 θ (7.53 (7.54 Lagrangian L L αi 1 ϕ βi 2 ψ 1 2 I 1 θ I (α β cos θ 2 2 Mgd cos θ β2 I1 2 (7.55 sin 2 θ 2I 2 L θ θ L β2 I I 2 2 I 1 θ I (α β cos θ Mgd cos θ E( (7.56 sin 2 θ 5.1

81 Legendre x f (x y d f /dx g(y f (x f (x y f (x x y x x(y g(y f (x(y f (x f (x x y f C (x f (x C (C : (8.1 (x,f(x y f C (x x x x C(y f C (x f (x C x C (y x(y+c (0,-g(y g C (y f C (x C (y f (x(y g(y (8.2 g C (y C f C (x C f (x Legendre g(y 8.1 Legendre Legendre g(y yx(y f (x(y (8.3 y d f /dx x x x(y Legendre y f (x f (x ( g x (8.4

82 78 8. f (x x y g Legendre g(y f (x g(y f (x (0, g(y y y f (x g(y g dg ydx + xdy d f xdy (8.5 ydx d f ydx Legendre x(y dg dy g (y (8.6 y f (x x x x(y g (y y g(y g(y f (x f (x g(y x(y g (y g(y x y x x(y g (y y y y(x x x (y(x xy(x g (y(x xy(x y(xx (y(x + f (x (y(x f (x (8.7 g(y f (x Legendre f (x f (x Legendre g(y f (x 0 Legendre 8.2 q ν (ν 1, 2,, f p ν L q ν (8.8 Lagrangian L(q, q, t Legendre Lagrangian q, p, t Hamiltonian H(q, p, t p q L(q, q, t (8.9

83 p L/ q q q q(q, p, t H q, p t ( 2 L det 0 (8.10 Hamiltonian H(q, p, t Lagrangian H(q, p, t Hamiltonian dh d p ν q ν L q ν dp ν + p ν d q ν dl ( q ν dp ν + p ν d q ν L dq ν p ν d q ν L t dt ( q ν dp ν L dq ν L dt (8.11 t q(t p(t Lagrange ṗ ν L/ q H p, ṗ H q (8.12 q p Hamilton Hamiltonian H(q, p, t Legendre Lagrangian q H p p p p(q, q, t (8.13 L(q, q, t p q H(q, p, t (8.14 ( 2 H det 0 (8.15 p ν ν

84 80 8. (q, p (q, p 2 f trajectory 8.3 Poisson u v Poisson [u, v] ( u v v u p ν p ν (8.16 [q ν, q ν ] 0, [p ν, p ν ] 0, [q ν, p ν ] δ νν (8.17 Poisson Poisson [u, v] [v, u] (8.18 [u, c 1 v + c 2 w] c 1 [u, v] + c 2 [u, w] (8.19 [uv, w] u[v, w] + [u, w]v (8.20 [q ν, F(q, p] F, [p ν, F(q, p] F p ν (8.21 [u, [v, w]] + [v, [w, u]] + [w, [u, v]] 0 (8.22 Poisson Jacobi Poisson *1 *1 u, v, w u [v, [w, u]] + [w, [u, v]] ( ( v w u u w q νν ν p ν p ν p ν + w ( u v v u p ν p ν p ν ( w u u w v p ν p ν p ν ( u v v u w p ν p ν p ν 2 u/ p ν p ν ( 2 u v w w v 0 p ν,ν ν p ν

85 (p, q, t F Poisson df dt F t F t F t + + ( F q ν + F ṗ ν p ν ( F H H F p ν p ν + [F, H] F(q, p (8.23 [F, H] 0 (8.24 df/dt 0 Hamiltonian [H, H] 0 F(q, p, G(q, p Poisson [F, H] [G, H] 0 Jacobi [[F, G], H] [G, [F, H]] [F, [G, H]] 0 ( Lagrange f Lagrange Lagrange q Q Q(q, t Q Q ν 1 q ν + Q t (8.26 Q Q(q, q, t Hamiltonian q p 2 f (q, p 2 f (Q, P Q Q(q, p, t, P P(q, p, t ( u/ 2 u/ p ν ( v w 2 u v w 2 u + v w 2 u + v w 2 u 0 p ν p ν p ν p ν p ν p ν p ν p ν νν u v, w

86 82 8. Hamilton (Q, P Hamiltonian H (Q, P, t Q H P, Ṗ H Q (8.28 H (Q, P, t H(q, p, t Legendre Lagrangian gauge Λ(q, t, Λ (Q, t p q H(q, p, t + dλ dt P Q H (Q, P, t + dλ dt (8.29 dt Λ 1 Λ Λ p dq H(q, p, tdt P dq H (Q, P, tdt + dλ 1 (8.30 Λ 1 (q, Q, t Λ (Q, t Λ(q, t H (Q, P, t Legendre Lagrangian L (Q, Q, t det( 2 H/ P ν P ν 0 Λ 1 (q, Q, t Λ (Q, t Λ(q, t Hamilton Lagrangian Λ 1 q, Q, t Λ 1 (q, Q, t dλ 1 p dq P dq + (H Hdt (8.31 p Λ 1 q, P Λ 1 Q, H H + Λ 1 t (8.32 Q Q Q(q, p, t P P(q, p, t det( 2 Λ/ Q ν 0 q P Λ 1 (q, Q, t Legendre Λ 2 (q, P, t Λ 1 + P Q dλ 2 p dq + Q dp + (H Hdt (8.33 p Λ 2 q, Q Λ 2 P, H H + Λ t (8.34

87 P det( 2 Λ 2 / P ν 0 Λ 3 (p, Q, t Λ 1 p q dλ 3 q d p P dq + (H Hdt (8.35 q Λ 3 p, P Λ 3 Q, H H + Λ 3 (8.36 t det( 2 Λ 3 / Q ν p ν 0 Λ 4 (p, P, t Λ 1 p q + P Q dλ 4 q d p + Q dp + (H Hdt (8.37 q Λ 4 p, Q Λ 4 P, H H + Λ 4 t (8.38 det( 2 Λ 4 / P ν p ν 0 q p (q, p 8.5 ϵ Λ 2 (q, P q ν P ν + ϵg(q, P (8.39 p Λ 2(q, P q Q Λ 2(q, P P G(q, P P + ϵ q q + ϵ G(q, P P (8.40 (8.41 G(q, p Q q + δq δq ν ϵ ϵ[q ν, G] (8.42 p ν P p + δp δp ν ϵ G(q, p p ν ϵ[p ν, G] (8.43 δq, δp ϵ G P p G(q, p F(q, p

88 84 8. δf F(q + δq, p + δp F(q, p F F δq + q p δp ( F G F G p ν p ν ϵ[f, G] (8.44 Hamiltonian δh 0 [H, G] 0 G(q, p N ( r (1, r (2,, r (N ( p (1, p (2,, p (N n n P N i1 p(i n r i r i + ϵn (8.45 p i p i (8.46 G n P n N p (i (8.47 i1 n L N i1 r(i p (i n r i r i + ϵn r i (8.48 p i p i + ϵn p i (8.49 G n L n N r (i p (i (8.50 i1 Hamiltonian n n P n n L dq ν dt H p ν [q ν, H], dp ν dt ϵ q ν p ν H [p ν, H] (8.51 δq ν ϵ[q ν, H], δp ν ϵ[p ν, H] (8.52 Hamiltonian

89 t Hamiltonian t t s a b 2 f (q(s, q(s (q(a, p(a (q(b, p(b (Q(s, P(s (Q(a, P(a (Q(b, P(b b p dq p(s dq(s b ds ds, P dq P(s dq(s ds (8.53 ds a a p dq P dq b a dλ 1 ds ds Λ 1(q(b, Q(b Λ 1 (q(a, Q(a 0 (8.54 p dq P dq (8.55 (q, p (Q, P p dq P dq Λ 1 (q a, p a (q b, p b C 1, C 2 (Q a, P a (Q b, P b C 1, C 2 p dq P dq p dq P dq C 1 C 1 C 2 C 2 p dq P dq C 1 +C 2 C 1 +C 2 p dq P dq 0 (8.56 C C Λ 1 p dq P dq (8.57 C 1 Λ 1 (q, Q dλ 1 (q, Q p dq P dq Λ 1 (8.55 (q, p (Q, P C 1

90 86 8. (u, v a u b, v c v d 2 f (q(u, v, p(u, v S p dq ( b b ( a a b p ν u a du + vc d du dv c v d du dv c d c ( qν u p ν v ( p ν u dv + ub + p ν v p ν u b a v a b p ν v d du dv c u du + vd ( p ν v c d p ν v S *2 wedge dv ua (8.58 dx 1 dx 2 dx 2 dx 1 (8.59 dx 1 (c 2 dx 2 + c 3 dx 3 c 2 dx 1 dx 2 + c 3 dx 1 dx 3 (8.60 n (x 1, x 2,, x n n wedge dx 1 dx 2 dx n n dx dx dx dx 0 v dv ( qν dq ν dp ν u du + ( qν p ν u v p ν u ( pν v u du + p ν v dv du dv (8.61 du dv dv du, du du du du 0, dv dv dv dv 0 du dv (u, v (8.58 p dq S dq ν dp ν (8.62 (8.58 (8.55 Lagrange {u, v} q,p ( qν u p ν v p ν u v (8.63 {u, v} q,p {u, v} Q,P (8.64 *2

91 8.7 Poisson 87 wedge dq ν dp ν dq ν dp ν (8.65 (8.55 (q, p (Q, P (8.64 (8.65 (8.65 f wedge dq ν dp ν dq ν dp ν dq ν dp ν f! dq 1 dp 1 dq 2 dp 2 dq f dp f (8.66 dq 1 dp 1 dq 2 dp 2 dq f dp f dq 1 dp 1 dq 2 dp 2 dq f dp f ( f Liouville D D dq 1 dp 1 dq 2 dp 2 dq f dp f dq 1 dp 1 dq 2 dp 2 dq f dq f (8.68 D D D D 2 f 8.7 Poisson Lagrange 2 f x, X 2 f 2 f ˆ J x q 1 q 2... q f p 1 p 2... p f, X Q 1 Q 2... Q f P 1 P 2... P f ( J ˆ 0 ˆ1 ˆ1 0 ˆ1 f f Lagrange (8.69

92 88 8. {u, v} q,p {u, v} Q,P ( qν u ( Qν u p ν v p ν u P ν v P ν u v Q ν v ( T x J ˆ x u u ( T X J ˆ X u u (8.70 (8.71 M µµ X µ x µ (µ, µ 1, 2,, 2 f (8.72 ˆM dx ˆMdx ( X x u ˆM u (8.73 (8.64 ˆM T ˆ J ˆM J (8.74 ˆM symplectic ˆM J ˆ ˆM 1 Jˆ symplectic ˆ J 2 1 ˆM ˆ J ˆM T ˆ J (8.75 x X ˆM symplectic ( Qν ( Qν Q ν p ν P ν p ν ( Pν Q ν p ν ( Pν P ν p ν Q ν Q ν p ν P ν Q ν p ν Q ν P ν p ν P ν P ν p ν Poisson 0 (8.76 δ ν,ν (8.77 δ ν,ν ( (8.79 [Q ν, Q ν ] 0, [P ν, P ν ] 0, [Q ν, P ν ] δ νν (8.80 (q, p (Q, P

93 8.7 Poisson 89 Poisson [u, v] q,p ( u v v u p ν p ν ( T u J ˆ v x x (8.81 u x µ 2 f µ 1 u X µ X µ x µ µ M µ µ u X µ ( u x ˆM T u X (8.82 [u, v] q,p ( T u J ˆ v ( T u x x ˆM Jˆ ˆM T v X X (8.83 x X ˆM symplectic [u, v] q,p ( T u J ˆ v X X [u, v] Q,P (8.84 Poisson Poisson

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.

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. 1.1 1. 1.3.1..3.4 3.1 3. 3.3 4.1 4. 4.3 5.1 5. 5.3 6.1 6. 6.3 7.1 7. 7.3 1 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

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