Chapter Fgure.: x x s T = 2 mv2 mgx = 0 (.) s = X 0 x 0 x x v = 2gx + + ( ) 2 2 y x 2 ( ) 2 2 y x 2 /2gx (.2) y(x) 2
. S = L(t, r, ṙ) r(t) ṙ = r L(t, r, ṙ) t, r, ṙ L x t r, ṙ r + δr, ṙ + δṙ δs δs = δr + δṙ (.3) r ṙ r, ṙ δs = 0 r, ṙ r, ṙ δr + r ṙ δṙ = 0 r, ṙ = 0 δṙ = δr t δr + r ṙ δṙ = 0 [ ] ṙ δr + r δr ṙ δr = 0 ( r ) δr = 0 ṙ δr ().2 r ṙ = 0 (.4) equaton.4 S T, t x, r y equaton.2 ( ) 2 2 y L = + x 2 /2gx y x ẏ x ẏ ẏ ẏ 2gx( + ẏ2 ) = 0 = 0 = C = C ẏ 2 C 2 = 2gx( + ẏ 2 ) ẏ 2 = 2gC 2 x 3 toppage
C 2 = ẏ 2 = y x = x y 4ga a ( ) 2 x = 2a y x (.5) y = a(θ sn θ) (.6) x = a( cos θ) (.7) Fgure.2: equaton.5 equaton.5 x y = x θ θ y = x θ ( ) 2 x = y = = = { x θ ( ) y θ ( ) } 2 y θ a 2 sn θ 2 a 2 ( cos θ) 2 cos θ2 ( cos θ) 2 ( cos θ)( + cos θ) ( cos θ) 2 = + cos θ cos θ 2a x = 2 cos θ = + cos θ cos θ 4 toppage
5 toppage
Chapter 2 2. equaton.4 L(r, ṙ) = T (ṙ) U(r) T U F r ṙ ṙ T (ṙ) ṙ T (ṙ) = 2 mṙ2 F = U r = 0 = r = U(r) r m 2 r 2 = F S = L(t, r, ṙ) r, ṙ F = U r S = L(t, r, ṙ) (2.) L(t, r, ṙ) = T (ṙ) U(r) S m r, ṙ 6
S = (T(ṙ) U(r)) T (ṙ) U(r) T (ṙ) + U(r) L L(t, r, ṙ) = T (ṙ) U(r) S L L(t, r, ṙ) = T (ṙ) U(r) F = U r U(r ) = k V ( x x k ) L = 2 m ẋ 2 V ( x x k ) (2.2) k L L L 2.2 x,y,z L (r, θ) m M G T = 2 mv2 = 2 m(ṙ2 + (r θ) 2 ) L(r, ṙ) = T (ṙ) U(r) U(r) = GmM r L(r, ṙ) = 2 m(ṙ2 + (r θ) 2 ) + GmM r 7 toppage
r θ = ṙ r (mṙ) = mr θ 2 GmM r 2 m 2 r 2 = mr θ 2 GmM r 2 (2.3) θ = θ (mr2 θ) = 0 (2.4) equaton2.4 8 toppage
Chapter 3 3. L(q, q) q q L(r, ṙ) = 2 m(ṙ2 + (r θ) 2 ) + GmM r θ = 0 q q L(q, q) q = 0 q q equaton2.4 3.2 q (q, t) = q (t) + εs (q, t) ε S (q, t) L(q, q) q(t) q (q, t) S ((q, t)) = 0 (3.) q 9
q (q, t) = q (t) + δq δq q t ε δq = εs (q, t) L(q, q) q(t) q (q, t) L(q, q) q(t) q (t) L = 2 mẋ2 x = x + ε L = 2 mẋ 2 (3.2) = 2 m (x + ε) (3.3) = 2 mẋ2 (3.4) δl = L(q, q ) L(q, q) L(q, q ) = L(q + δq, q + δ q) = L(q, q) + δq + δq q q δl = δq + δq q q δl = δq = ( ) δq δq q q ( q ( ) q ( )) δq + ( ) δq q q equaton.4 0 δl = ( ) δq q 0 toppage
δl = 0 ( ) δq = 0 q δq = εs (q, t) ε equaton3. ( ) S (q, t) = 0 q 3.3 3.3. x x = x + a L = ( ) 2 2 m x V ( x x k ) (3.5) L = ( ) 2 2 m x V ( x x k ) (3.6) = ( ) 2 2 m (x + a) V ( x + a (x k + a) ) (3.7) = 2 m ( x ) 2 V ( x x k ) = L (3.8) a = εs, S = (,, ) ( ) S (q, t) = 0 (3.9) q 3.3.2 (mv x + mv y + mv z ) = 0 (3.0) mv = Const (3.) x x = x cos θ + y sn θ y cos θ x sn θ z ε 0 sn ε = ε, cos ε = x x = x + εy y εx (3.2) z toppage
ẋ 2 ẋ 2 = (ẋ + εẏ) 2 + (ẏ εẋ) 2 + ż 2 (3.3) = ẋ 2 + ẏ 2 + ż 2 + (εẏ) 2 + (εẋ) 2 = ẋ 2 x x k x x k = {x + εy (x k + εy k )} 2 + {y εx (y k εx k )} 2 + (z z k ) 2 = (x x k ) 2 + ε 2 (y y k ) 2 + (y y k ) 2 + ε 2 (x x k ) + (z z k ) 2 = x x k L = 2 mẋ 2 V ( x x k ) = L k δq = x x = εy εx 0 = εs S = y x 0 ( ) S (q, t) = 0 (3.4) q m(yẋ xẏ) = 0 (3.5) m(x ẋ) z = 0 (3.6) m(x ẋ) z = Const (3.7) 3.3.3 S L S L t = t + ε q(t), q(t) L(q(t ), q(t ), t ) = L(q(t), q(t), t) L q(t), q(t) q (t ) = q(t ε) = q(t) (3.8) q (t ) = q(t ε) = q(t) (3.9) 2 toppage
S L t q q L = Const (3.20) http://hooktal.sakura.ne.jp/analytc/energydefntonconservaton/ L = 2 m ( ) 2 x V ( x x k ) q L = ( m ẋ 2 q 2 m ẋ 2 ) V ( x x k ) (3.2) = 2 m ẋ 2 + V ( x x k ) (3.22) 3 toppage