m r = F m r = F ( r) m r = F ( v) F = F (x) m dv dt = F (x) vdv = F (x)dx d dt = dx dv dt dx = v dv dx

m r = F m r = F ( r) m r = F ( v) x F = F (x) m dv dt = F (x) d dt = dx dv dt dx = v dv dx vdv = F (x)dx

2 mv2 x 2 mv2 0 = F (x )dx x 0 K = 2 mv2 W x0 x = x x 0 F (x)dx K K 0 = W x0 x x, x 2 x K 2 K = W x x 2 x x 0 x (x) = x 0 F (x )dx x, x 2 x x2 x x2 x x 0 F (x)dx = F (x)dx F (x)dx = (x ) (x 2 ) x 0 K 2 K = 2 K + = K 2 + 2

E = K + (x) = 2 mv2 + (x) z m F = mgˆk z0 < z 2 mv2 2 mv2 0 = z z 0 mgdz = mg(z 0 z ) z z 0 = h v = 0 h = v2 0 2g F ( r) c = r(t) n r () r (2) n s i,i+ r (i) r (i+) r (i) = r (i+) r (i) r (i) F ( r(i) ) r (i) W F ( r () ) r () + F ( r (2) ) r (2) + = n i= F ( r (i) ) r (i) n n F ( r (i) ) r (i) n i= c F d r

F....... r.......... F r. ri + r. i r n. ri- F F c = r(t) W = F d r. t c = r(t) c F d r = tn t c F d r tn ( r) dt d t = F ( r) v dt t t t n c m d v dt = F ( r)

d r m d v dt d r = F ( r) d r d r = vdt m d v v = F ( r) d r P, P 2 2 m v 2 2 P2 2 m v 2 = F ( r) d r P P P 2 P2 W 2 = F ( r) d r P K 2 K = K = 2 m v 2 2 F d r = W 2 F m l ϕ 0 B = m g N r = l dr = 0 d r = ê r dr + ê ϕ rdϕ = êϕldϕ

N N d r = 0 W = B d r g d r = g ê ϕ l dϕ = gl (ϕ π 2 ) ϕ W = mgl ϕdϕ = mgl( ϕ 0 ϕ) ϕ 0 v 0 = 0 2 mv2 = mgl( ϕ 0 ϕ) v = 2mgl( ϕ 0 ϕ) N B=mg

ê r v 0 = v 0r ê r + v 0ϕ ê ϕ v = v r ê r + v ϕ ê ϕ ê ϕ v x v x = 0ê r + v ϕx ê ϕ = v ϕx ê ϕ K x K x K 0 = F d r K = 2 m v 2 F = G Mm r 2 êr R r O

d r = ê r dr + ê ϕ rdϕ R r r F d r = GMm R ( dr = GMm r2 r ) R 2 m v 2 ( 2 m v 0 2 = GMm r ) R v 2 v 0 2 = 2 GM ( ) ( ) R R R 2 R r 2gR r v 0 2 = v 2 0r + v2 0ϕ v x 2 = v 2 ϕ x ( ) R vϕ 2 x (v0r 2 + v0ϕ 2 ) = 2gR r x r x v ϕx r x L = r p = mrv ϕˆk ˆk ê r, ê ϕ r = R r = r x L = mrv 0ϕ = mr x v ϕ v ϕ v ϕ = R r x v 0ϕ R 2 ( ) R rx 2 v0ϕ 2 (v2 0r + v0ϕ 2 ) = 2gR r x

R r x r x g, R h r x = R + h R r x = R R + h h R, R 2 (R + h) 2 2 h R h = v2 0r 2g v 0ϕ = 0 v2 0ϕ gr h = v2 0r 2g

F = yî + axĵ (0, 0) (, ) (xy) x : (0, 0) (, 0) y : (, 0) (, ) x = y x = y dx = dy, d r = îdx + ĵdy = (î + ĵ)dx F = x(î + aĵ) W O A (C ) = 0 x(î + aĵ) (î + ĵ)dx = + a 2 y = 0, dy = 0 F d r = 0 x = dx = 0, dr = ĵdy F = yî + aĵ W O A (C 2 ) = a W O A (C ) W O A (C 2 ) 0 dy = a a =

a F = yî+xĵ F F W U(r) = r r 0 F d r r 0 r r, r 2 U(r ) U(r 2 ) = r2 r F d r U(r ) U(r 2 ) = K 2 K U(r i ) = U i U + K = U 2 + K 2 E = K + U = 2 mv2 + U F ϵξ

P 2 P F ϵξ d r = 0 c a c b P 2 P 2 F ϵξ d r = Fϵξ d r + Fϵξ d r c a,p 2 c b,p 2 Fϵξ d r c b P 2 = Fϵξ d r c b,p 2 Fϵξ d r c b,p 2 = Fϵξ d r c a,p 2 f m f = µmg W = µmgl l U( r) r 0 F

U r 0 r r U( r) = F d r r 0 F ( r) = U( r) U( r) F = G Mm r 2 êr r ( U( r) = F d r = GMm ) r 0 r 0 r 0 r 0 = U r = r = x 2 + y 2 + z 2 U(r) = G Mm r U(r) U(r) = k G Mm r = k r = k GMm

x 2 + y 2 + z 2 = c 2 U(r) U(r) F ( r) = U(r) h R R = mgh r = R + h r = R + h ( h ) R R (r) = U(r) U(R) = G Mm r + G Mm R GMm R 2 h = mgh m F δ F µδ F = F δ + F µδ W 2 = = 2 2 F d r = K 2 K 2 F δ d r + F µδ d r

2 F δ d r = U U 2 W µδ, 2 = 2 F µδ d r E = K 2 + U 2 (K + U ) = W µδ, 2 ϕ v 0 µ B = mg N = mg ϕ f = µ N E = W τρ. K = 2 mv2 0, U = mgh K 2 = 2 mv2, U 2 = 0 f N h B s

E = (K + U) = 2 m(v2 v0) 2 mgh s W τρ. = f d r = fs = µmg ϕ s s ϕ = l = h ϕ E = W τρ. v 2 v 2 0 = 2gh( µ ϕ) µ ϕ < v 2 = v 2 0 + 2gh( µ ϕ) µ ϕ = v = v 0 µ ϕ > h c = v 2 0 2g(µ ϕ ) ( hhc ) v 2 = v 2 0 h < h c h > h c (x) 3 4 5 2 x

x W 2 = 2 F x dx F x U(x) = F x dx F x = du dx U(x) h = mgh U = 0 U = 0 U < 0 U = 0 U > 0 E > U K = E U /r 2 /r

(r) = a r 6 + r 0 < r < r 0 U r b r 2 (r 0 ) = 0 r 0 = 6 2a b (r 0 ) = a2 4b a = b = E > 0 r min 0 < E < U E = U E < 0 r min, r max E = U r 0 U(r 0 ) = U min E > 0 E < 0

r 0.2 0..0.2.4.6 r 0. 0.2 s K = 2 mṡ2 F = (F t, F n ) F t F t U = F t ds E = K + U F ( r) = f( r) ê r f( r) r f( r)

f( r) = f(r) F ( r) = A r 2 êr A = Gm m 2 kq q 2 F = U U U = ê r r + ê U θ r θ + ê U ϕ r θ ϕ ê r U θ = U ϕ = 0 U r U = U(r) F = du(r) ê r = F dr (r) F = f(r)ê r

M ρ( r) M i M = i M i M i r i M i m R F i = G mm i R r i ( R r i ) = G mm i 3 r i R ( r 3 i R) F = i G mm i r i R ( r 3 i R) i M i dm = ρd r i r F = Gm r R r R 3 ρ( r)d g = F /m g = G r R r R 3 ρ( r)d dm du = G mdm r R U( R) = Gm ρ( r) r R d (r)

U( R) F ( R) = R U( R) = Gm ρ( r) R r R d (r) ( r = Gm ρ( r) ) R r R d (r) 3 R R r m u( R) = U( R)/m u( R) = G ρ( r) r R d (r) u( R) S F S Φ F = F ds S U( R) = c S S U

s R = R(s) du ds = 0 du ds = U ê t ê t U ê t = 0 U S U ê t F ( R) = U( R) F F S M ρ( r) Φ = F ds F = GMm r r 3 S Φ = GMm S r r 3 d S r Ω = r 3 d S = 4π S

Φ = 4πGMm F ds = S F d F d = 4πGMm M M = ρ( r)d F = m g( r) m ( g + 4πGρ( r)) d = 0 g = 4πGρ( r) g( r) = u( r) 2 u( r) = 4πGρ( r), r 0 2 u( r) = 0, r / 0

x = (3 t 2 )at, y = 3at 2, z = (3 + t 2 )at t = [0, t 0 ] k = τ = 3a ( + t 2 ) 2 F = 2xyî + x 2 ĵ m y = x 2 (0, 0) (, ) y = x F = kx k x (x, y) = (0, 0) v 0 y = y(x) 30 40 0 5 0 20 0 0 40 30 20 0 0

m F = f(r)ê r F 2 = k v L 0 L = L 0 e kt/m v = ± m 2 m m + m 2 g r, r r r r r 3 r r r r r r r, r r r r r 3, 2 r r r ρ( r) P oisson 2 u( r) = 4π G ρ( r) ρ( r) = A e r r Laplace P oisson g( r) = F /m ρ( r) r g( r) = 4π G ρ( r)

r r r g( r) = G r r 3 ρ( r ) d r M r r r ( r ) r r g( r) = G r r r 3 ρ( r ) d { } 2 r r ρ( r ) d = 4πρ( r) r 2 r r r = 4πδ3 ( r r ) δ 3 ( r r )ρ( r ) d = ρ( r) r r r = r ; R xy z R M σ

m r { G mm U(r) = r G mm R r > R r < R m ρ(r) r m U(r) = G mm r, r > R U(r) = G mm(r) r Gm4π R r ρ(r )r dr, r < R M(r) r F (r) = G mm(r) r 2 m R ; N m i U = Mgz cm M z cm v 0 = 0

z a σ 0 ( ) U(z) = 2πG N σ 0 z 2 + a 2 z R z F π = G Mm (xî + yĵ 2 zˆk) R3 a) ρ( r) b) x 2 + y 2 + (z ) 2 < 2 z m 0 Ioannina Athens r R

m, m 2 v, v 2 v, v 2 m v + m 2 v 2 = m v + m 2 v 2

U(r) = G m m 2 r r r 2 F ( r r 2 ) 0 K 2 = K 2 m v 2 + 2 m 2v2 2 = 2 m v 2 + 2 m 2v 2 2 Q K 2 = K + Q 2 m v 2 + 2 m 2v2 2 = 2 m v 2 + 2 m 2v 2 2 + Q m m 2 2 r R r 2 2

Q > 0 Q = 0 Q < 0 v, v 2 v, v 2 m v + m 2 v 2 = m v + m 2 v 2 2 m v 2 + 2 m 2v2 2 = 2 m v 2 + 2 m 2v 2 2 + Q Q = 0 m (v v ) = m 2 (v 2 v 2 ) m (v 2 v 2 ) = m2 (v 22 v 2 2 ) v v 0 v 2 v 2 0 v + v = v 2 + v 2

v 2 = v + v v 2 v = (m m 2 )v + 2m 2 v 2 m + m 2 v 2 = (m 2 m )v 2 + 2m v m + m 2 v = m v + m 2 v 2 (m m 2 (v v 2 )) + 2 m m 2 (m + m 2 )Q m + m 2 m 2 (m + m 2 ) v 2 = m v + m 2 v 2 (m m 2 (v v 2 )) 2 m m 2 (m + m 2 )Q m + m 2 m (m + m 2 ) v = v 2 (m m 2 (v v 2 )) 2 m m 2 (m + m 2 )Q = 0 Q = m m 2 (v v 2 ) 2 m + m 2 M = m + m 2 v = m v + m 2 v 2 m + m 2 M m R > r h m = m 2 v = v 0, v 2 = 0 m v 0 + 0 = m( v + v 2) 2 mv2 0 = 2 mv 2 + 2 mv 2 2

v 0 v 0 = ( v + v 2) ( v + v 2) v 2 0 = v 2 + v 2 2 + 2 v v 2 v v 2 = 0 R = m r + m 2 r 2 = m r + m 2 r 2 m + m 2 M = m v + m 2 v 2 M v v 2 v c = v m 2 = ( v v 2 ) m + m 2 v 2c = v 2 = m m + m 2 ( v v 2 ) r r 2

p c = m v c = m m 2 m + m 2 ( v v 2 ) p 2c = m 2 v 2c = m m 2 m + m 2 ( v v 2 ) µ = m m 2 m + m 2 p c = µ ( v v 2 ) p 2c = µ ( v v 2 ) p = p c + p 2c = 0 p = m v + m 2 v 2 = (m + m 2 ) = constant v 2c, v c v 2c, v c

v ic, v ic v 2c = m m 2 v c v 2c = m m 2 v c 2 m vc 2 + 2 m 2v2c 2 = 2 m v c 2 + 2 m 2v 2c 2 ( ) ( ) m + m2 vc 2 = m + m2 v 2 c m 2 m 2 v c = v c, v 2c = v 2c ` 2c c 2c ` c c 2 2c 2

v c = m 2 m v 2c, v c = m 2 m v 2c v 2c = 0 m = v m + m 2 m 2 v c = v m + m 2 v 2c = v c v c Θ v = + v c θ v 2, v 2c θ = v c Θ + v c Θ = v c = v c = m 2 m + m 2 v, = Θ /v c + Θ m m + m 2 v m 2c c

/v c = m /m 2 Θ θ = m m 2 + Θ m m 2 < m m 2 + Θ θ (, ) θ m m 2 m m 2 > + Θ θ θ = θ max v v c θ max = v c v, 2 = v 2 + v 2 c ` ` c c θ m m 2 < ` ` c max c θ max m m 2 >