On the relations between the spherical and planar two ... · Workshop to honour Prof. Orlando...

On the relations between the spherical and planartwo-center problems

M. A. González León1, J. Mateos Guilarte2 and M. de la Torre Mayado2

1Departamento de Matemática Aplicada (Universidad de Salamanca). U.I.C. MathPhys-CyL

2Departamento de Física Fundamental (Universidad de Salamanca). U.I.C. MathPhys-CyL

Nonlinear Integrable SystemsWorkshop to honour Prof. Orlando Ragnisco in his 70th anniversary.

Burgos, October 2016.M.A.G.L. - J. Mateos - M. de la Torre (USAL) Spherical two-center problem Nonlinear Integrable Systems 1 / 26

Outline

1 The two-fixed center problem in S2

2 The Gnomonic projection

3 The two-fixed center problem in R2

4 From planar solutions to spherical orbits

M.A.G.L. - J. Mateos - M. de la Torre (USAL) Spherical two-center problem Nonlinear Integrable Systems 2 / 26

The two-fixed center problem in S2

The two Newtonian centers problem in S2:

U(θ1, θ2) = −γ1

Rcotan θ1 −

γ2

Rcotan θ2 (1)

Rθ1 and Rθ2 are the orthodromic distances from F1 and F2 to a given point in S2.Standard approach: separability in sphero-conical coordinates.Refs: Killing (1885), Kozlov & Harin (1992), Vozmischeva (2000, 2003), . . .


Sphero-Conical coordinates (Neumann (1859)): Given three arbitrary distinctnon-negative real numbers: a, b, c, sphero-conical coordinates (λ0, λ1, λ2) in R3 aredefined as :

x21 =

λ0(a− λ1)(a− λ2)

(a− b)(a− c), x2

2 =λ0(b− λ1)(b− λ2)

(b− a)(b− c), x2

3 =λ0(c− λ1)(c− λ2)

(c− a)(c− b)(2)

where: λ0 = x21 + x2

2 + x23, and (λ1, λ2) are the roots of equation:

x21

λ− a+

x22

λ− b+

x23

λ− c= 0

λ1 = constant and λ2 = constant⇒ elliptic cones. λ0 = constant⇒ spheres.M.A.G.L. - J. Mateos - M. de la Torre (USAL) Spherical two-center problem Nonlinear Integrable Systems 4 / 26

Fixing λ0 = R2, then (λ1, λ2) constitute a system of orthogonal coordinates in S2.Two sets of parameters (a, b, c) and (a′, b′, c′) such that: c−b

b−a = c′−b′b′−a′ lead to

equivalent coordinate systems. Thus, we have not loss of generality if we fix: a = 0,b = σ2 and c = 1, with:

σ = cos θf , σ =√

1− σ2 = sin θf

Finally:

x21 =

R2

σ2 λ1 λ2 ; x22 =

R2

σ2σ2 (σ2 − λ1)(λ2 − σ2) ; x23 =

R2

σ2 (1− λ1)(1− λ2)

with: 0 < λ1 < σ2 < λ2 < 1.

The metric in sphero-conical coordinates reads:

ds2S2 = g11(λ1, λ2) · dλ2

1 + g22(λ1, λ2) · dλ22

g11 = g−111 =

4λ1(σ2 − λ1)(1− λ1)

−R2(λ1 − λ2), g22 = g−1

22 =4λ2(σ2 − λ2)(1− λ2)

−R2(λ2 − λ1)


Potential in sphero-conical coordinates:

U(λ1, λ2) =−1

R (λ2 − λ1)

((γ1 + γ2)

√λ2 (1− λ2) + (γ2 − γ1)

√λ1 (1− λ1)

)Standard separation process:

H =1

R2(λ2 − λ1)(H1 + H2)

H1 = 2λ1(σ2 − λ1)(1− λ1)π21 − R(γ2 − γ1)

√λ1 (1− λ1)

H2 = 2λ2(λ2 − σ2)(1− λ2)π22 − R(γ2 + γ1)

√λ2 (1− λ2)

Introducing the local time: dt = (λ2 − λ1) dτ , the separated first-order equations are:

dλ1

dτ= ±2

√2

R2

√λ1(σ2 − λ1)(1− λ1)(−I1R2λ1 + R(γ2 − γ1)

√λ1(1− λ1) + I2)

dλ2

dτ= ±2

√2

R2

√λ2(λ2 − σ2)(1− λ2)(−I1R2λ2 + R(γ1 + γ2)

√λ2(1− λ2)− I2)


Are the corresponding quadratures convertible into elliptic integrals?

Even in the explicit solutions are not known, some of the qualitative properties ofthe orbits, analysis of the bifurcation diagrams, etc. can be performed (see forinstance Vozmischeva (2003) and references therein).

Alternative approach: the gnomonic projection converts (a part of) the sphericalproblem into an associated two-fixed center problem in R2.Borisov & Mamaev (2007), Albouy (2003) and Albouy & Stuchi (2004).


Gnomonic projection from S2 to R2

Gnomonic projection (Thales of Miletus (6th century BC)) of S2 to the tangent plane atthe point (0, 0,R).

P : S2 −→ U ⊂ R3 , U ∼= R2 , P(x, y, z) =

(Rz

x,Rz

y, R)≡ (x, y, z)

The projection is well-defined only in the half-sphere S2+ =

{(x, y, z) ∈ S2 / z > 0

}.


A simple trigonometrical analysis leads to the following equations for the projection:

x =Rz

x , y =Rz

y , z = R (3)

and the corresponding ones for the inverse projection:

x =R x√

x2 + y2 + z2, y =

R y√x2 + y2 + z2

, z =R z√

x2 + y2 + z2(4)

with obviously z = R.


Trajectory isomorphimWriting the expressions of cartesian coordinates of S2 in terms of θ1 and θ2:

x =R

2 sin θf(cos θ2 − cos θ1)

y = ±R

√1− 1

4 cos2 θf(cos θ1 + cos θ2)

2 − 14 sin2 θf

(cos θ2 − cos θ1)2

z =R

2 cos θf(cos θ1 + cos θ2)

The potential reads:

U(x, y, z) = − 1R

(γ1(cos θf z− sin θf x)√

R2 − (cos θf z− sin θf x)2+

γ2(cos θf z + sin θf x)√R2 − (cos θf z + sin θf x)2

)


And correspondingly the Newton equations for the problem are:

x = −∂U∂x

+ λx , y = −∂U∂y

+ λy , z = −∂U∂z

+ λz (5)

being λ the Lagrange multiplier, i.e. the Euler-Lagrange equations for theLagrangian:

L =12(x2 + y2 + z2)− U(x, y, z) +

λ

2(x2 + y2 + z2 − R2)

We introduce the local time:

dτ =R2

z2 dt

and a few calculations with derivatives give:

x′′ =z2

R3 (xz− xz) , y′′ =z2

R3 (xz− xz) , z′′ = 0 (6)


Thus we can write the Newton equations in variables (x, y) and “time” τ :

x′′ =z2

(x2 + y2 + z2)32

(−z∂U∂x

+ x∂U∂z

)(7)

y′′ =z2

(x2 + y2 + z2)32

(−z∂U∂y

+ y∂U∂z

)(8)

Finally, the z coordinate is obviously fixed in the projection plane z = R, i.e.:

x′′ =R2

(x2 + y2 + R2)32

(−R

∂U∂x

+ x∂U∂z

)(9)

y′′ =R2

(x2 + y2 + R2)32

y∂U∂z

(10)


If we introduce the affine transformation (Borisov & Mamaev 2007):

x1 ≡ x , x2 ≡y

cos θf

thus equations (9, 10) can be viewed as the Newton equations for the potential:

V(x1, x2) = −sec2 θf γ1√

(x1 + R tan θf )2

+ x22

−sec2 θf γ2√

(x1 − R tan θf )2

+ x22

(11)

i.e. (9, 10) are written as:

x′′1 (τ) = − ∂V∂x1

, x′′2 (τ) = − ∂V∂x2

(12)

But (12) are nothing but the Newton equations for the planar two-fixed centerproblem:

V(x1, x2) = − α1√(x1 + a)

2+ x2

2

− α2√(x1 − a)

2+ x2

2

(13)

with inter-center distance: a = R tan θf , and strengths: α1 = γ1cos2 θf

and α2 = γ2cos2 θf

.


Brief review on the planar two-center problemEuler (1760), Lagrange (1766), Jacobi (1866), Charlier (1902), . . . . . .

L =m2(x2

1 + x22

)− V(x1, x2)

H =1

2m

(p2

1 + p22

)+ V(x1, x2)

Adimensionalization:

x1 = ax , x2 = ay , pi =

√m(α1 + α2)√

api , τ =

√ma3

√α1 + α2

τ , α =α1

α1 + α2

H =p2

1 + p22

2− α√

(x + 1)2

+ y2− 1− α√

(x− 1)2

+ y2=

a√α1 + α2

H

Elliptic (Euler) coordinates:

x = uv , y = ±√

u2 − 1√

1− v2


Performing again a standard separation process:

H =1

u2 − v2 (Hu + Hv) , Hu =u2 − 1

2p2

u − u , Hv =1− v2

2p2

v − (1− 2α)v

with a new local time:ds =

dτu2 − v2

we obtain the separated first-order equations:(duds

)2

= 2(u2 − 1)(hu2 + u + g) ,

(dvds

)2

= 2(1− v2)(−hv2 + (1− 2α)v + g)

in terms of the energy H = h, the value of the second invariant: G = g and therelative intensity of the centers (α). All the involved quadratures are obviouslyelliptic, Demin (1961), Strand & Reinhardt (1979), . . . (see Mathuna (2008)).


Radial Integral:

s− s0 =

∫ u

u0

dξ√(ξ2 − 1)(h ξ2 + ξ − g)

(ξ2 − 1)(h ξ2 + ξ − g) = h(ξ − ξ1)(ξ − ξ2)(ξ − ξ3)(ξ − ξ4)

ξ1 = 1 , ξ2 = −1 , ξ3 =−12h

+

√gh

+1

4h2 , ξ4 =−12h−√

gh

+1

4h2

Two different regimes are allowed:

Case 1) −1 < 1 < ξ4 < u < ξ3

Case 2) −1 < ξ4 < 1 < u < ξ3


Angular Integral:

s− s0 =

∫ v

v0

dη√(1− η2)(−h η2 + (1− 2µ)η + g)

η1 = 1 , η2 = −1 , η3 =1− 2α

2h+

√gh

+(1− 2α)2

4h2 , η4 =1− 2α

2h−√

gh

+(1− 2α)2

4h2

Five allowed cases:

Case i) η4 < −1 < η3 < v < 1Case ii) −1 < η4 < η3 < v < 1Case iii) −1 < v < η4 < η3 < 1Case iv) −1 < v < 1, η3, η4 ∈ CCase v) η4 < η3 < −1 < v < 1


Three types of orbits: tl (lemniscatic orbits), tp (planetary orbits), ts and ts’ (satelitaryorbits). W.P Waalkens, Dullin & Richter (2004).Bifurcation diagram for α = 1

3 :


From planar solutions to spherical orbitsRecapitulation:

Problem in S2, with radius R and inter-center "distance" 2θf . Strengths: γ1 andγ2. Variables x, y, z and time t ⇒

Gnomonic projection gives a planar system of differential equations in variablesx, y and local time τ ⇒

Affine transformation: x1 ≡ x, x2 = sec θf y. These equations become aNewtonian two-fixed center problem with inter-center distance a = R tan θf andstrengths: αi = sec2 θf γi.

The transformation is an isomorphism between the spherical orbits lying in thehalf-sphere (z > 0) and the bounded orbits of the planar problem.

Going back we can obtain explicitly all the orbits of the problem in S2 that not crossthe equator.


A concrete example of Planetary typeLet us choose the following set of parameters:

Spherical problem: R =√

3, θf = π6 , γ1 = 3

4 , γ2 = 32

Planar problem: a = 1, α1 = 1, α2 = 2⇒ α = 13 .

And values for h and g:

h = −14, g =

45⇒ tp

that determine a planetary type orbit.

x1(s) = −2

√83−

√43 +

(√83 +

√43)

cn(cvs|k2

v)

√83 +

√43 +

(√83−

√43)

cn (cvs, k2v)

3 +√

5− 2√

5 sn2 (cus, k2u)

5− 2√

5 + 4√

5 sn2 (cus, k2u)

x2(s) =

2 4

√3569

(201 + 88

√5)

sn(cvs|k2

v)

√83 +

√43 +

(√83−

√43)

cn (cvs|k2v)

dn(cus|k2

u)

2√

5 + 5− 4√

5 sn2 (cus|k2u)

cu =√

1120 + 1√

5, cv =

√√83√

432√

15, k2

v = 12 −

332√

83√

43, k2

u = 841 (11

√5− 20)


The corresponding orbit in S2 is:


with parametric equations:

x(s) = −2((√

129 +√

249)

cnv +√

249−√

129)√(√

3569− 126)

cn2v −80 cnv−

√3569− 126

×

(−4√

5 sn2u +2√

5 + 5)(

2√

5 sn2u−√

5− 3)

(4√

5 sn2u−2√

5− 5)√−320 sn4

u +8(

40 + 21√

5)

sn2u−84

√5− 191

where: snu = sn(cu s|k2

u

)and so on.

And similar formulas for y(s) and z(s).

The "time" parameter s is the local time introduced in the separation of variables inelliptic coordinates.


Finally, it is interesting to search for periodic solutions that appear when the periodsof the elliptic functions are commensurable.

Two examples of planetary closed orbits:


Thank you very much


On the relations between the spherical and planar two ... · Workshop to honour Prof. Orlando...

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