^{1,a)}and A. V. Turbiner

^{1,b)}

### Abstract

The classical mechanics of two Coulomb charges on a plane (*e* _{1}, *m* _{1}) and (*e* _{2}, *m* _{2}) subject to a constant magnetic field perpendicular to the plane is considered. Special “superintegrable” trajectories (circular and linear) for which the distance between charges remains unchanged are indicated and their constants of motion are presented. The number of the independent constants of motion for the special trajectory is larger than for generic ones and hence they can be called “superintegrable.” A classification of pairs of charges for which special trajectories occur is given. The special trajectories are analyzed for three particular cases, namely that of two electrons, an electron-positron pair, and an electron-α-particle pair.

The authors are grateful to J. C. López Vieyra for his interest in the present work, helpful discussions, and the assistance with computer calculations. A.V.T. thanks P. Winternitz for a valuable remark. This work was supported in part by the University Program FENOMEC, by the PAPIIT (Grant No. IN109512) and CONACyT (Grant No. 166189) (Mexico).

I. INTRODUCTION

II. TWO CHARGES IN A MAGNETIC FIELD

III. SPECIAL TRAJECTORIES

A. Configuration I ()

B. Configuration II

C. Configuration III

D. Configuration IV

E. Configuration V

IV. CONCLUSIONS

### Key Topics

- Magnetic fields
- 20.0
- Kinematics
- 13.0
- Angular momentum
- 6.0
- Poisson's equation
- 4.0
- Charge coupled devices
- 3.0

## Figures

Configuration I: Both particles rotate symmetrically on the same circle of diameter ρ with frequency ω. If *e* _{ c } ≠ 0, the frequency and the center of the orbit is fixed in the space. For identical particles (*e* > 0, *m*) for a given ρ, the frequency takes two values corresponding to different v; the center of the orbit rotates around the guiding center ** ρ **

_{0}with frequency , see text.

Configuration I: Both particles rotate symmetrically on the same circle of diameter ρ with frequency ω. If *e* _{ c } ≠ 0, the frequency and the center of the orbit is fixed in the space. For identical particles (*e* > 0, *m*) for a given ρ, the frequency takes two values corresponding to different v; the center of the orbit rotates around the guiding center ** ρ **

_{0}with frequency , see text.

Both particles rotate on concentric circles with relative phase π with angular frequency ω. If *e* _{ c } = 0, the frequency , if *e* _{ c } ≠ 0 the frequency .

Both particles rotate on concentric circles with relative phase π with angular frequency ω. If *e* _{ c } = 0, the frequency , if *e* _{ c } ≠ 0 the frequency .

Both particles rotate with the zero relative phase on concentric circles with angular frequency .

Both particles rotate with the zero relative phase on concentric circles with angular frequency .

A dipole with fixed dipole moment moves with constant velocity.

A dipole with fixed dipole moment moves with constant velocity.

Circular motion in the Born-Oppenheimer approximation (*m* _{2} → ∞).

Circular motion in the Born-Oppenheimer approximation (*m* _{2} → ∞).

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