Dynamics of constrained many body problems in constant curvature two-dimensional manifolds

Abstract

In this paper, we investigate systems of several point masses moving in constant curvature two-dimensional manifolds and subjected to certain holonomic constraints. We show that in certain cases these systems can be considered as rigid bodies in Euclidean and pseudo-Euclidean three-dimensional spaces with points which can move along a curve fixed in the body. We derive the equations of motion which are Hamiltonian with respect to a certain degenerated Poisson bracket. Moreover, we have found several integrable cases of described models. For one of them, we give the necessary and sufficient conditions for the integrability.

This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

1. Introduction

Recently, a lot of work has been devoted to study N body problem in spaces of constant curvature. Usually, it is assumed that points interact according to a certain generalization of the Newton Law of gravitation or the Hooke Law of elasticity. For a detailed overview of these works, we refer the reader to [1].

Investigations of such systems show that even very simple models integrable in the Euclidean space are not integrable when generalized to a space with constant non-zero curvature.

In this paper, we propose to study simple models of several points in a constant curvature two-dimensional space M² where M² is Euclidean plane , sphere or hyperboloid .

Motion of points is restricted by certain holonomic constraints. We assume that there are no other forces of interactions between points except those induced by constraints.

We notice that a dumbbell on a sphere is just a rigid body in Euclidean space and a two-chain system on a sphere can be interpreted as a rigid body which has one point inside which can move along a circle fixed in the body frame. We derive general form of equations of motion for this type of systems.

In the same way, we show that a dumbbell in a hyperboloid can be considered as a rigid body in a pseudo-Euclidean space. Similar to the Euclidean case, we derive equations of motion for a rigid body with one movable point inside. The problems of rigid bodies in non-Euclidean spaces are discussed in [2].

The obtained equations have Hamiltonian form with respect to certain degenerated Poisson structures.

We investigate integrability of these system. For this purpose, we use methods of differential Galois theory.

2. A chain in Euclidean plane

A n chain is a system of (n + 1) points . Point P_i is connected with point P_i+1 by a massless perfectly rigid rod and the junction is a spherical hinge. Thus, the points form an open chain with n links. If n = 1, then it is a dumbbell, and an example of a 2-chain is shown in figure 1.

Figure 1.

Geometry of the linear 2-chain. (Online version in colour.)

Further description in this section is just a shorter version of that presented in [3]. Let m_i and q_i denote the mass and the radius vector of point P_i, respectively. The holonomic constraints have the following form:

2.1

Hence

2.2

where

2.3

and

2.4

Clearly, (q₀, e₁, …, e_n) specify the configuration of the system. Hence, its configuration space is . Instead of q₀ one can take an arbitrary point q_i with 0 ≤ i ≤ n. However, it is natural to specify a configuration by (r, e₁, …, e_n), where r is the radius vector of the centre of the mass

2.5

Then, from (2.2) we obtain

2.6

and

2.7

for i = 0, …, n. Direct calculations show that the kinetic energy of the system is

2.8

where

2.9

It can be shown that

2.10

The total kinetic energy is the Lagrange function of the system, so the Lagrange equations of motion have the form

2.11

where λ_α are the Lagrange multipliers. Their dependence on dynamical variables r, e₁, …, e_n and velocities , , …, , can be deduced from constraints

2.12

Clearly, the total momentum of the system is a first integral. Moreover, the total angular momentum c = r × p + g, where

2.13

is also a conserved quantity. Thus, the chain centre of mass can be taken as the origin the inertial frame. Then, the equations of motion are reduced to system (2.12). In the considered planar case, it is convenient to parametrize the problem just putting

2.14

In these variables, the Lagrangian of the system reads

2.15

Lemma 2.1 —

System defined by Lagrangian (2.15) is integrable for n < 3.

Proof. —

The system has n degrees of freedom and has the energy integral, so it is evidently integrable for n = 1. It is also integrable for n = 2 because it has an additional integral g defined by (2.13) which in terms of variables reads

2.16

One can also show it in another way. The Lagrangian depends on the differences of angles. Thus, we can choose as generalized coordinates q₁ = φ₁ and q_α = φ_α − φ₁, for α = 1, …, n. Then q₁ is a cyclic coordinate and the corresponding canonical momentum is an additional first integral. ▪

Numerical experiments given in [4] allow one to conjecture that for n > 2 the n-chain problem is not integrable.

Remark 2.2 —

If we fix one end of the chain, then we obtain just a multiple pendulum. For n > 2 this system is seemingly not integrable, see [4].

3. A chain in a sphere

As in the previous section, we consider a system of (n + 1) points P₀, …, P_n, but now these points move in the unit sphere with the centre at the origin of . The sphere will serve as a two-dimensional Riemann manifold of constant curvature κ = 1.

Let m_i and r_i denote the mass and the radius vector of point P_i, respectively. The imposed constraints now have the form

3.1

where d denotes the spherical distance. Equivalently, we can put

3.2

where α_i∈( − 1, 1). The kinetic energy of the system is

3.3

The Lagrange equations have the following form:

3.4

where λ_i and γ_i are the Lagrange multipliers and γ₀ = γ_n+1 = 0.

Group is a symmetry group of this system. In fact, its dynamics does not depend on the orientation of the inertial frame because all constraints are invariant with respect to the natural action of in . Thus, the total angular momentum

3.5

is a first integral of equations (3.3).

It is difficult to perform efficient analysis of the described system because even for small n the analytical form of equations of motion in terms of any reasonable parametrization is very complicated. This is why we decided to derive equations of motion in a different way.

At first let us consider the case n = 1, that is two masses m₀ and m₁ which form a dumbbell in a sphere. As r₀ · r₁ = α₁∈( − 1, 1), vectors r₀ and r₁ are not parallel. Thus, we can define orthonormal frame with

3.6

Hence, the configuration space of spherical dumbbell is . In fact, we can consider it as a rigid body with tensor of inertia

Eigenvalues of I give the following principal moments of inertia:

Finally, the spherical dumbbell equations of motion can be written in the following form:

3.7

where Ω = (Ω₁, Ω₂, Ω₃) is the angular velocity of the frame projected to the axis of this frame, is the orientation matrix of frame with respect to the inertial frame with entries a_i,j: = e_i · b_j, for i, j = 1, 2, 3. Matrix has the form

3.8

Summarizing, the spherical dumbbell is equivalent to a rigid body with one fixed point and with the tensor of inertia satisfying I₁ + I₂ − I₃ = 0.

Now, let us consider three point masses m₀, m₁ and m₂ on the unit sphere (figure 2a). If they form an open chain, then their motion is restricted by two constraints

3.9

Similar to the case of spherical dumbbell, we attach an orthonormal frame , with b_i defined in (3.6), to the subsystem consisting of masses m₀ and m₁. In this frame, mass m₂ moves along a circle (figure 2b).

Figure 2.

Three points in a sphere. (a) Geometry in the inertial frame and (b) geometry in the body frame. (Online version in colour.)

As result we can consider a spherical 2-chain as a rigid body with an additional point moving freely inside it along a circle.

4. Rigid body with a movable point inside

We assume that N points with masses m₁, …, m_N form a rigid body, i.e. mutual distances between them are constant. One point of the body is fixed at the origin O of inertial frame . With the rigid body, we fix orthonormal frame . Its origin coincides with O. Inside the body moves a point with mass m₀.

The orientation of the body frame with respect to the inertial frame is given by matrix with entries that are inner products of base vectors A_i,j = e_i · b_j.

For a vector x, its coordinates in the inertial frame are denoted by x = [x₁, x₂, x₃]^T and by corresponding capital letter X = [X₁, X₂, X₃]^T we denote its coordinates in the body frame . The mapping

is the standard isomorphism between Lie algebra with the vector product as the Lie bracket and . We recall a few basic properties of this mapping that will be applied systematically:

for any .

If r is a vector radius to an arbitrary point in rigid body, then r = AR and . This is why

4.1

where and Ω is the body angular velocity. The time evolution of r₀ is given by

4.2

Kinetic energy of the our system T consists of kinetic energy of rigid body T_B and kinetic energy T_P of the movable point r₀. The kinetic energy of the body is given by

4.3

where we introduced the tensor of inertia of the rigid body

4.4

In a similar way, we calculate the kinetic energy of moving point

4.5

Hence, the kinetic energy of the system can be written in the following form:

4.6

where we introduced the tensor of inertia of whole system

4.7

Let us notice that because R₀ = R₀(t) tensor of inertia is not constant.

As local coordinates on we take the Euler angles of the type 3-1-3, i.e. we put A = A₃(ϕ)A₁(θ)A₃(ψ), where

4.8

From equality , we obtain

4.9

The form of R₀ depends on the considered problem. When we assume that our material point moves along a certain curve fixed in rigid body frame, then

where z = z(t) is a local parameter on the curve. In order to obtain equations of motion, we can use Lagrange or Hamiltonian formalism. As the generalized coordinates one can use

4.10

The Lagrange function is

where components of Ω_i in T given in (4.6) are expressed by q and according to (4.9). Then we just write the Lagrange equations

4.11

The Hamilton function of the system is given by

4.12

where are expressed in terms of canonical momenta by solving the Legendre equations

4.13

Unfortunately, the presented classical derivation gives an extremely complicated form of equations of motions. This is why we decided to derive them in another form.

The angular momentum of the system is

4.14

where M is the angular momentum in the body frame. It is given by

4.15

where I is tensor of inertia for whole system given by (4.7). The Lagrange function can be written in the following form:

4.16

Our aim now is to derive the Poincaré equations for the considered system (for a short overview of the Poincaré equations, see [5, ch. 1]). The generalized coordinates are q = (φ, θ, ψ, z), and we take ξ = (Ω₁, Ω₂, Ω₃, v) as the quasi-velocities. Then

4.17

where vector fields w_i(q) are given by

4.18

Formulae for w₁, w₂ and w₃ are obtained by solving relations (4.9) with respect to , and . These vector fields satisfy the following commutation relations:

4.19

where

4.20

and c^k_ij = 0 for remaining values of indices.

The Poincaré equations have the form

4.21

As L does not depend on the Euler angles the above system reads

4.22

This system has to be supplemented by equations (4.17). However, for the considered system to make (4.22) closed we need to add only one equation .

We can transform system (4.22) into the Hamiltonian form using procedure of Chetaev (see [5]). At first we define the momenta associated with the quasi-velocities

4.23

In our case η = (M₁, M₂, M₃, p). Then, we perform Legendre transformation

4.24

After this transformation, the equations of motion read

4.25

For the considered system, the explicit form of the Legendre transformation is the following:

4.26

where

4.27

The Hamilton function has a simple form

4.28

The equations of motion are the following:

4.29

These equations are Hamiltonian with respect to a degenerated Poisson bracket. In fact, denoting x = (M₁, M₂, M₃, z, p) and defining

4.30

we can write system (4.29) in the form

4.31

Matrix P defines a degenerated Poisson structure of rank four with one Casimir function

For the integrability of system (4.29), we need one additional first integral which is functionally independent with H and K.

5. Point mass on a circle in a rigid body

We consider dynamics of a rigid body with point mass moving in a circle lying in a plane perpendicular to third principal axis b₃ (figure 3). We assume that the centre of the circle lies on the principal axis. Thus,

For this choice of R₀, we obtain

5.1

and

5.2

We chose such units of the length and the mass that a = 1 and m₀ = 1.

Figure 3.

Geometry of a point mass on a circle in a rigid body.

Under these assumptions, dynamics of the system in variables [M₁, M₂, M₃, z, p_z] is described by equations (4.31) with Hamilton function

5.3

For integrability of this system, one more first integral independent of Hamiltonian H and Casimir K is necessary.

In our integrability analysis, we used physical conditions satisfied by principal moments of inertia. They should be non-negative I₁≥0, I₂≥0 and I₃≥0 and satisfy triangle inequalities

5.4

If one of the principal moments is zero, e.g. I₁ = 0, then these inequalities imply that remaining two principal moments are equal to each other, in this case I₂ = I₃.

The main integrability result is the following.

Theorem 5.1 —

System (4.31) with Hamiltonian (5.3) is integrable in class of meromorphic functions of variables if and only if

• — I₂ = I₁ with additional first integral F = M₃ − p, or

• — I₁ = 0 and I₃ = I₂, with or

• — I₂ = 0 and I₃ = I₁ and additional first integral is F = M²₂ + M₃p − M₁(bp + M₂sinz)secz.

To prove this theorem, we apply the Morales–Ramis theory that gives necessary integrability conditions in the class of meromorphic functions of variables. The main idea of this method is to investigate the variational equations along a particular solution. The integrability of the system is reflected by the properties of the differential Galois group of these equations. The fundamental theorem of this theory was formulated in [6].

Theorem 5.2 —

If a Hamiltonian system is meromorphically integrable in the Liouville sense in a neighbourhood of a particular solution, then the identity component of the differential Galois group of the normal variational equations along this solution is Abelian.

Detailed description of this theory with many examples can be found in the book [6] or review article [7] and references therein. After selection of values of parameters such that necessary integrability conditions are satisfied, we search for a first integral.

Proof. —

Equations of motion (4.31) have quite complicated form and we do not write them explicitly. System possesses an invariant plane on which equations of motion simplify to

5.5

This motion corresponds to a uniform rotation of the rigid body around its third principal axis. If [X₁, X₂, X₃, Z₁, Z₂]^T denote variations of [M₁, M₁, M₂, z, p]^T, then variational equations along this particular solution take the form

and z satisfies (5.5). As according to the last equation , we can fix Z₂(t) = 0. Then equations for X₁ and X₂ give closed subsystem that can be rewritten as a second-order differential equation for X₁

5.6

Next, we introduce new independent variable y = cos(2z). Then (5.6) transforms into equation

5.7

with rational coefficients

5.8

where we introduced new parameters

5.9

After transformation

5.10

this equation takes its reduced form

5.11

with coefficient r(y)

where we denote singularities y₁ = 1, y₂ = − 1, y₃ = α, y₄ = β and y₅ = ∞. Constants b_i depend only on parameters (α, β, γ).

Differences of exponents at finite singularities are , Δ₃ = 0 and Δ₄ = 2, respectively. If (1 + α)(α − β)( − 1 + α( − 1 + γ) − γ)( − 1 + γ)≠0, then the order of infinity is 3.

To investigate the differential Galois group of our normal variational equation (5.11) with rational coefficient r(y), we will use the Kovacic algorithm [8]. It allows us to decide if an equation of the form (5.11) possesses a Liouvillian solution. But applying it, we also obtain information about the differential Galois group of this equation which is an algebraic subgroup of . There are four possible cases. This group is either reducible (traingularizable), imprimitive, finite non-imprimitive, or it is whole when equation (5.11) has no a Liouvillian solution. If the last case occurs, then the system is not integrable.

We start analysis from the generic case when the variational equation (5.11) has four different singularities in complex plane , and the order of infinity is 3. In terms of parameters, it means that

5.12

One can check easily that local solutions around y₁ = α contain logarithmic terms. This implies that is either a subgroup of the Weil triangular group, or . In order to check whether differential Galois group can be triangular, we apply the first case of the Kovacic algorithm [8]. In this case, sets of exponents are the following:

Now we select elements e = (e₁, e₂, e₃, e₄, e₅) in the Cartesian product E₁ × E₂ × E₃ × E₄ × E₅ for which . There are only two such elements e and both give d(e) = 0:

According to the algorithm, we have to check if there exists a polynomial of degree d(e) which is a solution of appropriate differential equation. For e = e⁽¹⁾, this requirement gives the following conditions:

and

and for e = e⁽²⁾ these ones:

and

We notice that in both cases the second equality cannot be satisfied, because we assumed that degree of infinity is three and in generic case .

Analysis of all non-generic cases proceeds in a similar way but is lengthy, so we do not present it here. ▪

Remark 5.3 —

The physical restrictions imposed on the principal moments of inertia are very important. Without them, the analysis is more complicated and in general does not give definite answer. For example, let us assume that I₁ = 0 but I₂≠I₃. Then the normal variational equation simplifies to the Riemann P equation which is solvable iff

where d is a non-negative integer. This statement one can prove by applying Kimura theorem [9]. For the above values of parameters, necessary conditions for the integrability are fulfilled.

If I₂ = 0 but I₁≠I₃, then the normal variational equation depends on two parameters

and reduces to a Heun equation which is solvable iff

where d is a non-negative integer.

In both cases, the problem of the integrability of the system for the specified exceptional values of parameters remains open.

6. A chain in a hyperbolic plane

As a model of the hyperbolic plane , we choose a hyperboloid

6.1

The hyperbolic distance between two points is defined by

6.2

where

6.3

is the pseudo-Euclidean bilinear form.

A hyperbolic n-chain is a system of n + 1 points with n holonomic constraints

6.4

Let . We define map requiring that z = C(x, y) is a vector satisfying

6.5

As it is easy to show

6.6

This is why one can interpret it as a vector product in pseudo-Euclidean space . More importantly, one can easily check that with bracket [x, y]: = C(x, y) is a Lie algebra.

Later we will need the following facts.

Lemma 6.1 —

Let and z = C(x, y)≠0. Then

• (i) If J(x, x) = 1 and J(y, y) = 1, then J(z, z) < 0.

• (ii) If J(x, x) = 1, J(y, y) = − 1, and J(x, y) = 0, then J(z, z) = − 1.

Proof of this lemma is left to the reader.

At first, we consider a hyperbolic dumbbell consisting of two points with masses m₀ and m₁, respectively. The kinetic energy of this system is given by

6.7

In this form, the kinetic energy as well as constraint J(r₀, r₁) = l > 1 are invariant with respect to natural action of group SO(1, 2) on . Hence, as in the case of the spherical dumbbell, we define a frame {b₁, b₂, b₃} in which the dumbbell rests. As we assumed that l > 1, vectors r₀ and r₁ are not parallel, so z = C(r₀, r₁)≠0. So, we put

6.8

Notice that by lemma 6.1, J(z, z) < 0 and J(b₂, b₂) = − 1. Hence, matrix A = [b₁, b₂, b₃] is an element of SO(1, 2), that is A^TJA = J. If A is time dependent matrix, then

6.9

where matrix satisfies

6.10

Thus, is an element of Lie algebra SO(1, 2). A matrix B belongs to SO(1, 2) iff it is of the form

6.11

Now, assignment

6.12

defines an injective map. In fact, this map is a Lie algebra isomorphism of SO(1, 2) and considered as Lie algebra with Lie bracket defined by C; see (6.6).

If for we have , then where

6.13

Let us set r_i = AR_i with , for i = 0, 1. Then

6.14

and the kinetic energy can be rewritten in the following form:

6.15

where

6.16

is the tensor of inertia.

As coordinates on the group SO(1, 2), we take (q₁, q₂, q₃) which are similar to the Euler angles. Namely, for A∈SO(1, 2) we put

6.17

where matrix A₁(x) is defined in (4.8). Explicit form of vector Ω is the following:

6.18

Components of this vector can be taken as quasi-velocities and we can write equations of motion in the Chetaev form

6.19

which have two first integrals

6.20

Hence, the equations of motion of the hyperbolic dumbbell are integrable. Moreover, from the form of these equations we see that the hyperbolic dumbbell can be considered as a rigid body with one fixed point in the pseudo-Euclidean space.

The above considerations show that instead of a 2-chain in the hyperbolic plane we can investigate a more general problem of a rigid body in the pseudo-Euclidean space with a point inside it which can move along a specified curve. The derivation of these equations follows the steps presented in §4. Here we also assume that a point of mass m₀ is restricted to move along a curve R₀(z) in the body frame.

The kinetic energy of the system can be written in the form

6.21

where A: = R₀′^TJR₀′, , and I is the inertia tensor of the system, that can be written as

6.22

Here I_B is the inertia tensor without movable point, so it is constant. The Lagrange function of the system is just the kinetic energy L = T, so the Legendre transformation has the form

6.23

Notice that up to the definition of A, I and S, these equations are the same as those given in §4. Hence we have

6.24

where

6.25

The Hamilton function has a simple form

6.26

To write equations of motion in explicit form, we need vector fields w_i(q) which allow us to express the generalized velocities in terms of quasi-velocities ξ = (Ω₁, Ω₂, Ω₃, v); see (4.17). For the considered case, we obtained

6.27

Non-vanishing commutators of these fields are

6.28

With these relations, we obtain

6.29

These equations are Hamiltonian with respect to a degenerated Poisson bracket. In fact, denoting x = (M₁, M₂, M₃, z, p) and

6.30

we can write system (4.29) in the form

6.31

Matrix P defines a degenerated Poisson structure of rank four with one Casimir function

7. Pseudo-euclidean rigid body with a moving point

In this section, we consider a rigid body in the pseudo-Euclidean space with a point with mass m₀ which can move in the body frame along a hyperbola. More precisely, we assume that vector R₀(z) is

To simplify formulae, we rescale variables in such a way that m₀ = 1.

Dynamics of the system is described by means of variables [M₁, M₂, M₃, z, p] evolving according to Poisson equations of motion (6.31) with Hamilton function

7.1

Casimir function for this Poisson structure is

and for integrability just one additional first integral is necessary.

The integrability of this system is summarized in the following theorem.

Theorem 7.1 —

System (6.31) with Hamiltonian (7.1) is integrable in class of meromorphic functions of variables in the following cases:

• — if I₂ = − I₁ then the additional first integral is F = M₃ − p, 

• — if I₁ = 0 and I₃ = I₂, then F = M²₁ − M₃p − M₁M₂cothz, 

• — if I₂ = 0 and I₃ = − I₁ and additional first integral is F = M²₂ + M₃pz − M₁M₂tanhz.

The above theorem gives only sufficient conditions for the integrability and it can be checked by direct calculations. However, we selected integrable cases performing detailed analysis of variational equations for a particular solution which is located in invariant plane on which equations of motion simplify to

7.2

Let us consider the other example when a point mass is restricted to move along a circle

To simplify formulae, we rescale variables in such a way that m₀ = a = 1. For this case, our analysis of the integrability is summarized in the following theorem.

Theorem 7.2 —

System is integrable in class of meromorphic functions in the following cases:

• — if I₃ = I₂ with additional first integral F = M₁ − p_z, 

• — if I₂ = 0 and I₃ = − I₁, then

• — if I₃ = 0 and I₂ = − I₁, then

In this case, equations of motion possess an invariant plane on which equations of motion simplify to

7.3

Analysis of variational equations along the corresponding particular solution allowed us to distinguish integrable cases given in the above theorem.

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Author's contributions

Both authors made substantial contributions to the conception of this article, proof of the theorems, analysis of presented applications, drafting the article, revising it critically and the final approval of the version to be published.

Competing interests

The authors declare that they have no competing interests.

Funding

This paper was partially supported by grant no. DEC-2013/09/B/ST1/04130 of National Science Centre of Poland.