Mecanica Celeste Una Introduccion

8/18/2019 Mecanica Celeste Una Introduccion.

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n









n





M

m

M

m x ∈ R3 \ 0

M

x = (x1, x2, x3)

mM

|x

|

|F | = GM m

|x|2 ,

G

F = −GMm

|x|2x

|x| .

F m M

m M m M m

M

m t x(t) ∈ R3

t

F = mx(t).

F x(t)

x = −GM x

|x|3 ,

m

x = f (|x|) x

|x| ,



f (r) = − µr2 µ = GM

M >> m m

M m M

m

F : R3/0 → R3

F (x) = f (|x|) x

|x| ,

f : (0,∞

)→

R

f > 0

f < 0

f F

f F

x(t)

x(t)

x = f (|x|) x

|x| .

f (r) = − µr2 x = (x1, x2, x3) ∈ R

3

m

xi = −µ

xi

(x21 + x2

2 + x23)3/2 i = 1, 2, 3.

x(t) = aeiθ,

a > 0

θ = θ0 + ωt ω = µ1/2

a3/2.

µ = 1

x(t) = (cos t, sen t) (θ0 = 0, a = 1),

y(t) = 2(cos t√

8, sen

t√ 8

) (θ0 = 0, a = 2).

y(t)

x(t)

z(t) = y(t) − x(t)

M



x = f (|x|) x

|x| x ∈ R3 \ 0.

x = x(t) x = x(t + c) c ∈ R

y(t) = (cos(t + c), sen(t + c), 0) t ∈ R,

x = − x

|x|3 .

x = x(t)

x = f (|x|) x

|x| x ∈ R3 \ 0,

x(−t)

−t

x(−t)

dx

dt = −x(−t)

d2x

dt2 = x(−t).

R3

A ∈ M 3(R), AAT = AT A = I 3.

O(3) = A ∈ M 3(R) : A

A

A

|Ax| = |x| ∀x ∈ R3.

detA > 0

detA < 0

x : I → R3 \ 0

x = f (|x|) x

|x| ,

A ∈ O(3)

y : I −→ R3 \ 0t −→ y(t) = Ax(t)





x

|x|

d

dt

x

|x|

=x|x| − x( xx

|x| )

|x|2 = 0.

v ∈ R3

x/|x| ≡ v I |v| = 1

x(t) = |x(t)|v,

t ∈ I

x = (x1, x2, 0)

xi = µ xi

|x|, i = 1, 2.

π x y

I = [t0, t1] α : I → R2 C k k ≥ 1

α(t) = 0,

t ∈ I r, θ : I → R

C k r(t) > 0

α(t) = r(t)(cos θ(t), sen θ(t)),

t ∈ I r(t) θ(t) 2hπ h ∈ Z 2π

|α(t)| = r(t) |α(t)| = 1

α(t)

θ0

α1(t0) = cos θ(t0)

α2

(t0

) = sen θ(t0

)

θ(t)

θ(t0) = θ0

i)

α1(t) = −θ sen θ.

ii)

α1(t)α2(t) = −θ sen2 θ.

ii)

α2(t)α1(t) = θ cos2 θ.



α

2(t)α1(t) − α

1(t)α2(t) = θ

cos2

θ + θ

sen2

θ = θ

(cos2

θ + sen2

θ) = θ

. t1

t0

θ(s)ds =

t1

t0

(α2(s)α1(s) − α1(s)α2(s))ds,

θ(t) = θ0 +

t1

t0

(α2(s)α1(s) − α1(s)α2(s))ds,

θ0 = θ(t0) θ(t)

(x(t), y(t)) : I → R2 \ 0

(x(t), y(t)) = (cos θ(t), sen θ(t)),

θ(t) (x, y) x = −θy

y = θx

x(t0) = α1(t0)

y(t0) = α2(t0).

(α1(t), α2(t))

|α(t)| = 1

α1α1 + α2α2 = 0.

θ

α1α2 − α1α2 = θ.

α1α

1 + α2α

2 = 0

α1α2 − α1α2 = θ,

α1 = −θα2, α2 = θα1.

(α1, α2) (x, y)

α1 = cos θ, α2 = sen θ.

α(t) = r(t)(cos θ(t), sin θ(t))

t ∈ [t0, t1]

C 1

θ(t) > 0 t ∈ [t0, t1], θ(t1) − θ(t0) < 2π.

Ω = sα(t) : t ∈ (t0, t1), s ∈ (0, 1).

|Ω| = 1

2

t1

t0

r2(s)θ(s)ds,

|Ω| Ω



θ > 0 θ(t1) =θ(t0) + 2π α(t)

x : I → R2/0

x = f (|x|) x

|x| .

x(t) = r(t)cos θ(t) + ir(t)sin θ(t) = r(t)eiθ(t) t ∈ I.

dx

dt = r(t)eiθ(t) + r(t)iθ(t)eiθ(t)

d2x

dt2 = r(t)eiθ(t) + 2r(t)iθ(t)eiθ(t) + r(t)iθ(t)eiθ(t) − r(t)θ2(t)eiθ(t).

r(t)eiθ(t) + 2r(t)iθ(t)eiθ(t) + r(t)iθ(t)eiθ(t) − r(t)θ2(t)eiθ(t) = f (r)eiθ(t).

r − rθ2 = f (r)

2rθ + rθ = 0.

r

2rrθ + r2θ = 0,

(r2θ) = 0,

t ∈ I

r2θ = J,

J

c = x(t) ∧ x(t) = c,

c x(t) x(t) R3

x3 = 0

c = r2θe3 e3 = (0, 0, 1).

c = J e3 |c| = |J |



x(t) F (x) = f (|x|) x|x|

[t0, t1] Ω(t)

x(t0) x(t)

d

dt(|Ω(t)|) =

1

2J,

c = J e3

d

dt(|Ω(t)|) =

d

dt

1

2

θ(t)

θ0

r2(s)ds

=

1

2r2θ(t) =

1

2J.

x = x(t)

x = f (|x|) x

|x| , x ∈ R3 \ 0.

c 0

x(t)

x(t) = |x(t)|v, |v| = 1.

r(t) = |x(t)| x(t) r(t)

r = f (r), r > 0.

f (r) < 0 r ∈ (0, ∞)

r(t) < 0 r(t)

x(0) = 0 r(0) = 0

x(t)

f (

|x

|)

x

|x| f : (0, +∞) → (−∞, 0)

x = f (|x|) x|x|

x(0) = x0

x(0) = 0

x = 0

x = 0



r(t)

r

= f (r)r(0) = r0r(0) = 0,

r0 = |x(0)| > 0

r(t)

(α, w) [0, w) (α, 0]

τ = −t

f (r) < 0 r(0) = 0 r(t) < 0 t [0, w) r(t)

0 < r(t) < r0 = r(0)

t ∈ (0, w)

t1 ∈ (0, w)

r(t)

r(t) < r1 + r

1(t − t1)

t > t1 r1 = r(t1), r1 = r (t1) < 0 0 < w < ∞ w = +∞

lımt→∞

r(t) = −∞,

rw := lımt→w−

r(t), rw := lımt→w−

r(t),

r(t) r(t)

rw = lımt→w−

r(t) =

w

0

f (r(s))ds,

rw > −∞

rw rw rw = 0

r(w) = rw r(w) = r w r(t) [0, w1) w < w1

[0, w)

x(t)

I = (α, ω)

x = 0 −∞ < α < ω < +∞ lımt→α+ r(t) =lımt→ω− r(t) = 0 t0 ∈ (α, ω) r(t) > 0 α < t < t0 r(t) < 0

t0 < t < ω

x = 0 ∞ −∞ < α < w = ∞ lımt→α r(t) = 0

lımt→+∞ r(t) = +∞ r(t) > 0 t ∈ (α, +∞)

∞ −∞ = α < ω < +∞ lımt→−∞ r(t) = +∞

lımt→ω r(t) = 0 r(t) > 0 t ∈ (−∞, ω)

x(t)

x(t) = r(t)v,

r(t)

r(t) = f (r(t))r(t0) = r0r(t0) = r0



t0 ∈ I r0 = 0

t0 I

r0 = 0 r0 > 0

[t0, ω)

t1 > t0 r(t1) = 0 r(t) = 0 ∀t > t0

r(t) = 0

0 < r(t) < r0

r(t) r(t)

lımt→ω

r(t) = rω ≤ +∞, lımt→ω

r(t) = rω ≥ 0.

ω = +∞ lımt→ω r(t) = ∞

lımt→ω(r(t) + r(t)) = +∞,

ω < ∞

r(t) < r0 + r0(t − t0) ∀t > t0.

lımt→ω

r(t) = rω < ∞.

r(t) = r0 +

t

t0

f (r(s))ds ⇒ rω = r0 +

ω

t0

f (r(s))ds < ∞.

ω = +∞

rω < ∞

rω − δ < r(t) < rω

t > tδ

f (rω) − ε < f (r(t)) < f (rω) + ε

ε

f (rω) ± ε < 0

r(t) = r0 +

tδ

t0

f (r(s))ds +

t

tδ

f (r(s))ds.

(f (rω) − ε)(t − tδ) <

t

tδ

f (r(s))ds < (f (rω) + ε)(t − tδ)

t > tδ lımt→∞ r(t) = −∞

0 ≤ rω < r 0.

α > −∞ lımt→α+ r(t) = 0

r(t) > 0

t ∈ (α, t0]

0 < r(t) < r0 + r0(t − t0),



t ∈ (α, t0] α > −∞

lımt→α+ r(t) := rα > 0

rα = lımt→α+ r(t) = r0 + α

t0f (r(s))ds < ∞

r(t) = f (r(t))r(α) = rα

r(α) = rα

r(t) [α − ε, α]

r0 > 0

r0 < 0 ϕ(t) = r(−t)

ϕ

(t) = ϕ(t)ϕ(−t0) = r0ϕ(−t0) = −r0 > 0

r(t) = ϕ(−t)

Ω RN

F : Ω → RN

V : Ω → R

C 1 F

F = −∇V Ω.

V (x)

F E V (x) + E F Ω F

F (x) = f (|x|) x

|x| x ∈ R3 \ 0,

f : (0, +∞) → R V 0 : (0, +∞) → R f

V 0(r) = r

1

f (s)ds.

V (x) = −V 0(|x|) F

∇V = −f (|x|) x

|x| = −F.

F (x) = − µ

|x|2x

|x|



V (x) =

− µ

|x|,

F : Ω ⊂ RN → R

N V (x)

m x(t)

mx = F (x), x ∈ Ω.

E = 1

2mx(t)2 + V (x(t))

t

E m

F = −∇V x

mx, x + ∇V, x = 0.

dE dt E

12

mx2 T

M

E = 1

2x(t)2 − µ

|x| ,

µ = GM

x(t)

x = −µ x

|x|3 , x ∈ R20,

c = 0 I = (α, w) E

x = x(t)

E < 0

r(t) = |x(t)|

E ≥ 0 r(t)

t0

E = 1

2x(t0)2 − µ

|x(t0)| = − µ

|x(t0)| < 0, µ > 0.

r(t) r(t)

(0, ∞)

E = 1

2x(t)2 − µ

|x(t)| ≥ − µ

|x(t)| , t ∈ (α, w),

E ≥ 0



M

r1(t), r2(t) ∈ R3

C (t) = (1 − γ )r1 + γr2 γ ∈ (0, 1),

γ = m2

m1 + m2=

m2

M

1 − γ = m1

m1 + m2=

m1

M ,

M = m1 + m2

m1r1 = Gm1m2r2 − r1

|r2 − r1|3m2r2 = Gm1m2

r1 − r2|r2 − r1|3 .

C

(t) = (1 − γ )r

1 + γr

2 = 0,

C (t)

r1 = r1 − C (t) r2 = r2 − C (t).

r1 + C (t) = Gm2r2 − r1

|r2 − r1|3

r2 + C (t) = Gm1r1 − r2

|r2 − r1|3 ,



C (t) = 0 C (t)

r1 = Gm2 r2 − r1|r2 − r1|3

r2 = Gm1r1 − r2

|r2 − r1|3 .

m1

m2

m1

γ = m2

m1 + m2=

m2

m1

1 + m2

m1

≈ 0

1 − γ = 1 −m2

m1

1 + m2

m1

≈ 1.

C (t) ≈ r1 m1 >> m2

r1 = −γ (r2 − r1)r2 = (1 − γ )(r2 − r1).

(r2 − r1) = −GM r2 − r1|r2 − r1|3 .

z = r2 − r1 = r2 − r1

z = −M G z

|z|3

.

r1 = −γz r2 = (1 − γ )z

r1 = C (t) + r1

r2 = C (t) + r2

z

R3

z3 = 0

µ = M G

p(x, y)

x A, O ∈ R2

d1 = d(x, O) d2 = d(x, A)

d1 + d2 = c,



c > 0

|O

−x

|+

|A

−x

|= c

⇒ |A

−x

|= c

− |x

|.

|A − x|2 = (c − |x|)2 ⇒ |A|2 − 2A, x + |x|2 = c2 − 2c|x| + |x|2

⇒ 2c|x| − 2A, x = c2 − |A| ⇒ |x| + −A

c , x =

c2 − |A|22c

.

e = −A

c , k =

c2 − |A|22c

|A| < |x| + |A − x| = c

k > 0

|e| < 1.

|x| + e, x = k.

c

x A O

d2 − d1 = c,

c > 0

|A − x| − |O − x| = c ⇒ |A − x| = c + |x|.

|A − x|2

= (c + |x|)2

⇒ |A|2

− 2A, x + |x|2

= c2

+ 2c|x| + |x|2

⇒

2c|x| + 2A, x = |A|2 − c2 ⇒ |x| + A

c , x =

|A|2 − c2

2c .

e = A

c , k =

|A|2 − c2

2c

|x| + e, x = k.

d2 < |A| + d1 ⇒ c = d2 − d1 < |A|,

k > 0

|e| > 1.

L

O L x d1 = d(x, O) d2 = d(x, L)

d1 = d2.

c > 0



v ∈ R2

|v| = 1 L v

L

x, v + c = 0,

c > 0

d( p, L) =

p, v + c p

− p, v − c

|x| = x, v + c.

e = −v, k = c,

|x| + e, x = k,

k > 0

|e| = 1.

|x| + e, x = k, e ∈ R2, k > 0,

|e|

|e| < 1

|e

|= 1

|e| > 1

x = x(t), t ∈ I

x =−

µ

|x|3x, x

∈R3

0

,

c = 0 x(t)

I

u,v,w ∈ R3



(u ∧ v) ∧ w = u, wv − v, wu,

u ∧ v, w = u, v ∧ w

x(t) x3 =0

d

dt

x

|x|

= x|x|2 − x, xx

|x|3 = 1

|x|3 [(x ∧ x) ∧ x],

x(t) ∈ C 1

µ x

d

dt

µ

x

|x|

= −

(x ∧ x) ∧ −µx

|x|3

= −c ∧ x = d

dt(−c ∧ x).

c = x ∧ x

µ

x(t)

|x(t)| + e

= −c ∧ x(t), t ∈ I,

e ∈ R2

x(t)

µ

x(t)

|x(t)| , x(t) + e, x(t)

= |c|2, t ∈ I,

−c, x(t) ∧ x(t) = |c|2

|x(t)| + e, x(t) = |c|2

µ , t ∈ I.

x(t)

e

ε = |e|

x(t) |e|

e = 0

0 < |e| < 1

|e

|= 1

|e| > 1

ε = |e|

E = 1

2x

2 − µ

|x| x2 = |x|2,



ε E ε

µ

x|x| + e

= −c ∧ x

µ2

x(t)

|x(t)| + e

2

= |c ∧ x|2.

c

x

|c ∧ x| = |c||x|,

µ2 x2

|x|2 + 2

e,

x

|x| + ε2 = |

c

|2x2.

|x| + e, x = |c|2/µ E = x2/2 − µ/|x|

µ2|e|2 − 1

= 2E |c|2.

c = 0

ε < 1 ⇔ E < 0,

ε = 1 ⇔ E = 0,

ε > 1 ⇔ E > 0,

|x| + x, e = k x ∈ R2,

e ∈ R2

k > 0

e = ε(cos ω, sen ω) ε = |e| ≥ 0 ω ∈ R.

ε ω

O = (0, 0)

ω 2π

ε = 0 ω

x ∈ R2 \ 0

x =ρ(cos θ, sen θ), ρ > 0

θ ∈ R

ρ + ρε(cos θ, sen θ), (cos ω, sen ω) = k.

ρ[1 + ε cos(θ − ω)] = k ⇒ ρ = k

1 + ε cos(θ − ω).

ρ r



k ρ > 0 θ ∈ R x

1 + ε cos(θ

−ω) > 0

ρ = k1 + ε cos(θ − ω)

.

z(t) = x1(t) + ix2(t)

z = − µ

|z|3 z, z ∈ R2 \ 0.

z = ρeiθ ρ > 0 θ ∈ R t

z = eiθ(ρ + 2ρiθ + ρiθ − ρθ2).

d2ρdt2

− ρdθdt

2= − µ

ρ2

2dρ

dt

dθ

dt + ρ

dθ

dt = 0.

ρ2dθ

dt = J,

J

c J = 0

dθ

dt =

J

ρ2 ⇒ dt

dθ =

ρ2

J .

d2ρ

dt2 − ρ

J 2

ρ4 = − µ

ρ2.

t = t(θ)

ρ = ρ(θ)

dρ

dθ =

dρ

dt

dt

dθ =

dρ

dt

ρ2

J ⇒ dρ

dt =

dρ

dθ

J

ρ2.

ρ

d2ρ

dt2 =

d

dθ J

ρ2

dρ

dθ dθ

dt

.

d

dθ

J

ρ2dρ

dθ

dθ

dt − J 2

ρ3 = − µ

ρ2 ⇒

J d

dθ

J

ρ2dρ

dθ

− J 2

ρ = −µ ⇒ d

dθ

1

ρ2dρ

dθ

− 1

ρ = − µ

J 2 ⇒

− d

dθ

d

dθ

1

ρ

− 1

ρ = − µ

J 2 ⇒ d2

dθ2

1

ρ

+

1

ρ =

µ

J 2.



E E

u = 1

ρ

u + u = µ

J 2,

u = A cos θ + B sen θ + µ

J .

A = |w| cos ω B = |w| sen ω w = (A, B)

u = |w|(cos ω cos θ + sen ω sen θ) + µJ 2

= |w| cos(θ − ω) + µJ 2

.

u = 1ρ

ρ = 1

|w| cos(θ − ω) + µJ 2

.

ρ = J 2/µ

1 + (J 2/µ)|w| cos(θ − ω),

ρ = k

1 + ε cos(θ − ω),

ε = J 2

µ |w| > 0, k =

J 2

µ .

θ = ω θ = ω + π



x(t) = −µ x(t)

|x(t)|3 x(t) = (x1(t), x2(t), x3(t)) ∈ R3.

x(0), x(0) π

π x3 = 0

x(t) = (x1(t), x2(t), 0) = x1(t) + ix2(t) = ρ(t)eiθ(t),

ρ

− θ2

ρ = −µ

ρ2

ρ2θ = J

J

(x1(t), x2(t))

ρ = k

1 + ε cos(θ − ω),

k = J 2

GM z = 0

0 < ε < 1

E

E

T

T a

(x1(t), x2(t))

E

4

z = v

v

= −µ

z

|z|3

z = (x1(t), x2(t)), v(t) = (x1(t), x2(t))

(z(t), v(t))

z(t) = ρ(t)(cos θ(t), sen θ(t)),

(ρ, θ)

I = (α, w) R



t ∈ [0, w)

(0, 0) z(t) v(t)

ρ

= k

1 + ε ≤ |z(t)| ≤ k

1 − ε = ρ

.

µρ

ρ3

≤ |v(t)| ≤ µρ

ρ3

.

|v(t) − v(0)| = | t

0

v(s)ds| ≤ t

0

|v(s)|ds,

µ ρ

ρ3

t ≤ |v(t) − v(0)| ≤ µ ρ

ρ3

t.

|v(t)| [0, ω) ω < +∞

(α, 0]

z(t) (α, ω) = R ρ(t) θ(t) R

J > 0

lımt→±∞

θ(t) =±∞

.

z(t)

ρ2θ(t) = J,

ρ(t) = ρ(θ(t))

ρ(θ)

J > 0

θ = J

ρ2 ,

J

ρ2

t + θ(0) ≤ θ(t) ≤ J

ρ2

t + θ(0),

t ∈ R

θ(t) → ±∞

t → ±∞

J < 0



T > 0

θ(t)

θ(t + T ) = θ(t) + 2π,

t ∈ R

J > 0 J < 0

θ : R → R T > 0

θ(T ) = θ0 + π,

θ0 = θ(0) θ1(t) = θ(t + T )θ2(t) = θ(t) + 2π.

θ(t) =

J

ρ2(θ(t))

θ(0) = θ0 + 2π.

θ1(t) = θ2(t)

θ(t + T ) = θ(t) + 2π,

t ∈ R

z(t) z(t) = (x1(t), x2(t)) T

x1(t) = ρ(θ(t))cos θ(t)x2(t) = ρ(θ(t))sen θ(t),

ρ

ρ(θ + 2π) = ρ(θ)

x1(t + T ) = ρ(θ(t + T ))cos θ(t + T ) = ρ(θ(t))cos θ(t) = x1(t).

x2(t + T ) = x2(t).

z(t) T

T

J

z(t)

T

T = 2π

J ab,

a > b > 0



A(t)

θ(0) θ(t)

dA

dt =

1

2ρ(θ)2θ(t) ⇒ dA

dt =

1

2J.

θ(t)

θ(0) = θ(T )

A(T )

t = 0 t = T

A(T ) = 1

2JT.

A(T ) = πab

T = 2π

J ab.

3/2

2a = k

1 − ε +

k

1 + ε ⇒ a =

k

1 − ε2.

d C O

d = a − k

1 + ε = a − a(1 − ε) = aε.

b = √ a2 − d2 b = a√ 1 − ε2

T = 2π

J ab =

2π

J a2

1 − ε2.

k = J 2

GM ⇒ J =

√ GMk.

k = (1 − ε2)a J =

GM (1 − ε2)a

T = 2π

√ 1 − ε2a2

GM (1

−ε2)a

= 2π√

GM a3/2.

T

M

b

a =

1 − ε2.

ε

a



θ P

OP

θ t

t

a



u

P

x = x1

y = x2

(0, 0)

x21

a2 +

x22

b2 = 1.

x1 = a cos ux2 = b sen u

u ∈ [0, 2π],

u

(x1, x2) = (x1, a

bx2)

x21 + (x2)2 = a2.

x1 = a cos ux2 = a sen u,

x1 = a cos u

x2 = b sen u.

P = (x1, x2) → P = (x1, x2)

x2

a

ρ = k

1 + ε cos(θ − ω),

ω = 0

C = (−d, 0)



(x1 + d)2

a2 + x

22

b2 = 1.

u x1 + d = a cos u

x2 = b sen u.

b = a√

1 − ε2

(x1, x2) = a(cos u − ε,

1 − ε2 sen u),

u

θ = θ(t) θ(t)

k2

(1 + ε cos(θ − ω))2θ = J.

θ

0

ds

(1 + ε cos(s − ω))2 =

J

k2(t − t0),

ω = 0 t0

F (θ)

F (θ) = J

k2(t − t0),

θ = θ(t) F (θ)

u t

ρ2θ = x1x2 − x2x1.

ρ2θ = J

x1 = a(cos u − ε)x2 = a

√ 1 − ε2 sen u,

a2

1 − ε2(1 − ε cos u)du

dt = J,

u

ω = 0 t0

u − ε sen u = J

a2√

1 − ε2(t − t0),



ζ

u − ε sen u = ζ

J =

GMa(1 − ε2),

u − ε sen u =

√ µ

a3/2(t − t0) µ = GM.

u = 2π t = t0 + T T

T = 2π

√ µa3/2,

x1 = a(cos u − ε)x2 = a

√ 1 − ε2 sen u

u − ε sen u = 2π

T (t − t0),

t0



n

n

n

ri : J → R3, ri = (xi(t), yi(t), zi(t)), 1 ≤ i ≤ n.

miri = −n

j=1,j=i

Gmimj (rj − ri)

|rj − ri|3 ,

F i(r1, r2, . . . , rn)

Ω

∆ij = (r1, . . . , rn) ∈ Rn : ri = rj i = j.

∆ij R

3n

3n − 3

∆ = ∪1≤i<j≤n∆ij .

∆ R3n

Ω = R3n \ ∆

∆

n C ∞

r : I ⊂ R → R3n \ ∆, r = (r1, . . . , rn),

Mr = F (r),



M= (m1I 3, . . . , mnI 3) F = (F 1, . . . , F n),

mi i I 3

3 × 3 C ∞

F

C 1 V : Ω → R

F = −∇V,

∇V = (∂ r1V , . . . , ∂ rnV ) V = V (r1, . . . , rn),

∂ riV = ∂V

∂ri V xi, yi, zi ri

V = −

1≤i<j≤n

Gmimj

|ri − rj| r ∈ Ω,

V

n

T = 1

2

n

i=1

mi|ri(t)|2,

r(t)

E = T + V,

t

dE

dt =

ni=1

miri , ri +n

i=1

dV

dri, ri

=

ni=1

miri +

dV

dri, ri

= 0.

n

Ω ⊂ Rd

f : Ω → R

C 1 p f (tx) = t pf (x) t > 0 x ∈ Ω

∇f (x), x = pf (x), x ∈ Ω.

Ω tx ∈ Ω t > 0 x ∈ Ω



x ∈ Ω C 1 t ∈ (0, +∞) → f (tx) = t pf (x)

t

x, ∇f (tx) = p t p−1f (x).

t = 1 x, ∇f (x) = p t p−1f (x)

n

r(t) = r∗

∂V

∂ri= 0 i = 1, . . . , n ,

r = r∗ Ω = R

3n \∆

V (r) −1

V (tr1, . . . , t rn) = t−1

V (r1, . . . , rn) t > 0.

r∗ ∈ Ω

0 =n

i=1

∂V

∂riri = −V (r∗),

V (r∗) = 0

r(t) t ∈ I := (α, ω) n

(r(t), v(t))

r = v

v = M−1F (r).

ω < ∞

ω < ∞

ρ(t) = dist (r(t), ∆),

∆

lımt→ω

ρ(t) = 0.

f : B(x0, R) ⊂ Rm → R

m M = supB(x0,R) |f |

x(t) x = f (x)

x(0) = x0,

[−M/R, M/R]



E δ > 0 τ = τ (E, δ ) r(t)

E ρ(0) > δ [

−τ, τ ]

tn tn → ω ρ(tn) > δ > 0 δ E

δ [tn − τ, tn + τ ] τ

|tn − ω| < τ /2 [ω, ω + τ /2]

M R τ = M/R

|(r, v)|∞ = max|rj |, |vj |.

B((r(0), v(0)), R) R = δ/4

M−1F (r)

|M−1F (r)|∞ ≤ C 1δ 2

,

C 1 B((r(0), v(0)), R)

|v|∞ ≤ |v − v(0)|∞ + |v(0)|∞ ≤ δ

4 + |v(0)|∞.

|v(0)|∞

ms|vs(0)|22

≤ E − V (r(0)) ≤ E +i<j

Gmimj

|ri(0) − rj (0)| ≤ E +i<j

Gmimj

δ .

|v(0)|∞ ≤ C 2√ δ

,

C 2 E δ

|(v, M−1F (r))|∞ ≤ C 1δ 2

+ δ

4 +

C 2√ δ

:= M,

M

E

δ

r = (r1, . . . , rn) : I

→ R

3n

n

I

I (t) = 1

2

ni=1

mi|ri(t)|2, t ∈ I.

t

I (t) =

ni=1

miri(t), ri(t).

I (t) =n

i=1

mi|ri|2 +n

i=1

miri(t), ri (t).



T (t) = 12

ni=1 mi|ri|2

I (t) = 2T (t) +n

i=1

miri(t), ri (t)

−∂V

∂ri= miri (t)

I (t) = 2T (t) −n

i=1

ri(t), ∂V

∂ri = 2T (t) + V (r).

I (t)

I (t) = 2T + V = 2E − V,

E

r : (α, ω) → R3n

E

ω = ∞

lımt→+∞

I (t) = +∞.

I (t) = 2E − V V < 0 I (t) ≥ 2E I (t)

t

t0

I (s)ds >

t

t0

2Eds ⇒ I (t) − I (t0) > 2E (t − t0) ⇒ I (t) > 2E (t − t0) + I (t0).

t

t0

I (s)ds >

t

t0

(2E (s − t0) + I (t0))ds ⇒ I (t) > E (t − t0)2 + I (t0)(t − t0) + I (t0).

lımt→+∞

I (t) > lımt→+∞

(E (t − t0)2 + I (t0)(t − t0) + I (t0) = +∞.

r = (r1, . . . , r2) : I

→R3n

C (t)

C (t) =n

i=1

mi

M ri(t),

M = n

i=1 mi

t

C (t) = 1

M

ni=1

miri (t) = 1

M

ni,j=1,i=j

Gmimjrj − ri

|rj − ri|3 =



1

M

n

i,j=1,i>j

Gmimjrj − ri

|rj − ri|3

+n

i,j=1,i<j

Gmimjrj − ri

|rj − ri|3 = 0.

C (t) = α + βt α, β ∈ R

ni=1

miri(t) = M C (t),

r(t) = (r1(t), . . . , rn(t)) n

c =

ni=1

mi(ri ∧ ri),

c

dc

dt =

ni=1

mi(ri ∧ ri + ri ∧ ri ) =n

i=1

(ri ∧ ri ) =n

i=1

(ri ∧ miri )

=n

i=1

ri ∧ n

j=1,j=i

Gmimj

|rj − ri|3 (rj − ri) = n

i,j=1,i=j

Gmimj

|rj − ri|3 ri ∧ rj

=

ni,j=1,i<j

Gmimj

|rj − ri|3 ri ∧ rj +

ni,j=1,i>j

Gmimj

|rj − ri|3 ri ∧ rj = 0,

n ≥ 3

ri(0) vi(0) = ri(0)

x − p0, v = 0 p0 ∈ R3, v ∈ R

3,

r(t) n t ∈ I

ri(t) − p0, v = 0 ∀t ∈ I,

i = 1, . . . , n



r = (r1, . . . , rn) n I = (α, ω)

n

∃ξ ∈ R3

lımt→t0

ri(t) = ξ i = 1, . . . , n .

t0 t0 = α t0 = ω

C (t0) = 1

M

ni=1

miri = ξ,

t0 = ω

ri(t) = ri(t) − C (t)

C (t) =n

i=1

miri(t) = 0,

ξ = 0

r(t) n t = ω

ω < ∞

V (r(t)) = −G

1≤i≤j≤n

mimj

|ri(t) − rj (t)| → −∞,

I (t) = 1

2

ni=1

mi|r(t)i|2 → 0,

t → ω

I (t) = 2E − V (r(t)), V < 0,

∃t0 < ω I (t) ≥ 1 t ∈ (t0, ω)

I (t)≥

(t

−t0)2

2 + I (t

0)(t

−t0

) + I (t0

) t∈

[t0

, ω).

ω = ∞

lımt→∞

I (t) = ∞,

lımt→w I (t) = 0

n = 3



z = x + iy

rj (t) = λ(t)zj j = 1, 2, 3 z1, z2, z3

z1 |z1| = 1

z2 = eiϕz1 z3 = eiϕz2 ϕ = 2π

3 .

z = eiϕ

z2 + z + 1 = 0 ⇒ z1 + z2 + z3 = 0,

|zj − zk| =√

3 j = k

m1 = m2 = m3 = m ri(t) = λ(t)zi

i = 1, 2, 3

λ(t)zi = Gm3

j=1,i=j

λ(t)(zj − zi)

λ3(t)|zj − zi|3 = −Gmzi√ 3λ2

,

|zi − zj | = √ 3

λ = − Gm√ 3λ2

c =3

i=1

mi(ri ∧ ri) = mλλ3

i=1

zi ∧ zi = 0.

r = (r1, . . . , rn) n t → ω

r = (r1, . . . , rn) n

|c|2 ≤ 4I (I − E ).

c =n

i=1

mixi ∧ xi,

|c| ≤n

i=1

mi|xi||xi| =n

i=1

(√ mi|xi|)(√ mi|xi|).

|c| ≤√

2T √

2I.

I = 2T + V E = T + V

I = T + E T = I − E

|c| ≤ 2√

I √

I − E.



ψ ∈ C 2[t0, w)

ψ(t) > 0, ψ(t) < 0, ψ(t) ≥ α + β ψ(t)

, t ∈ [t0, ω).

1

2ψ(t)2 − αψ(t) − β ln ψ(t) ≤ H, ∈ [t0, w),

H = 12ψ(t0)2 − αψ(t0) − βlnψ(t0)

ψψ − αψ − β ψ

ψ ≤ 0,

ddt

12 (ψ)2 − αψ − β ln ψ ≤ 0.

t0

t

t ∈ [t0, w)

|c|2 ≤ 4I (I − E ) ⇒ I ≥ |c|24I

+ E.

I → ∞ I > 0 t ω I → 0

t → ω I < 0 t ω

1

2I − EI − |c|2 ln I ≤ H t ∈ [t0, ω).

|c| = 0 c = 0

m1r1 + m2r2 + m3r3 = 0.

ω z1, z2, z3

rj (t) = eiωtzj j = 1, 2, 3.

z1, z2, z3 ∈ R2

rj (t) = eiωtzj j = 1, 2, 3,



m1z1 + m2z2 + m3z3 = 0.

z1, z2, z3 d > 0

|ω| =

GM

d3 M = m1 + m2 + m3

m1, m2, m3 d > 0

d

d =√

3

z1 = λ + eiθ, z2 = λ + weiθ, z3 = λ + weiθ,

λ θ

m1z1 + m2z2 + m3z3 = 0 ⇒ λ = −eiθ(m1 + m2w + m3 w)

M .

z1 = eiθ

(1 − m1 + m2w + m3 w

M )

z2 = eiθ(w − m1 + m2w + m3 w

M )

z3 = eiθ( w − m1 + m2w + m3 w

M )

|ri(t) − rj (t)| = |zi − zj | := ri,j.

r = eiωt(z1, z2, z3)

−wz1 = Gm2

r31,2

(z2 − z1) + Gm3

r31,3

(z3 − z1)

−wz2 = Gm1

r32,1

(z1 − z2) + Gm3

r32,3

(z3 − z2)

−wz3 = Gm1

r33,1

(z1 − z3) + Gm2

r33,2

(z2 − z3).

m1

m2

m3

m1z1 + m2z2 + m3z3 = 0,



−wz1 = Gm2r31,2

(z2 − z1) + Gm3r31,3

(z3 − z1)

−wz2 = Gm1

r32,1

(z1 − z2) + Gm3

r32,3

(z3 − z2)

m1z1 + m2z2 + m3z3 = 0

zi = (x1, yi) x =(x1, x2, x3) y = (y1, y2, y3) x, y

Aξ = 0

A=

w − Gm2

r31,2

− Gm3

r31,2

Gm2

r31,2

Gm3

r31,3

Gm1

r31,2 w − Gm1

r31,2 − Gm3

r32,3

Gm3

r32,3

m1 m2 m3

Aξ = 0 2 r(A) = 1

Gm1

r31,2

Gm3

r32,3

m1 m3

Gm2

r31,2

Gm3

r32,3

m2 m3

,

r1,2 = r1,3 = r2,3 = d r(t) z1, z2 z3

d

A

det A = m3

w − GM

d3

2= 0 ⇒ ω2 =

GM

d3 ,

|ω| =

GM

d3

z1, z2

z3

rj (t) = eiωtzj j = 1, 2, 3

m1 ≥ m2 > 0 m3

r1 = Gm2

1

|r2 − r1|3 (r2 − r1) + ε 1

|r3 − r1|3 (r3 − r1)

r2 = Gm1

1

|r1 − r2|3 (r1 − r2) + εm2

m1

1

|r3 − r2|3 (r3 − r2)

r3 = Gm1

|r1 − r3|3 (r1 − r3) + Gm2

|r2 − r3|3 (r2 − r3),



ε = m3

m2 m3 m2 ≤ m1 ε → 0

r1 = Gm2

|r2 − r1|3 (r2 − r1)

r2 = Gm1

|r1 − r2|3 (r1 − r2)

r3 = Gm1

|r1 − r3|3 (r1 − r3) + Gm2

|r2 − r3|3 (r2 − r3).

r3

r1

r2

r1(t) r2(t)

r3 = Gm1

|r1(t) − r3|3 (r1(t) − r3) + Gm2

|r2(t) − r3|3 (r2(t) − r3).

r1(t) r2(t)

rj (t) = eiωtzj ,

(1 − µ)z1 + µz2 = 0 µ = m2

M M = m1 + m2.

z1 = −µ∆z

z2 = (1 − µ)∆z ∆z = z2 − z1

rj (t) = eiωtzj

j = 1, 2

ω2|∆z|3 = GM.

∆z ∆z = |∆z|

r1(t) = −eiωtµ|∆z| r2(t) = eiωt(1 − µ)|∆z|.

r3(t) = |∆z|eiωtζ (t),

ζ + 2ωiζ − ω2ζ = GM

|∆z|3−µ − ζ

|µ + ζ |3 + GM

|∆z|31 − µ − ζ

|ζ + µ − 1|3 .

t = ωτ

ζ + 2iζ − ζ = −µ − ζ

|ζ + µ|3 + 1 − µ − ζ

|ζ + µ − 1| ,



ω2|∆z|3 = GM

z + 2Jz = z + −µ − z|z + µ|3 + 1 − µ − z|z + µ − 1|3 ,

z = (x, y)

J =

0 −11 0

.

Φ(z) = 1

2|z|2 +

1 − µ

|z + µ| + µ

|z + µ − 1| + 1

2µ(1 − µ).

z + 2Jz = ∇Φ(z), z ∈ R2 \ −µ, 1 − µ.

J (z, v) = 2Φ(z) − |v|2,

v = z

J (z(t), z(t))

dJ dt

= 2

∇Φ(z), z

−2

z, z

= 4

Jz, z

= 0.

J

J

z

z

z = (1 − µ) z + µ

|z + µ

|3

+ µ z + µ − 1

|z + µ

−1

|3

.

ρ1 = |z + µ| ρ2 = |z + µ − 1|

z = (x, y)

y = 0

x = (1 − µ) x + µ

|x + µ|3 + µ x + µ − 1

|x + µ − 1|3 .

g(x)



g I 1 = (−∞, −µ) I 2 = (−µ, 1 − µ) I 3 = (1 − µ, ∞)

g

I 1

I 3

|g| → ∞

x → −µ

x → 1 − µ

x = g(x),

xi ∈ I i

i = 1, 2, 3

g(0) = 0

g(0) > 0

g(0) < 0

µ = 12

µ < 12

µ > 12

x2 = 0 x2 ∈ (0, 1 − µ) x2 ∈ (−µ, 0)

µ = 12 µ < 1

2 µ > 12

rj (t) = eiωtzj j = 1, 2

xi

Li = (xi, 0) i = 1, 2, 3

1 − 1 − µ

ρ31− µ

ρ32

z =

µ(1 − µ)

ρ31− µ(1 − µ)

ρ32.

z = (x, y) y = 0

1 −

1−

µ

ρ31 − µ

ρ32

x = µ(1

−µ)

ρ31 − µ(1

−µ)

ρ32

1 − 1 − µ

ρ31− µ

ρ32= 0.

µ(1 − µ)

ρ31− µ(1 − µ)

ρ32= 0,

ρ1 = ρ2 ρ1 = ρ2 = 1

L4 L5

(−µ, 0) (1 − µ, 0)





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