the maslov index of periodic orbits in the linearized ...€¦ · a property of periodic orbits of...

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International Journal of Mechanical Engineering and Technology (IJMET) Volume 8, Issue 12, December 2017, pp. 1121–1136, Article ID: IJMET_08_12_122

Available online at http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=8&IType=12

ISSN Print: 0976-6340 and ISSN Online: 0976-6359

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THE MASLOV INDEX OF PERIODIC ORBITS IN

THE LINEARIZED RHOMBUS FOUR-BODY

PROBLEM

Abdalla Mansur

Al Ain University of Science and Technology, United Arab Emirates

ABSTRACT:

This paper concerns a two parameter family of periodic solutions and

computations of the Maslov index in the linearized Rhombus four body problem. For

given four masses, there exists a family of periodic solutions, each body traveling

along an elliptic kepler orbit. By means of symmetry, we reduce the dimensions of our

problem from 16 to 4 dimensions. After making a clever change of coordinates a 4

dimensional system is obtained. We show that this system is G-Hamiltonian where G

can be found by the restriction of an invariant subspace to the linear system. In

particular, we prove that the system is no longer Canonical due to the fact that the

linear transformation is not Hamiltonian. Also we computed that the Maslov index

around the periodic solutions of the system is equivalent to the reduced Maslov index

and this index is equal to 2.

Keywords: Hamiltonian, n-body problem, equivariant action integral, reduced space

Cite this Article: Abdalla Mansur, The Maslov Index of Periodic Orbits in the

Linearized Rhombus Four-Body Problem, International Journal of Mechanical

Engineering and Technology 8(12), 2017, pp. 1121–1136

http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=8&IType=12

1. INTRODUCTION

In this paper, we introduce an interesting example which comes from Celestial Mechanics and

the rhombus four body problem. Kepler orbits of the rhombus four body problem was studied

by Ouyang Tiancheng and Xie Zhifu in 2005[1]. Given any four masses, there exists a family

of periodic solutions for which each body with the proper mass is at the vertex of a rhombus

and travels along an elliptic Kepler orbit. The question of stability for these solutions has

studied using numerical techniques by Ouyang Tiancheng and Xie Zhify in [1] while

geometric and analytic results were obtained by A. Mansur and D. Offin in [5]. The system is

non-linear with four degrees of freedom. By means of symmetry, we reduce the dimensions of

the system into a 2-degrees of freedom. Linearizing the 2-degrees of freedom system along

the periodic solution gives rise to a linear system of Hamiltonian equations. After

linearization and changing of coordinates a new 2-degrees of freedom coordinate system is

obtained, where the new system is no longer canonical, due to the fact that the transformation

Abdalla Mansur


system is not symplectic. In particular, the new system is G-Hamiltonian system where the

matrix G can be found by the restriction of the symplectic form to an invariant subspace of the

linear system. To study the stability of the periodic solution of the G-Hamiltonian system, we

will consider the circular solutions (the system will be time independent) first. By determining

the eigenvalues of the new system, a 2-parameter family of periodic solutions exist which are

isolated on an energy surface. The energy of the new system is conserved thus solutions are

restricted to 3-dimensonal integral surface

The qualitative properties of the flow near an unstable periodic solution can be considered

by introducing the stable and unstable manifolds of the orbit. The stable and unstable

manifolds are G-Lagrangian where the matrix G is anti-Hermitian and non-singular. The

tangent spaces to these manifolds along the periodic solution then describe a closed curve of

G-Lagrangian planes in R4.

An important tool used to study the stability and instability of a Hamiltonian and a G-

Hamiltonian systems, is a topological invariant called the Maslov index. The Maslov index is

a property of periodic orbits of Hamiltonian and G-Hamiltonian systems that is required in a

wide range of physical applications: semiclassical quantization[2], and classical mechanics[3].

The Maslov index of a linear periodic Hamiltonian system was introduced by Gel’fand

Lidskii[7], who called it index of rotation. Under the present name, the Maslov index was

defined by Maslov in 1965, who described as an integer associated to closed curves of

lagrangian subspaces of a symplectic vector space[8].

Daniel Offin reproduced several results of classical Flouqet theory for low dimensional

linear periodic systems using the Maslov index. For higher dimensional systems with

symmetry he generalized some results on instability[11]. In terms of the morse index of the

variational problem with periodic and anti-periodic boundary conditions, he gave necessary

and sufficient conditions for stability in 2001[9]. He also studied hyperbolicity for symmetric

periodic orbits in the isosceles three body problem as well in the parallelogram four body

problem using symmetric varitional method and the Maslov index[4, 11, 10].

As we mentioned early, the stable and unstable manifolds along the periodic solutions

describe a closed curve of G-Lagrangian subspace in R4, to which we can associate a maslov

index defined to be the intersection number of the tangent stable and unstable manifolds with

a fixed G-lagrangian subspaces. As a primary result of this work, we will show that in the

circular case this Maslov index is equivalent to the reduced index, which defined by reducing

the equations by the flow direction along the periodic orbit within an energy surface. By

explicit calculation of the index we show that this index is equal to 2.

The organization of this paper is as follows; in section 2, Kepler orbits and a complete

derivation of the rhombus four body problem are given. A more detailed discussion is given

in section 3, in which we describe the linearization about the periodic solutions and its

geometric properties. In section 4, we compute the Maslov index of the periodic orbit in the

circular case by describing the tangent stable and unstable manifolds as closed curves within

the G-lagrangian subspaces, and how this index is equivalent to the reduced Maslov index.

The Maslov Index of Periodic Orbits in the Linearized Rhombus Four-Body Problem


2. KEPLER ORBIT IN THE RHOMBUS FOUR BODY PROBLEM

We let the mass and position of the N-bodies be given by mi and qi ∈ R2, respectively.

Interaction between the masses is determined by the Newtonian potential function

(1)

On the set of non-collision configuration (where qi ≠ qj, i ≠ j). The Hamiltonian for the N-

body problem is

(2)

The Hamiltonian equations of the N-body problem are

(3)

Where the function U(q) = − V(q) is called the force function. These equations are

equivalent to the second-order differential equation system

� (4)

We begin by looking for solutions of the form

� (5)

Where q0i is a constant vector and ∅(t) is a scalar function. Identify R2 with the complex

plane by considering the qi, qi0, ∅(t) as a complex numbers. Substituting this into equation (4),

we get

Dividing both sides by ∅ and multiplying by |∅|3, we obtain

(6)

Since there is no time dependence on the right hand side, that means there is a constant λ

so that

The above equation (6) can be split into an equation for the scalar function ∅(t)

(7)

Abdalla Mansur


And an equation for the initial positions (q01, ...., q0N)

(8)

For the case N = 4, the position vectors (q1, q2, q3, q4) for the rhombus four body problem

are functions of the four masses m1, m2, m3, m4 as shown in Figure 1. Note that summing

equations (8) over all i, one obtain ∑ �� 0�� so that the center of mass is at the origin.

It can be shown that central configurations are uniquely determined up to rotations and

uniform dilations. For simplicity, we choose q1 = (−α, 0), with α > 0, and q3 = (0, β) with

β > 0. Similarly by the symmetry of rhombus, the other two coordinates q2, q4 are also

determined. Long and Ouyang [12] prove that m1 = m2 and m3 = m4 when α, β are given and

1/(√(3))α < β < √(3)α. We then scale the masses so that the parameter λ in equation (8) is set

to one. This fixes a pair of unique values of the masses m1, m3 as a function of the parameters

α, β. By checking equation (8), we find

(9)

(10)

Figure 1 The rhombus configuration.

Kepler’s equation (7) is solvable up to quadrature [14]. The solution to (7) in polar

coordinates (r, θ) with λ = 1 is given by

(11)

Where e, the eccentricity of the ellipse, and ω the angular momentum, are two parameters.

In polar coordinates, if we can write the central configuration,

Where



Then the position component of the periodic orbit is written as

(12)

The period of the orbit T, which is equivalent to the period of Kepler orbit, is given by

For the planar four body problem, system (3) is a 16 dimensional system. As is well

known, the N-body problem is a Hamiltonian system with several first integrals, therefore we

can reduce the dimension by eliminating all 8 standard first integrals. But the remaining

system still has 8 dimensions even after eliminating all first integrals. For this reason, it is still

hard to analyze the behavior of the Kepler periodic orbits. Here we can apply another

approach to reduce these dimensions by means of symmetry constraint. We turn to the

variational method to construct the constrained Hamilton system on rhombus four body

problem [15].

2.1. Symmetry Constraints. For this particular rhombus four body problem, we define the

action functional AT (q(t)) to be one of the form

Where the Lagrangian L is defined by

The action functional AT (q) is defined for absolutely continues T-periodic curves q(t) in

the configuration manifold

We will look for symmetric solutions of the equations of the motion for rhombus four

body problem. Consider the following symmetric function space in polar coordinates

Under these constraints, we have the new Lagrangian in polar coordinates

Where δ = √(r12 + r3

2) and the Lagrangian depends on the variables (r1, r3, θ1, r1 ̇, r3 ̇, θ1 ̇). Then the

corresponding conjugate variables R1, R3, Θ1 with respect to r1, r3, θ1 are

(13)

Abdalla Mansur


This Hamiltonian will be independent of θ1 which means that the angular momentum Θ1

is an integral, and θ1 is an ignorable variable. Setting the angular momentum Θ1 = c and

substituting into the Hamiltonian H1 gives reduced Hamiltonian

(14)

This reduces the system to four dimensions, with the variables (r1, r3, R1, R3). The

equations of motion in these new variables are

Using equations (11) and (12), a short calculation reveals that the relative periodic

solution, denoted in general as Γ(t), is written as

Where m1 = m2, m3 = m4 and m1, m3 satisfy the equations (9), (10). Recall that

Is the periodic solution to Kepler’s problem mentioned earlier. In addition to the four

masses, the two parameters in this solution are the eccentricity e and the angular momentum

ω of the elliptic orbits. The angular momentum integral for the full problem has the value

Θ1 = c = 2(m1α2 + m3β

2)ω.

3. LINEARIZATION AT THE PERIODIC SOLUTIONS

The linearized equation at the periodic solution Γ(t) is given by

(15)

where ξ = (u, v) represents the new coordinates for the linearized system, Hess H denotes

the Hessian, the matrix of second derivatives of H and J is the canonical symplectic form. The

Hessian matrix at Γ(t) will has the form



Where

(16)

The coefficients aij, i, j  ∈ {1, 2} satisfy the important relation

We are now in a position to give the quadratic Hamiltonian function for the linearized

equations, denoted by H0:

(17)

Letting X0 denote the Hamiltonian vector field corresponding to H0, the linearized

equation of motion (15) can then be written in Hamiltonian form:

(18)

Where the coefficients a11, a12, a22 are given in (16).

Following the Xie Zhifu discussion in [1], we give a complete proof and develop the

method to obtain the 4 × 4 G-Hamiltonian system.

A linear time-dependant periodic Hamiltonian system is one of the form

(19)

Where J is the standard symplectic matrix, and H′′

(t + T) = H′′(t). When such a system

results from linearizing about a periodic solution, there are at least two +1 multipliers. One of

these is attributable to the periodic orbit and another arises from the existence of an integral,

which in this case is the Hamiltonian H. This fact is easily proven via differentiation [13].

Indeed, given a periodic solution Γ(t) to a Hamiltonian system ξ ̇= J ∇H(ξ), plugging in Γ(t)

and differentiating with respect to t yields

Thus, Γ(̇t) is a solution of the linear system and is periodic, since Γ(t) is periodic. This

relation is very important, because it suggests a useful change of coordinates and to obtain a

new differential equation. This follows from a standard result in theory of Hamiltonian

system.

Abdalla Mansur


Define the skew-inner product of two vectors u, v ∈ C4n

as

A key trait of linear Hamiltonian systems is that the skew-orthogonal complement of an

invariant subspace is also invariant, where the skew-orthogonal complement is defined as

Lemma 3.1. Suppose Φ is an invariant subspace of the matrix JH′′(t), then the skew-

orthogonal complement Φ⊥, defined above is also an invariant subspace of the matrix JH′′(t).

Proof. Suppose u ∈ Φ⊥. Then, for any v ∈ Φ we have

Where v̂ = JH′′(t)v ∈ Φ. Thus, JH

′′(t)u ∈ Φ⊥.

The above lemma shows that a simple linear change of variables will decouple the system.

The characteristic multipliers remain the same since the transformation is linear. We need to

find an invariant subspace for A(t), to apply the same idea. We make use of the fact that the

Kepler periodic solution r(t) satisfies

(20)

Differentiating this with respect to t yields

(21)

From this we have

As expressions for the first and second derivatives of the periodic orbit. A short

calculation gives

Applying the equalities m1 = m2 and m3 = m4 to the above expressions, the vectors

Φ1: = [α, β, 0, 0], Φ2: = [0, 0, 2m1α, 2m3β], which are the initial conditions for Γ ̇ and Γ̈

respectively, will span an invariant subspace Φ for A(t). This is the basis for an application of

lemma (3.1).



Consider the change of variables determined by

(22)

(Recall that c/ω = 2(m1α2 + m3β

2). The columns Φ3 = [2m3β,  − 2m1α, 0, 0],

Φ4 = [0, 0, β,  − α] of the above matrix are chosen to form a basis for Φ⊥. Consequently, this

change of variables decouples our linear system and gives a final form which is no longer

canonical. The new coordinates then can be denoted by ξ = (x, y) and defined by

(23)

(24)

Combining equations (18) and (23), we find that ξ satisfies

(25)

(26)

Where

(27)

An important observation her is that this system (25) is no longer Hamiltonian (due to the

fact that the transformation (24) is not canonical). To study the behavior of the periodic orbits,

we first show that the system is G-Hamiltonian where the matrix G will be found by the

restriction of Φ to the invariant subspace Φ⊥ or equivalently by transforming the symplectic

form ω into a G-form ω, that is, by considering the following calculations. Let x = Du and

y = Dv, then

We proceed by considering this calculation, the skew symmetric form can be determined

from the following:

Abdalla Mansur


More explicitly, the matrix G has the form

(28)

Note that the matrix G is anti-symmetric, and non-singular.

The following lemma states the G-Hamiltonian structure of the matrix B in equation (26).

The explicit formula for BG is given by B

G = G − 1B

*G so that

(29)

Theorem 3.2. The 4 × 4 system given in equation (25) is G-Hamiltonian.

Proof. In order to prove this theorem, we need to check only that BG =  − B. This is clear

by comparing (29) with (26).

Corollary 3.3. The matrizant X(t) of the G-Hamiltonian system given in (25) is a G-unitary

for all time t.

Proof. The proof is a straightforward and is left to the reader.

We are now going to consider the circular case of the G-Hamiltonian system defined in

(25), that is, the matrix B in (26) will be time independent. This G-Hamiltonian system can be

written in time independent case as

(30)

Where the matrix B defined in (26) will be of the form

(31)

And a1, a2 are the time independent values of the parameters a34 and a43 defined in (27)

Expanding equation (30) explicitly in terms of x and y, the equations of motion are given

by

(32)

We are now in a position to give the quadratic G-Hamiltonian function for the new

equations

(33)



Where

Solution to equation (30) can be expressed in the form

Where Mt defines the flow of the G-Hamiltonian system. In coordinates the flow can be

interpreted as a fundamental matrix solution of the corresponding equation, satisfying the

initial condition M(0) = I. Since the equations of motion is autonomous, it follows that the

flow is given by the matrix exponential

We will now determine the eigenvalues of the matrix B. The characteristic polynomial of

the matrix B is given by

(34)

Where a1a2 > 0.

Proposition 3.4. The matrix B has two real, and two imaginary eigenvalues.

Proof. It is clear that from equation (34), λ2 has the form

Hence, the eigenvalues of B are ±λ1, ±iλ2, where

(35)

Now we can easily compute the eigenvectors of B. The eigenvectors of the matrix B with

corresponding eigenvalues ±λ1, ±iλ2 respectively, have the form

(36)

The general real solution to the equation of motion given in (32) has the form

(37)

For α1, α2 ∈ R, β ∈ . The most important interest is that the existence of a 2-parameter

family of periodic solutions about the periodic solutions. In G-Hamiltonian variables (x, y),

the periodic solutions are of the form γ(t) = 2Re(βη1eit),  where β is complex.

We have seen previously that the linearized system about Γ(t) of equation (30) gives rise

to a 2-parameter family of periodic solutions described in terms of the flow direction X0(γ)

and its normal ∇H0(γ), defined by

(38)

Abdalla Mansur


Let the fundamental matrix Mt defines the flow of the system on the tangent space Tγ(t)R4,

the matrix MT is called the monodromy matrix of the system, and its eigenvalues are called

the characteristic multipliers of the periodic solutions. The stability of the periodic orbit γ(t) is

determined by the characteristic exponents λ1, ...., λn − 1, the eigenvalues of the matrix B or by

the characteristic multipliers eλ1T

, ..., eλn − 1T, the eigenvalues of the monodromy matrix MT. As

we have seen previously, the eigenvalues of B consist of a real pair, and a purely imaginary

pair eigenvalues, hence the multipliers of the orbit consist of the double +1 eigenvalue, and

the non-trivial pair.

4. CALCULATION OF THE INDEX

Definition 4.1.A G-Lagrangian subspace of a 2n-dimensional symplectic vector space (V, ω)

is defined to be an n-dimensional subspace on which the G-form vanishes, that is

ω(x, y) = xTGy = 0, where G is nonsingular and anti-symmetric matrix.

Definition 4.2. The Maslov index is the number of intersections of a closed curve u(t) with a

fixed G-Lagrangian subspace. In other words, for a given closed curve of a G-Lagrangian

subspace u(t),

To start computing the Maslov index of the periodic orbit for the circular solutions, we

begin with the following proposition:

Proposition 4.3. For time independent case, the stable and unstable manifolds (Ws, Wu

)

defined previously are G-Lagrangian submanifold.

Proof. In order to prove this proposition, we need only to show that the G-form vanishes.

Using the bases of the tangent spaces which described as the span of the stable or unstable

subspace and the flow direction, to show the G-form vanishes, it suffices to show

Consider the unstable case first, that is when i = 1, and expand the flow direction in terms

of η1(t) and η2(t), we compute

By direct computation, we can immediately see that ω(ζ1, ηi(t)) = 0, i = 1, 2. Hence

ω(ζ1, X0(γ)) = 0

Consideration of the stable case when i = 2 leads to the same result.

The stable and unstable manifolds are invariant under the flow M(t), hence,

Where W denotes either the stable or the unstable manifold. Since the flow is linear, it

follows that

Thus the translation of the tangent space of the subspace W along the periodic orbit

describes a curve of a 2-dimensional G-Lagrangian subspace. Furthermore, it follows that this

curve is in fact closed curve under one period T, since



It was proven by Bondarchuk [16] that the maslov index of hyperbolic closed extremals

can be computed by considering the rotation through the vertical of the stable or unstable

subspace W

We then can ask the question what is the Maslov index of the circular periodic solutions

of the rhombus four body problem computed in the same way, where Vt has the following

form:

• The transformed subspace Vt, which can be determined by transforming the vertical

space using the linear transformation matrix described in equation (24) to obtain

(39)

The following proposition shows the above subspace is G-Lagrangian.

Proposition 4.4.The subspace Vt which we defined in (39) is G-Lagrangian subspace.

Proof. The only way to check Vt is G-Lagrangian is to show the G-form vanishes, that is,

ω(u, v) = 0, where u, v  ∈ Vt. By direct computation, it is clear that

The intersection of G-Lagrangian planes must formally be considered in the four-

dimensional symplectic space TR4, however, by the following proposition it will be sufficient

to restrict our attention to the tangent space TγΣ.

Proposition 4.5. The 2-dimensional transformed G-lagrangian subspace Vt1 intersects the

tangent space Tγ(t)Σ in a 2-dimensional subspace for two values of t and in a 1-dimensional

otherwise.

Proof. Since Tγ(t)Σ is in ℝ3,  that is spanned by ζ1, ζ2 and X0(γ). Then the dimension of the

intersection with the G-Lagrangian subspace is the number of linearly independent solutions

to

(40)

Solving this equation for C1, C2 and C3, it is clear that

We have from the first equation that C1 = −C2 and from the second equation that if

C3 ∈ ℝ, then −β1sint = β2cost, or equivalently tan t = −β2/β1 = β. for any value of the constant

β, this equation has two solutions t over one period [0, 2π], which conclude that the

intersection with the G-Lagrangian subspace is a 2-dimensional subspace for those values of t

and a 1-dimensional for others.

Abdalla Mansur


It will now be shown that determining the Maslov index is equivalent to counting the

intersections of the stable or unstable manifold with the G-Lagrangian subspace projected in

the two dimensional reduced space

The projections of the stable and unstable manifolds in the reduced space are the 1-

dimensional stable and unstable subspaces defined as:

We regard these as subspaces of equivalence classes in the quotient space. We shall now

consider the G-anti-symmetric form on the full tangent space to be equivalent to the reduced

G-anti-symmetric form on the reduced space.

Lemma 4.6. The G-anti-symmetric form ω on TR4 induces a G-anti-symmetric form ωred on

the reduced space Tγ(t)Σ ∕  < X0(γ) > , given by

(41)

Proof. Since we have ω is non-degenerate, then we must verify that ωred is also non-

degenerate. Assume that

Since we have shown that ω(ζi, ηj(t)) = 0 for all i, j and the flow direction X0(t) is in the

span of η1 and η2, hence

Since ω(ζi, ζj) is anti-symmetric form for i = j, and a bilinear form for i ≠ j, it follows that

Since ω(ζ1, ζ2) ≠ 0, hence we conclude that ω([x1], [x2]) = 0 for all [x2] if and only if

α1 = β1 = 0, or equivalently [x1] = 0.

Thus the reduced space is a 2-dimensional subspace, any 1-dimensional subspace is then

G-lagrangian with respect to the induced G-form. In particular the stable and unstable

subspaces are G-Lagrangian subspaces.

Now we shall consider the reduced G-Lagrangian subspaces in the following proposition:

Proposition 4.7. We can verify that for all time t, Vt intersect Tγ(t)Σ ∕  <  X0(γ) in a 1-

dimensional subspace, denoted

Proof. We have proved in proposition (4.5) that the G-Lagrangian subspace intersects the

tangent space in a 2-dimensional subspace for two values of t and in a 1-dimensional for other

values of t. In the case of the 2-dimension of intersection with the G-Lagrangian subspace Vt,

the flow direction is vertical, since the first component of the flow is zero, that is,

( − β1sint − β2cost) = 0. In the reduced space therefore this intersection becomes 1-dimension.



For the case of 1-dimension of intersection, the flow is never vertical and hence the

intersection with Tγ(t)Σ ∕  < X0 >  becomes 1-dimension.

We can now describe the subspace Vt ∕< X0(γ), ∇H0(γ) >  as a curve of G-Lagrangian

subspaces in the reduced space. In addition, it is clear that this curve is in fact closed, hence

we can investigate the reduced Maslov index, defined to be

(42)

Where the fixed G-Lagrangian subspace E describes either the stable or unstable subspace

at γ(t)

From here we can prove our result in the following theorem:

Theorem 4.8. (Mansur) The Maslov index is equivalent to the reduced index of the orbit

γ(t), where the G-Lagrangian subspace is defined in (39) and this index is equal to 2.

Proof. By definition, we have

Let W = WU describes the unstable subspace. The intersection is non-empty, if there exists

solutions c ∈ R such that

(43)

This is equivalent to asking if there exists a solution (c1, c2, c3) of equation (40). The

dimension of the intersection is equal to the number of solutions c, either zero or one.

In the reduced space, the intersection E∩Vt ∕  < X0(γ), ∇H0(γ) >  is at least 1-dimensional

subspace, where E denotes the 1-dimensional unstable subspace. Moreover, this equivalent to

requiring that

(44)

The conditions in (43), (44) are equivalent, and hence iγ = ired(γ).

We can now determine the intersection number by expanding equation (43) in the

configuration coordinates:

(45)

A solution c exists if and if only

Or equivalently

For any value of the constant β, this equation has two solutions t over the period [0, 2π]. For these values of t, equation (45) has a solution and the intersection number is increased by

1. hence we conclude that the index is equal to 2.

Abdalla Mansur


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the maslov index of periodic orbits in the linearized ...€¦ · a property of periodic orbits of...

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