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

Sofiane Bouarroudj; Semyon E. Konstein

doi:10.1080/14029251.2020.1683997

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

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

Authors

Sofiane Bouarroudj

New York University Abu Dhabi, Division of Science and Mathematics, P.O. Box 129188, United Arab Emirates,sofiane.bouarroudj@nyu.edu

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,konstein@lpi.ru

^*Corresponding author

Corresponding Author

Semyon E. Konstein

Received 30 May 2019, Accepted 20 July 2019, Available Online 25 October 2019.

DOI: 10.1080/14029251.2020.1683997 How 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 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).
Copyright: © 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 (3d mechanical system) with the Hamiltonian

H=p22−1r, where p2:=∑i=1,2,3pi2, r:=(∑i=1,2,3qi2)1/2,(1.1)

the symmetry group of canonical transformations^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 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: “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)?” 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 𝔬(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 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(q_i, p_i), where i = 1, 2, 3, be a Hamiltonian of some mechanical system. We will also denote the whole set of the q_i and p_i for i = 1, 2, 3 by z_α, where α = 1,...,6. Let the first integral F of this system be a real function on some domain U_F ⊂ ℝ⁶. Let (q, p) ∈ U_F; the case F = H is not excluded. Then F generates a 1-dimensional Lie group ℒ_F of canonical transformations (q, p) ↦ (q^F(τ | q, p), p^F(τ | q, p)) leaving the Hamiltonian H and the domain U_F invariant if

(qF(τ|q,p),pF(τ|q,p))∈UF for any τ∈ℝ.

The transformations are defined by the relations

dqiFdτ={qiF,F}=∂F(qF,pF)∂piF,(2.1)

dpiFdτ={piF,F}=−∂F(qF,pF)∂piF,(2.2)

qiF(0|q,p)=qi, piF(0|q,p)=pi,(2.3)

where {·,·} is the Poisson bracket^c in ℝ⁶:

{F,G}:=∑i=1,2,3(∂F∂qi∂G∂qi−∂F∂pi∂G∂qi)=∑α,β=1,…,6∂F∂zαωαβ∂G∂zβ.(2.4)

Here the symplectic form ω is of shape ω=(0313−1303), where 1₃ and 0₃ are 3 × 3 matrices. We call the transformations Eq. (2.1)–(2.3) the Hamiltonian flow, generated by the Hamiltonian F, and denote it ℒ_F. If a certain finite set of first integrals ℱ = {F_α | α = 1, 2,...} has the same domain U invariant under the action of all Hamiltonian flows ℒ_{F_α}, then these flows generate a Lie group. The Lie algebra of this group coincides with the Lie algebra generated by the set ℱ 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=p22−1r, where p2:=∑i=1,2,3pi2, r:=(∑i=1,2,3qi)1/2.(3.1)

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_i, defined in the domain

U={z∈ℝ6|H(z)<0},

or in any of the domains E_min < H < E_max < 0, for any pair of numbers E_min < E_max < 0 by the formulas

Li:=∑j,k=1,2,3εijkqjpk,(3.2)

Ri:=(−2H)−1/2(∑j,k=1,2,3εijkLjpk+qir),(3.3)

where ε_ijk is an anti-symmetric tensor such that ε₁₂₃ = 1.

These first integrals satisfy the following commutation relations:

{H,Li}=0,{H,Ri}=0,(3.4)

and

{Li,Lj}=∑k=1,2,3εijkLk,{Ri,Rj}=∑k=1,2,3εijkLk,{Li,Rj}=∑k=1,2,3εijkRk.(3.5)

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 𝔬(4).

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

Gi:=12(Li+Ri), where i=1,2,3,G3+i:=12(Li−Ri), where i=1,2,3,(3.6)

satisfying the commutation relations

{Gi,Gj}=∑k=1,2,3εijkGk, where i,j=1,2,3,{G3+i,G3+j}=∑k=1,2,3εijkG3+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 G_α satisfying the commutation relations Eq. (3.7).

Consider two 6 × 6 matrices: the Jacobi matrix J with elements

Jαβ:=∂Gα∂zβ, where α,β=1,…,6,(4.1)

and the matrix P with elements

Pαβ:={Gα,Gβ}, where α,β=1,…,6.(4.2)

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

Pαβ=∑γ,δ=1,…,6JαγωγδJβδ.(4.3)

Suppose that G12+G22+G32≠0 and G42+G52+G62≠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 ω is non-degenerate, the relation Eq. (4.3) and degeneracy of the matrix P imply that

rank(P)≤rank(J)<6.(4.5)

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+ωi22qi2(5.1)

and where the ω_i for 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

qi=2ωirisin(ωit+ϕi), pi=2ricos(ωit+ϕi), for i=1,2,3,(5.2)

where the r_i and φ_i are constants specifying the trajectory. Since every trajectory given by Eq. (5.2) is everywhere dense on the torus

T(r1,r2,r3):={z∈ℝ6|12pi2+ωi22qi2=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 r_i. This implies F = F(H₁, H₂, H₃).

Now suppose that the system has 6 first integrals G_α satisfying commutation relations (3.7) of the Lie algebra 𝔬(4). Then, since G_α = G_α(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+G32≠0 and G42+G52+G62≠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 −q₃. This is a system with 3 degrees of freedom with Hamiltonian

H=p22−q3.(6.1)

Let

U:={z∈ℝ6|p12<a12,p22<a22},(6.2)

where each a_s is any smooth function of Hamiltonian H. We denote the boundary of U by ∂U and the closure of U by Ū.

Then the real functions

G1=p1,G2=a12−p12cos(q1−p1p3),G3=a12−p12sin(q1−p1p3),G4=p2,G5=a22−p22cos(q2−p2p3),G6=a22−p22sin(q2−p2p3),(6.3)

are the first integrals defined in Ū and smooth in U. Let 𝒜 be the space generated by G_α. The space 𝒜, 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:=∑i=1,2,3Gi2, K2=∑i=1,2,3G3+i2

are equal to

K1=a12, K2=a22

and do not define the Hamiltonian only if the a_s are constant. In the case where the a_s 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 ℒ_G

For λ₂ and λ₃ real, such that λ22+λ32=1, λ₂ = cosφ, and λ₃ = −sinφ, we see that G := λG₁ + λ₂G₂ + λ₃G₃ is of the shape

G=λp1+Qcos(q1−p1p3+ϕ), where Q:=a12−p12.(6.4)

Set

QH:=∂Q∂H=a1Qda1dH

so that

{zα,Q}=QH{zα,H}−p1Q{zα,p1},{zα,H}=∑i{zα,pi}pi−{zα,q3}.

Introduce a new variable u instead of q₁:

u:=q1−p1p3+ϕ.(6.5)

Let z(τ₀) ∈ U. The equations of the Hamiltonian flow ℒ_G are then of the form

ddτzα={zα,G}, i.e.,ddτp3=QHcos(u)ddτq3=Qp1sin(u)+QHp3cos(u)ddτp2=0, ddτq2=QHp2cos(u)ddτp1=Qsin(u),ddτq1=λ+Qp3sin(u)−p1Qcos(u)+QHp1cos(u).(6.6)

Since {G, H} = 0, it is clear that dHdτ=0 and dasdτ=0 along the trajectories z(τ) defined by Eqs. (6.6).

Proposition 6.1.

For any z(τ₀) ∈ U, there exists a first integral G_z ∈ 𝒜 such that the Hamiltonian flow ℒ_{G_z} leads the point z(τ₀) to the boundary of U for a finite time.

Proof.

We have

ddτu=λ−p1Qcos(u),(6.7)

ddτp1=Qsin(u),(6.8)

ddτQ=−p1sin(u),(6.9)

and hence

d2dτ2p1=−p1sin2(u)+Qcos(u)(λ−p1Qcos(u))=−p1+λQcos(u)

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

d2dτ2p1=−p1.(6.10)

and

p1=p1maxsin(τ+ψ),(6.11)

where p1max≥0 and ψ are constant on the trajectories.

We have

(p1max)2=p12+(ddτp1)2=p12+(a12−p12)sin2(u)=a12sin2(u)+p12cos2(u)=a12−(a12−p12)cos2(u)

and

(p1max)2=a12−(a12−p12(τ))cos2(u(τ))(6.12)

for any τ since p1max is constant on every trajectory.

If cos(u(τ₀)) ≠ 0 and p12(τ0)<a12(τ0), then

(p1max)2=a12−(a12−p12(τ0))cos2(u(τ0))<a12.(6.13)

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

p12(τ)<a12(τ) for any τ(6.14)

i.e., z(τ) ∈ U for any τ ∈ ℝ. Besides, conditions (6.12) and (6.13) imply that

cos(u(τ))≠0 for any τ.

Now, observe that for every z(τ₀) it is possible to choose λ₂ and λ₃ (i.e., φ) so that cos(u(τ₀)) = 0. Then, for this φ, we have (p1max)2=a12 and Q(π/2 − ψ) = 0, i.e., z(π/2 − ψ) ∈ ∂U.

Remark 6.1.

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

Uϕ:={z∈U|cos(q1−p1p3+ϕ)≠0}(6.15)

is invariant under the action of Hamiltonian flow ℒ_{Qcos(q₁−p₁p₃+φ)} acting on U_φ as 1-dimensional Lie group.

Remark 6.2.

There is no domain U_common ⊂ U invariant under Hamiltonian flows ℒ_{Qcos(q₁−p₁p₃+φ)} for all φ ∈ [0, 2π). Indeed, U_common ⊂ ∩_φU_φ, and ∩_φU_φ = ∅ since for any z ∈ U there exists φ ∈ [0, 2π) such that cos(q₁ − p₁p₃ + φ) = 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

a

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.

b

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

c

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].

References

[1]V.I. Arnold, Mathematical methods of classical mechanics, 2nd edition, Springer-Verlag, NY, 1989. Graduate texts in mathematics v. 60 xvi+520pp.

[2]P. Appell, Traité de méchanique rationelle, Gauthier-Villars et Fils, Paris, 1896. t. I–IV available at https://archive.org/details/traitdemcanique03appegoog, Tome Deuxième, Sixième Édition, Paris, Gauthier-Villars, Éditeur, 1953.

[3]M.B. Balk, Elements of dynamics of a flight through space, Nauka, Moscow, 1965. (in Russian).

[4]A.O. Barut, Dynamical symmetry group based on Dirac equation and its generalization to elementary particles, Phys. Rev, Vol. 135, No. 3B, 1964, pp. 839. 2 ser.

[5]G. Beutler, Methods of Celestial Mechanics, Textbook. Volume I: Physical, Mathematical, and Numerical Principles

[6]V.A. Fock, Zur Theorie des Wasserstoffatoms, Z. Phys., Vol. 98, No. 3–4, March 1935, pp. 145-154.

[7]F.R. Gantmacher, Lectures in Analytical Mechanics, Mir Publisher, Moscow, Russia, 1970. (in Russian).

[8]L.D. Landau and E.M. Lifshitz, Mechanics, Pergamon Press, New York, 1960.

[9]N. Mukunda, L. O’Raifeartaigh, and E. Sudarshan, Characteristic noninvariance groups of dynamical systems, Phys. Rev. Lett, Vol. 15, 1965, pp. 1041.

[10]N. Mukunda, Dynamical symmetries and classical mechanics, Phys. Rev., Vol. 155, No. 5, 1967, pp. 1383-1386.

[11]D.E. Okhotsimsky and Yu.T. Sikharulidze, Basics of dynamics of a flight through space, Nauka, Moscow, 1990. (in Russian).

[12]A. Szymacha and J. Werle, Problem of Dynamical Groups in Quantum Mechanics, Bull. Acad. Polon. Sci., Ser. Math, Astron et Phys, Vol. 14, No. 6, 1966, pp. 325-330.

[13]D. ter-Haar, Elements of Hamiltonian Mechanics, Second Edition, Pergamon Press, International Series of Monographs in Natural Philosophy, Oxford, Vol. 34, 1964.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 27 - 1
Pages: 162 - 169
Publication Date: 2019/10/25
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.1080/14029251.2020.1683997 How to use a DOI?
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/).

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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  -

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Journal of Nonlinear Mathematical Physics

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

1. Introduction

2. Generalities (following [1])

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

4. Restrictions on the rank

5. Not all 3d mechanical systems have 𝔬(4) symmetry

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

6.1. Non-invariance of the domain U under the flows ℒG

Proposition 6.1.

Proof.

Remark 6.1.

Remark 6.2.

Acknowledgements

Footnotes

References

Cite this article

6.1. Non-invariance of the domain U under the flows ℒ_G