Journal of Nonlinear Mathematical Physics

Volume 27, Issue 1, October 2019, Pages 162 - 169

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

Sofiane Bouarroudj
New York University Abu Dhabi, Division of Science and Mathematics, P.O. Box 129188, United Arab Emirates,
Semyon E. Konstein*
I.E.Tamm department of Theoretical Physics, P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Leninskij prosp. 53, RU-119991 Moscow, Russia,
*Corresponding author
Corresponding Author
Semyon E. Konstein
Received 30 May 2019, Accepted 20 July 2019, Available Online 25 October 2019.
10.1080/14029251.2020.1683997How to use a DOI?

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 𝔬(4) by means of the Poisson brackets? A system which is not centrally symmetric, but has 6 first integrals generating Lie algebra 𝔬(4), is presented. It is shown also that not every mechanical system with 3 degrees of freedom has first integrals generating 𝔬(4).

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 (

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 (3d mechanical system) with the Hamiltonian

H=p22βˆ’1r,   where   p2:=βˆ‘i=1,2,3pi2,   r:=(βˆ‘i=1,2,3qi2)1/2,(1.1)
the symmetry group of canonical transformationsa 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 𝔬(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: β€œdoes there exist a mechanical system with 3 degrees of freedom, except for the Coulomb system, which has 6 first integrals generating the Lie algebra 𝔬(4)?” posed, e.g., in [10, 12]. Two different answers to the question were given fifty years ago:

  1. 1)

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

  2. 2)

    Szymacha and Werle [12] claimed that there are no other mechanical systems with the same property, assuming that 𝔬(4) contains the Lie algebra of spatial rotations of ℝ3.

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

To prove that not for every system with 3 degrees of freedom its the first integrals generate 𝔬(4), we offer a simple necessary condition for existence of 𝔬(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 𝔬(4).

For each element of this 𝔬(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(qi, pi), where i = 1, 2, 3, be a Hamiltonian of some mechanical system. We will also denote the whole set of the qi and pi for i = 1, 2, 3 by zΞ±, where Ξ± = 1,...,6. Let the first integral F of this system be a real function on some domain UF βŠ‚ ℝ6. Let (q, p) ∈ UF; the case F = H is not excluded. Then F generates a 1-dimensional Lie group β„’F of canonical transformations (q, p) ↦ (qF(Ο„ | q, p), pF(Ο„ | q, p)) leaving the Hamiltonian H and the domain UF invariant if

(qF(Ο„|q,p),pF(Ο„|q,p))∈UF  for  anyβ€‰β€‰Ο„βˆˆβ„.

The transformations are defined by the relations

qiF(0|q,p)=qi,   piF(0|q,p)=pi,(2.3)
where {Β·,Β·} is the Poisson bracketc in ℝ6:

Here the symplectic form Ο‰ is of shape Ο‰=(0313βˆ’1303), where 13 and 03 are 3 Γ— 3 matrices. We call the transformations Eq. (2.1)–(2.3) the Hamiltonian flow, generated by the Hamiltonian F, and denote it β„’F. If a certain finite set of first integrals β„± = {FΞ± | 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

H=p22βˆ’1r,   where   p2:=βˆ‘i=1,2,3pi2,   r:=(βˆ‘i=1,2,3qi)1/2.(3.1)

This Hamiltonian has two well-known triples of first integrals: one consists of the coordinates Li of the angular momentum vector, the other one consists of the coordinates of the Runge-Lenz vector Ri, defined in the domain

or in any of the domains Emin < H < Emax < 0, for any pair of numbers Emin < Emax < 0 by the formulas
where Ξ΅ijk is an anti-symmetric tensor such that 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 Li and Ri.

The relations (3.5) show that these first integrals generate the Lie algebra 𝔬(4).

Since 𝔬(4) ≃ 𝔬(3) βŠ• 𝔬(3), we can introduce two commuting triples of first integrals

Gi:=12(Li+Ri),   where   i=1,2,3,G3+i:=12(Liβˆ’Ri),   where   i=1,2,3,(3.6)
satisfying the commutation relations
{Gi,Gj}=βˆ‘k=1,2,3Ξ΅ijkGk,   where  i,j=1,2,3,{G3+i,G3+j}=βˆ‘k=1,2,3Ξ΅ijkG3+k,   where  i,j=1,2,3,{Gi,G3+j}=0,   where  i,j=1,2,3.(3.7)

4. Restrictions on the rank

Let some 3d mechanical system have the Hamiltonian H and 6 first integrals GΞ± satisfying the commutation relations Eq. (3.7).

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

JΞ±Ξ²:=βˆ‚GΞ±βˆ‚zΞ²,   where  α,Ξ²=1,…,6,(4.1)
and the matrix P with elements
PΞ±Ξ²:={GΞ±,GΞ²},   where  α,Ξ²=1,…,6.(4.2)

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


Suppose that G12+G22+G32β‰ 0 and G42+G52+G62β‰ 0. Then the matrix P has two independent vectors in its kernel

(G1,G2,G3,0,0,0)   and   (0,0,0,G4,G5,G6)(4.4)
due to relations (3.7), and so rank(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 3d mechanical systems have 𝔬(4) symmetry

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

H=H1+H2+H3,   where  Hi=12pi2+Ο‰i22qi2(5.1)
and where the Ο‰i for i = 1,2,3 are incommensurable.

Evidently, each of the functions Hi is a first integral.

Let us show that each first integral of this system is a function of the Hi, 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

qi=2Ο‰irisin(Ο‰it+Ο•i),   pi=2ricos(Ο‰it+Ο•i),   for   i=1,2,3,(5.2)
where the ri and Ο†i are constants specifying the trajectory. Since every trajectory given by Eq. (5.2) is everywhere dense on the torus
T(r1,r2,r3):={zβˆˆβ„6|12pi2+Ο‰i22qi2=ri2   for   i=1,2,3},(5.3)
it follows that F is constant on every torus T (r1, r2, r3), and hence F is a function of the ri. This implies F = F(H1, H2, H3).

Now suppose that the system has 6 first integrals GΞ± satisfying commutation relations (3.7) of the Lie algebra 𝔬(4). Then, since GΞ± = GΞ±(H1, H2, H3), 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 G12+G22+G32β‰ 0 and G42+G52+G62β‰ 0, then rank(P) = 4.

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

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

Consider a particle in an homogeneous field with potential βˆ’q3. This is a system with 3 degrees of freedom with Hamiltonian



where each as is any smooth function of Hamiltonian H. We denote the boundary of U by βˆ‚U and the closure of U by Εͺ.

Then the real functions

are the first integrals defined in Εͺ and smooth in U. Let π’œ be the space generated by GΞ±. The space π’œ, with Poisson brackets as an operation, is the Lie algebra isomorphic to 𝔬(4). It is subject to a direct verification that the integrals (6.3) indeed satisfy the relations (3.7) for generators of 𝔬(4).

The Casimirs, defined by the formulas

K1:=βˆ‘i=1,2,3Gi2,   K2=βˆ‘i=1,2,3G3+i2
are equal to
K1=a12,   K2=a22
and do not define the Hamiltonian only if the as are constant. In the case where the as are constant, the Jacobi matrix for the functions (6.3) has rank 4 at the generic point. Otherwise, rank(J) = 5 at the generic point.

6.1. Non-invariance of the domain U under the flows β„’G

For Ξ»2 and Ξ»3 real, such that Ξ»22+Ξ»32=1, Ξ»2 = cosΟ†, and Ξ»3 = βˆ’sinΟ†, we see that G := Ξ»G1 + Ξ»2G2 + Ξ»3G3 is of the shape

G=Ξ»p1+Qcos(q1βˆ’p1p3+Ο•),   where   Q:=a12βˆ’p12.(6.4)


so that

Introduce a new variable u instead of q1:


Let z(Ο„0) ∈ U. The equations of the Hamiltonian flow β„’G are then of the form

ddΟ„zΞ±={zΞ±,G},   i.e.,ddΟ„p3=QHcos(u)ddΟ„q3=Qp1sin(u)+QHp3cos(u)ddΟ„p2=0,   ddΟ„q2=QHp2cos(u)ddΟ„p1=Qsin(u),ddΟ„q1=Ξ»+Qp3sin(u)βˆ’p1Qcos(u)+QHp1cos(u).(6.6)

Since {G, H} = 0, it is clear that dHdΟ„=0 and dasdΟ„=0 along the trajectories z(Ο„) defined by Eqs. (6.6).

Proposition 6.1.

For any z(Ο„0) ∈ U, there exists a first integral Gz ∈ π’œ such that the Hamiltonian flow β„’Gz leads the point z(Ο„0) to the boundary of U for a finite time.


We have

and hence

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

where p1maxβ‰₯0 and ψ are constant on the trajectories.

We have

for any Ο„ since p1max is constant on every trajectory.

If cos(u(Ο„0)) β‰  0 and p12(Ο„0)<a12(Ο„0), then


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

p12(Ο„)<a12(Ο„)   for  any  τ(6.14)
i.e., z(Ο„) ∈ U for any Ο„ ∈ ℝ. Besides, conditions (6.12) and (6.13) imply that
cos(u(Ο„))β‰ 0   for  any τ.

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 (p1max)2=a12 and Q(Ο€/2 βˆ’ ψ) = 0, i.e., z(Ο€/2 βˆ’ ψ) ∈ βˆ‚U.

Remark 6.1.

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

UΟ•:={z∈U|cos(q1βˆ’p1p3+Ο•)β‰ 0}(6.15)
is invariant under the action of Hamiltonian flow β„’Qcos(q1βˆ’p1p3+Ο†) acting on UΟ† as 1-dimensional Lie group.

Remark 6.2.

There is no domain Ucommon βŠ‚ U invariant under Hamiltonian flows β„’Qcos(q1βˆ’p1p3+Ο†) for all Ο† ∈ [0, 2Ο€). Indeed, Ucommon βŠ‚ βˆ©Ο†UΟ†, and βˆ©Ο†UΟ† = βˆ… since for any z ∈ U there exists Ο† ∈ [0, 2Ο€) such that cos(q1 βˆ’ p1p3 + Ο†) = 0.


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.



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.


See, for example, [3, 5, 11] and references therein.


The definition (2.4) has the opposite sign as compared with the one given in [8], but coincides with the definition of the Poisson bracket given in [1, 2, 7, 13].


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[11]D.E. Okhotsimsky and Yu.T. Sikharulidze, Basics of dynamics of a flight through space, Nauka, Moscow, 1990. (in Russian).
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Journal of Nonlinear Mathematical Physics
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162 - 169
Publication Date
ISSN (Online)
ISSN (Print)
10.1080/14029251.2020.1683997How to use a DOI?
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 (

Cite this article

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  -
DO  - 10.1080/14029251.2020.1683997
ID  - Bouarroudj2019
ER  -