Date post: | 07-Jul-2018 |
Category: |
Documents |
Upload: | carlos-vazquez-monzon |
View: | 223 times |
Download: | 0 times |
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 1/54
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 2/54
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 3/54
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 4/54
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 5/54
n
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 6/54
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 7/54
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 8/54
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 9/54
n
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 10/54
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 11/54
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| ,
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 12/54
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
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 13/54
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)
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 14/54
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 15/54
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.
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 16/54
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
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 17/54
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 |
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 18/54
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
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 19/54
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
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 20/54
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),
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 21/54
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|
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 22/54
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
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 23/54
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 ,
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 24/54
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,
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 25/54
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
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 26/54
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
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 27/54
(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,
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 28/54
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
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 29/54
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.
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 30/54
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
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 31/54
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
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 32/54
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
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 33/54
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
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 34/54
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
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 35/54
θ P
OP
θ t
t
a
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 36/54
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)
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 37/54
(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),
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 38/54
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
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 39/54
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),
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 40/54
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
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 41/54
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]
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 42/54
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).
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 43/54
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 =
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 44/54
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
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 45/54
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
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 46/54
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.
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 47/54
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,
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 48/54
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,
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 49/54
−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),
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 50/54
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| ,
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 51/54
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)
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 52/54
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)
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 53/54
8/18/2019 Mecanica Celeste Una Introduccion.
http://slidepdf.com/reader/full/mecanica-celeste-una-introduccion 54/54