# SO(4)-symmetry of mechanical systems with 3 degrees of freedom

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- DOI
- 10.1080/14029251.2020.1683997How to use a DOI?
- Abstract
We answered an old question: does there exist a mechanical system with 3 degrees of freedom, except for the Coulomb system, which has 6 first integrals generating the Lie algebra π¬(4) by means of the Poisson brackets? A system which is not centrally symmetric, but has 6 first integrals generating Lie algebra π¬(4), is presented. It is shown also that not every mechanical system with 3 degrees of freedom has first integrals generating π¬(4).

- Copyright
- Β© 2020 The Authors. Published by Atlantis and Taylor & Francis
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

## 1. Introduction

It is well-known (see, e.g., [8]) that in the Coulomb field, i.e., in the mechanical system with 3 degrees of freedom (3*d* mechanical system) with the Hamiltonian

^{a}keeping Hamiltonian (1.1) invariant has a subgroup isomorphic to SO(4) acting in the domain

*H*< 0. This fact, found by V. Fock [6], helps to explain the structure of the spectrum of the hydrogen atom. Sometimes this symmetry is called

*hidden*.

An important property of this SO(4) is that the Casimirs of its Lie algebra π¬(4) restore the Hamiltonian. The Hamiltonian in Eq. (1.1) describes, for example, the motion of two particles interacting via gravity, and the motion of two charged particles with the charges of opposite sign. The number of works investigating this Hamiltonian is huge.^{b} It is therefore astonishing that the literature does not give (at least, we could not find it) the definite answer to a natural question: βdoes there exist a mechanical system with 3 degrees of freedom, except for the Coulomb system, which has 6 first integrals generating the Lie algebra π¬(4)?β posed, e.g., in [10, 12]. Two different answers to the question were given fifty years ago:

- 1)
Mukunda [10] claimed that every mechanical system with 3 degrees of freedom has 6 first integrals, generating Lie algebra π¬(4) by means of Poisson brackets.

- 2)
Szymacha and Werle [12] claimed that there are no other mechanical systems with the same property, assuming that π¬(4) contains the Lie algebra of spatial rotations of β

^{3}.

In this note, we showed that some 3*d* systems have 6 first integrals generating Lie algebra π¬(4), and some have not.

To prove that not for every system with 3 degrees of freedom its the first integrals generate π¬(4), we offer a simple necessary condition for existence of π¬(4) symmetry, see Section 4, and in Section 5 we give an example for which this condition is violated.

In Section 6 we consider the Hamiltonian of a charged particle in an homogeneous electric field. For this Hamiltonian, there exists a family of sextuples of first integrals such that every sextuple generates (by means of the Poisson bracket) the Lie algebra π¬(4).

For each element of this π¬(4), we consider the corresponding hamiltonian flow (for details, see the next section) and show that the set of these flows does not constitute the Lie group SO(4) of canonical transformations.

To avoid misunderstanding, note that we consider the *symmetry algebra* (consisting of some first integrals) of the system, not the Lie algebra of *dynamical symmetry group* introduced in [4], which is also called *non-invariance group*, see [9].

## 2. Generalities (following [1])

Recall the definition of the symmetry group of canonical transformations keeping the Hamiltonian invariant and the Lie algebra of this group. Let *H*(*q _{i}*,

*p*), where

_{i}*i*= 1, 2, 3, be a Hamiltonian of some mechanical system. We will also denote the whole set of the

*q*and

_{i}*p*for

_{i}*i*= 1, 2, 3 by

*z*, where

_{Ξ±}*Ξ±*= 1,...,6. Let the first integral

*F*of this system be a real function on some domain

*U*β β

_{F}^{6}. Let (

*q*,

*p*) β

*U*; the case

_{F}*F*=

*H*is not excluded. Then

*F*generates a 1-dimensional Lie group

*β*of canonical transformations (

_{F}*q*,

*p*) β¦ (

*q*(

^{F}*Ο*|

*q*,

*p*),

*p*(

^{F}*Ο*|

*q*,

*p*)) leaving the Hamiltonian

*H*and the domain

*U*invariant if

_{F}The transformations are defined by the relations

^{c}in β

^{6}:

Here the symplectic form *Ο* is of shape _{3} and 0_{3} are 3 Γ 3 matrices. We call the transformations Eq. (2.1)β(2.3) the *Hamiltonian flow*, generated by the Hamiltonian *F*, and denote it *β _{F}*. If a certain finite set of first integrals

*β±*= {

*F*|

_{Ξ±}*Ξ±*= 1, 2,...} has the same domain

*U*invariant under the action of all Hamiltonian flows

*β*

_{FΞ±}, then these flows generate a Lie group. The Lie algebra of this group coincides with the Lie algebra generated by the set

*β±*by means of the bracket (2.4).

## 3. The case of the Coulomb field (following [8])

Here we briefly consider the mechanical system (1.1) with the Hamiltonian

This Hamiltonian has two well-known triples of first integrals: one consists of the coordinates *L _{i}* of the angular momentum vector, the other one consists of the coordinates of the Runge-Lenz vector

*R*, defined in the domain

_{i}*E*

_{min}<

*H*<

*E*

_{max}< 0, for any pair of numbers

*E*

_{min}<

*E*

_{max}< 0 by the formulas

*Ξ΅*is an anti-symmetric tensor such that

_{ijk}*Ξ΅*

_{123}= 1.

These first integrals satisfy the following commutation relations:

Due to relations (3.4) and by definition of the domain *U*, the later is invariant under the action of Hamiltonian flows generated by the first integrals *L _{i}* and

*R*.

_{i}The relations (3.5) show that these first integrals generate the Lie algebra π¬(4).

Since π¬(4) β π¬(3) β π¬(3), we can introduce two commuting triples of first integrals

## 4. Restrictions on the rank

Let some 3*d* mechanical system have the Hamiltonian *H* and 6 first integrals *G _{Ξ±}* satisfying the commutation relations Eq. (3.7).

Consider two 6 Γ 6 matrices: the Jacobi matrix *J* with elements

*P*with elements

Then definition (4.1) of Jacobi matrix and (2.4) of brackets imply that

Suppose that *P* has two independent vectors in its kernel

*P*) = 4.

Since the symplectic form *Ο* is non-degenerate, the relation Eq. (4.3) and degeneracy of the matrix *P* imply that

So either rank(*J*) = 4 or rank(*J*) = 5. Both these cases can be realized: rank(*J*) = 5 for the Coulomb system while rank(*J*) = 4 for some of the systems described in Section 6.

## 5. Not all 3*d* mechanical systems have π¬(4) symmetry

To give an example of a 3*d* mechanical system without π¬(4) symmetry, consider the Hamiltonian

*Ο*for

_{i}*i*= 1,2,3 are incommensurable.

Evidently, each of the functions *H _{i}* is a first integral.

Let us show that each first integral of this system is a function of the *H _{i}*, where

*i*= 1, 2, 3. Indeed, let

*F*be a first integral. So,

*F*is constant on every trajectory defined for the system under consideration by relations

*r*and

_{i}*Ο*are constants specifying the trajectory. Since every trajectory given by Eq. (5.2) is everywhere dense on the torus

_{i}*F*is constant on every torus

*T*(

*r*

_{1},

*r*

_{2},

*r*

_{3}), and hence

*F*is a function of the

*r*. This implies

_{i}*F*=

*F*(

*H*

_{1},

*H*

_{2},

*H*

_{3}).

Now suppose that the system has 6 first integrals *G _{Ξ±}* satisfying commutation relations (3.7) of the Lie algebra π¬(4). Then, since

*G*=

_{Ξ±}*G*(

_{Ξ±}*H*

_{1},

*H*

_{2},

*H*

_{3}), it follows that the Jacobi matrix

*J*in Eq. (4.1) is of rank 3, and so due to Eq. (4.3) the matrix

*P*, see Eq. (4.2), is of rank 3. But this fact contradicts the easy to verify fact that if

*P*) = 4.

So, no sextuple of the first integrals of the system under consideration generates π¬(4).

## 6. An example of non-Coulomb 3*d* mechanical system with Lie algebra π¬(4) of the first integrals

Consider a particle in an homogeneous field with potential β*q*_{3}. This is a system with 3 degrees of freedom with Hamiltonian

Let

*a*is any smooth function of Hamiltonian

_{s}*H*. We denote the boundary of

*U*by

*βU*and the closure of

*U*by

*Εͺ*.

Then the real functions

*Εͺ*and smooth in

*U*. Let

*π*be the space generated by

*G*. The space

_{Ξ±}*π*, with Poisson brackets as an operation, is the Lie algebra isomorphic to π¬(4). It is subject to a direct verification that the integrals (6.3) indeed satisfy the relations (3.7) for generators of π¬(4).

The Casimirs, defined by the formulas

*a*are constant. In the case where the

_{s}*a*are constant, the Jacobi matrix for the functions (6.3) has rank 4 at the generic point. Otherwise, rank(

_{s}*J*) = 5 at the generic point.

## 6.1. Non-invariance of the domain *U* under the flows *β*_{G}

_{G}

For *Ξ»*_{2} and *Ξ»*_{3} real, such that *Ξ»*_{2} = cos*Ο*, and *Ξ»*_{3} = βsin*Ο*, we see that *G* := *Ξ»G*_{1} + *Ξ»*_{2}*G*_{2} + *Ξ»*_{3}*G*_{3} is of the shape

Set

Introduce a new variable *u* instead of *q*_{1}:

Let *z*(*Ο*_{0}) β *U*. The equations of the Hamiltonian flow *β _{G}* are then of the form

Since {*G*, *H*} = 0, it is clear that *z*(*Ο*) defined by Eqs. (6.6).

### Proposition 6.1.

*For any z*(*Ο*_{0}) β *U, there exists a first integral G _{z}* β

*π such that the Hamiltonian flow β*

_{Gz}*leads the point z*(

*Ο*

_{0})

*to the boundary of U for a finite time.*

### Proof.

We have

The system (6.7)β(6.9) can be solved explicitly for any *Ξ»*, but further on we consider only the case *Ξ»* = 0. In this case

*Ο*are constant on the trajectories.

We have

*Ο*since

If cos(*u*(*Ο*_{0})) β 0 and

Eqs. (6.13) and equality (6.11) imply that

*z*(

*Ο*) β

*U*for any

*Ο*β β. Besides, conditions (6.12) and (6.13) imply that

Now, observe that for every *z*(*Ο*_{0}) it is possible to choose *Ξ»*_{2} and *Ξ»*_{3} (i.e., *Ο*) so that cos(*u*(*Ο*_{0})) = 0. Then, for this *Ο*, we have *Q*(*Ο*/2 β *Ο*) = 0, i.e., *z*(*Ο*/2 β *Ο*) β *βU*.

### Remark 6.1.

The proof of Proposition 6.1 shows also that for each fixed *Ο*, the domain

*β*

_{Qcos(q1βp1p3+Ο)}acting on

*U*as 1-dimensional Lie group.

_{Ο}### Remark 6.2.

There is no domain *U*_{common} β *U* invariant under Hamiltonian flows *β*_{Qcos(q1βp1p3+Ο)} for all *Ο* β [0, 2*Ο*). Indeed, *U*_{common} β β©_{Ο}*U _{Ο}*, and β©

_{Ο}*U*= β since for any

_{Ο}*z*β

*U*there exists

*Ο*β [0, 2

*Ο*) such that cos(

*q*

_{1}β

*p*

_{1}

*p*

_{3}+

*Ο*) = 0.

## Acknowledgements

Authors are grateful to I.V. Tyutin and A.E. Shabad for useful discussions. S.K. is grateful to Russian Fund for Basic Research (grant No. 17-02-00317) for partial support of this work. S.B. was supported in part by the grant AD 065 NYUAD.

## Footnotes

Recall, that the transformations of the phase space that preserve the Hamiltonian form of the Hamilton equations, whatever the Hamiltonian function is, are called *canonical*.

## References

### Cite this article

TY - JOUR AU - Sofiane Bouarroudj AU - Semyon E. Konstein PY - 2019 DA - 2019/10/25 TI - SO(4)-symmetry of mechanical systems with 3 degrees of freedom JO - Journal of Nonlinear Mathematical Physics SP - 162 EP - 169 VL - 27 IS - 1 SN - 1776-0852 UR - https://doi.org/10.1080/14029251.2020.1683997 DO - 10.1080/14029251.2020.1683997 ID - Bouarroudj2019 ER -