Đề tài " Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses " potx

Annals of Mathematics Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses By Kuo-Chang Chen... Existence and minimizing

Annals of Mathematics

Existence and minimizing

properties of retrograde orbits to the three-body

problem with various choices of

masses

By Kuo-Chang Chen

Existence and minimizing properties of retrograde orbits to the three-body

problem with various choices of masses

By Kuo-Chang Chen

Abstract

Poincar´e made the first attempt in 1896 on applying variational calculus

to the three-body problem and observed that collision orbits do not necessarilyhave higher values of action than classical solutions Little progress had beenmade on resolving this difficulty until a recent breakthrough by Chencinerand Montgomery Afterward, variational methods were successfully applied to

the N -body problem to construct new classes of solutions In order to avoid

collisions, the problem is confined to symmetric path spaces and all new planarsolutions were constructed under the assumption that some masses are equal

A question for the variational approach on planar problems naturally arises:Are minimizing methods useful only when some masses are identical?

This article addresses this question for the three-body problem For ious choices of masses, it is proved that there exist infinitely many solutionswith a certain topological type, called retrograde orbits, that minimize theaction functional on certain path spaces Cases covered in our work includetriple stars in retrograde motions, double stars with one outer planet, and somedouble stars with one planet orbiting around one primary mass Our resultslargely complement the classical results by the Poincar´e continuation methodand Conley’s geometric approach

Calculus of variations, in spite of its long history, should be considered

a relatively new approach to the three-body problem In 1896 Poincar´e [23]made the first attempt to utilize minimizing methods to obtain solutions forthe three-body problem, but found out the discouraging fact that existence

of collisions does not necessarily cause a significant increment in the value of

326 KUO-CHANG CHEN

the action functional As a result solutions were obtained only for the force potential, instead of the Newtonian case In 1977 Gordon [13] proved aminimizing property for elliptical Keplerian orbits, including the degeneratecase – collision-ejection orbit It turns out that the actions of these orbitsover one period depend only on the masses and the period, not on eccentricity.From this point of view the collision-ejection orbits and other elliptical orbitsare not distinguishable A common doubt at the time is: Are minimizing

strong-methods useful for the N -body problem? Concerning this question,

Chenciner-Venturelli [8] constructed the “hip-hop” orbit for the four-body problem withequal masses and, a few months later, Chenciner-Montgomery [7] constructedthe celebrated figure-8 orbit for the three-body problem with equal masses,

a solution numerically discovered in [20] Afterward, Marchal [16] found aclass of solutions related to the figure-8 orbit and made important progress

on excluding collision paths [17], [5] Inspired by the discovery of the

figure-8 orbit, a large number of new solutions [2], [3], [4], [11], [26] were proved

to exist by variational methods These discoveries attract much attentionnot only because they are not covered by classical approaches, but also due

to the amusing symmetries they exhibit On the other hand, these orbitswere constructed under the assumption that some masses are equal Except aclass of nonplanar solutions constructed by varying planar relative equilibria

in a direction perpendicular to the plane (see Chenciner [5], [6]), among the

discoveries for the N -body problem, none of the new solutions constructed

by variational methods can totally discard this constraint A question forthe variational approach, especially on planar problems, naturally arises: Areminimizing methods useful only when some masses are identical?

This article is concerned with variational methods on the existence ofcertain types of solutions to the planar three-body problem with various choices

of masses There is a natural way of classifying orbits by their topologicaltypes in the configuration space From the terminology normally used in lunar

theory, we call a solution retrograde if its homotopy type in the configuration

space (with collision set removed) is the same as those retrograde orbits inthe lunar theory Detailed descriptions are left to Section 2 and 3 Our maintheorem (Theorem 1) shows the existence of many periodic and quasi-periodicretrograde solutions to the three-body problem provided the mass ratios fallinside the white regions in Figure 1 The method used is a variational approachwith a mixture of topological and symmetry constraints The advantage of ourapproach, as Figure 1 indicates, is that it applies to a wide range of masses

In sharp contrast with the results obtained from the classical Poincar´econtinuation method [22] (see [24], [18] and references therein) and Conley’sgeometric approach [9], [10], our main theorem does not apply to Hill’s lu-nar theory and many satellite orbits, both of which treat the case with onedominant mass It is worth mentioning that Hill’s lunar theory can also be

200 50

50

150

100

Figure 1: Admissible mass ratios (the white region) for the main theorem

analysized by variational methods; see Arioli-Gazzola-Terracini [1] Cases weare able to cover include retrograde triple stars, double stars with one outerplanet, and some double stars with one planet orbiting around one primarymass See Section 2 and Figure 3 for details Moreover, due to the minimizingproperties the orbits we obtained do not contain tight binaries, and there areperiodic ones with very short periods in the sense that the prime periods aresmall integral multiples of their prime relative periods Classical approachesnormally produce orbits with very long periods

2 The Main Theorem

The planar three-body problem concerns the motion of three masses m1,

m2, m3 > 0 moving in the complex planeC in accordance with Newton’s law

There is no loss of generality to assume that the mass center is at the

origin; that is, assuming x stays inside the configuration space:

V := {x ∈ C3: m1x1+ m2x2+ m3x3 = 0}

ob tuse

2 1

3

1 3

1

3 2

3 2

3 1 2

Figure 2: The unit shape sphere

A preferred way of parametrizing V is to use Jacobi’s coordinates:

obtained by quotient out from V the rotational symmetry given by the action: e iθ · (z1, z2) = (e iθ z1, e iθ z2) The identification ˜V = V /SO(2) is via the

SO(2)-Hopf map

(u1, u2, u3) := (|z1|2− |z2|2, 2 Re(¯ z1z2), 2 Im(¯ z1z2))

(2)

Each single point in ˜V represents a congruence class of triangles formed by the

three mass points, and each point on its unit sphere{|u|2 = 1}, called the unit shape sphere, represents a similarity class of triangles The signed area of the

triangle is given by 12u3

Figure 2, due to Moeckel [19], relates the configurations of the three bodieswith points on the unit shape sphere In the figure Λj represents isosceles tri-

angles with jth mass equally distant from the other two The equator (u3= 0)

represents collinear configurations On the upper hemisphere (u3 > 0),

trian-gles with vertices{x1, x2, x3} are positively oriented; on the lower hemisphere

they are negatively oriented The poles correspond to equilateral triangles.Let Δ := {x ∈ C3 : x i = x j for some i = j} be the variety of collision

configurations It is invariant under rotations and its projection ˜Δ in ˜V is the

union of three lines emanating from the origin (the triple collision) Each line

represents a similarity class of one type of double collision Let S3 be the unit

sphere in V and S2 be the unit shape sphere The Hopf fibration (2) renders

S3\Δ the structure of an SO(2)-bundle over S2\ ˜Δ, whose fundamental group

is a free group with two generators For φ > 0, let α φ be the following loop in

π1(S2\ ˜Δ) The left side of Figure 2 depicts the path ˜α φ over t ∈ [0, 1].

A solution x of (1) is called relative periodic if its projection ˜ x in the

reduced configuration space ˜V is periodic The prime relative period of x

is the prime period of ˜x Our major result concerns the existence of relative

periodic solutions to the three-body problem that are homotopic to α φ in V \Δ

respecting the rotation and reflection symmetry of α φ A precise description

is given in (9) These types of solutions, called retrograde orbits, are of special importance in the three-body problem When 0 < m1, m2  m3, the searchfor this type of solutions is an important problem in lunar theory A typical

example is the system Sun-Jupiter-Asteroid When 0 < m3  m2, m1, thesetypes of solutions are sometimes called satellite orbits or comet orbits If allmasses are comparable in size and none of them stay far from the other two,then the system forms a triple star or triple planet Another interesting case is

0 < m2  m1, m3 The binary m1, m3 form a double star (or double planet)

and m2 is a planet (or satellite) orbiting around m1 There is no evidentborderline between these categories The dash lines in Figure 3 make a roughsketch of the borders between them

There is no loss of generality in assuming m3 = 1 Let M = m1+ m2+ 1

be the total mass Define functions J : [0, 1) → R+ and F, G :R2

J(s) :=

 10

The following is our main theorem

Theorem 1 Let m3 = 1, M = m1+ m2 + 1 be the total mass, and let

F , G be as in (5), (6) Then the three-body problem (1) has infinitely many

Double Star with one planet

1

A star with two planets

Double Star with one outer planet or comet

orbiting around one primary mass

Figure 3: Theorem 1 applies to the complement of the shaded region

is not attained To ensure that the minimizing problem is solvable, we selectthe following ground space:

condi-sequentially lower semicontinuous on H φ Following a standard argument inthe calculus of variations, the action functional A attains its infimum on H φ.Although it may appear as an easy fact, let us remark here that collision-free critical points of A restricted to H φ are classical solutions to (1) If H φ ∗

is the space H φ except that the configuration space V is replaced by (R2)3,

then on H φ ∗ the fundamental lemmas for the calculus of variations are clearly

applicable Now if x is a collision-free critical point of A restricted to H φ, fromthe first variation ofA constrained to H φ , at x we have

0 = δ h A(x) = −

 10 3

Therefore y i (t) = m i α(t) for some α : [0, 1] → R2 and for each i It can be

easily verified that 3k=1 y k (t) = 0, that is (m1+ m2+ m3)α(t) = 0 Then α and hence every y i is identically zero This proves that x is indeed a classical

solution of (1)

The conventional definition of inner product on the Sobolev space

H1([0, 1], V ) defines an inner product on H φas well:

332 KUO-CHANG CHEN

From these observations, any critical point x of A on H φ is a solution of (1),

but possibly with collisions If we can show that x has no collision on [0, 1), then there is no collision at all and x indeed solves (1) for any t ∈ R Moreover,

x is periodic if φ π is rational; it is quasi-periodic if φ π is irrational

Consider a linear transformation g on H φ defined by

(g · x)(t) := x(−t)

(8)

The space of g-invariant paths in H φ is denoted by H φ g That is,

H φ g:={x ∈ H φ : g · x = x}

Observe that g is an isometry of order 2, and the action functional A defined

on H φ is g-invariant By Palais’ principle of symmetric criticality [21], any

collision-free critical point of A while restricted to H g

φ is also a collision-freecritical point ofA on H φ, and hence solves (1)

Let α φ be as in (3) The space X φ of retrograde paths in H φ g is defined

as the path-component of collision-free paths in H φ g containing α φ In otherwords,

X φ:=



x ∈ H g

φ: x(t) ∈ Δ for any t, x is homotopic to α φ in V \ Δ

within the class of collision-free paths in H φ g

inf

x∈X φ

A(x)

(10)

As noted before, the action functional A is coercive and hence attains its

infimum on the weak closure ofX φ The boundary ∂ X φofX φconsists of paths

in H φ g that have nonempty intersection with the collision set Δ The next twosections are devoted to proving the inequality

for φ ∈ (0, π] sufficiently close to π, under the assumptions in Theorem 1.

4 Upper bound estimates for the action functional A

This section is devoted to providing an upper bound estimate for (10)

Assume m3 = 1, φ ∈ (0, π], and M = m1+ m2+ 1 Let

(M φ) 2/3 e φti ,

(m1+ m2)2/3 (2π − φ) 2/3 e (φ −2π)ti ,

Figure 4: The retrograde path x (φ).

The calculation for K( ˙x (φ)) is simple:

m1m2(2π − φ) 2/3 (m1+ m2)1/3 +

φ2M

1

m1J (m2ξ) + m2J (m1ξ)



.

Combining this with K( ˙x (φ)), we have proved

Lemma 2 Assume m3 = 1 Let J, ξ, ξ π be as in (4), (11), (12) Then

5 Lower bound estimates for A on collision paths

Let x = (x1, x2, x3) be any path in Hloc1 (R, V ) From the assumption onthe center of mass the action functional A can be written

1

2| ˙x i − ˙x j |2+ M

|x i − x j | dt

(13)

This formulation has been used to construct Lagrange’s equilateral solutions

by Venturelli [25] and Zhang-Zhou [27] Each integral in this expression will

be estimated by the formula in the first subsection below In the second

sub-section, we will provide lower bound estimates for collision paths in ∂ X φ

5.1 An estimate for the Keplerian action functional Given any φ ∈ (0, π],

T > 0, consider the following path space:

ΓT,φ:={r ∈ H1([0, T ], C) : r(0), r(T ) = |r(0)||r(T )| cos φ} ,

Γ∗ T,φ:={r ∈ Γ T,φ : r(t) = 0 for some t ∈ [0, T ]}

The symbol ·, · stands for the standard scalar product in R2 ∼=C Let μ, α

be positive constants Define a functional I μ,α,T : H1([0, T ], C) → R ∪ {+∞}

by

I μ,α,T(r) :=

 T0

μ

2( ˙r

2+ r2˙θ2) +α

r dt

This is actually the action functional for the Kepler problem with reduced

mass μ and some suitable gravitation constant, under the assumption that the

mass center is at rest Each integral in (13) is of this form In this sense,expression (13) is essentially treating the system as three Kepler problems.The proposition below is an extension of a result in [4, Th 3.1] It concerns

the minimizing problem for I μ,α,T over ΓT,φ and Γ∗ T,φ We reproduce it herebecause (15) is not contained in [4], and the proof below is shorter and makes

no use of Marchal’s theorem [17], [5]

Proposition 3 Let φ ∈ (0, π], T > 0, μ > 0, α > 0 be constants Then

which consists of paths that start from the positive real axis and end on

{re φi : r ≥ 0} Let

Δ∗ T,φ={r = re iθ ∈ Δ T,φ : r(t) = 0 for some t ∈ [0, T ]}

It is easy to show that both ΔT,φand Δ∗ T,φ are weakly closed

Given any r∈ Γ T,φ(resp Γ∗ T,φ ), there is an A ∈ O(2) and ˜r ∈ Δ T,φ (resp

Δ∗ T,φ ) such that ˜r = Ar and I μ,α,T (r) = I μ,α,T(˜r) This is because the space

ΓT,φ (resp Γ∗ T,φ) is actually the image of O(2) acting on ΔT,φ (resp Δ∗ T,φ).Therefore, we may just consider the minimizing problem over ΔT,φand Δ∗ T,φ

Let rφ ∈ Δ ∗

T,φ be a minimizer of I μ,α,T on Δ∗ T,φ Suppose ξ1 = rφ(0),

ξ2 = rφ (T ), then clearly r φ also minimizes I μ,α,T over paths with fixed ends ξ1,

ξ2 In particular, this implies rφis a Keplerian orbit with collision(s), and thushas zero angular momentum almost everywhere Now we recall a result byGordon [13, Lemma 2.1] that implies such a path with lowest possible action

The symbol ·, · stands for the standard scalar product in R2...

This is actually the action functional for the Kepler problem with reduced

mass μ and some suitable gravitation constant, under the assumption that the< /i>

mass center... of this form In this sense,expression (13) is essentially treating the system as three Kepler problems .The proposition below is an extension of a result in [4, Th 3.1] It concerns

the