New integrable two-centre problem on sphere in Dirac magnetic field
- PDF / 570,451 Bytes
- 15 Pages / 439.37 x 666.142 pts Page_size
- 4 Downloads / 179 Views
New integrable two-centre problem on sphere in Dirac magnetic field A. P. Veselov1,2
· Y. Ye1
Received: 14 February 2020 / Revised: 9 June 2020 / Accepted: 18 June 2020 © The Author(s) 2020
Abstract We present a new family of integrable versions of the Euler two-centre problem on two-dimensional sphere in the presence of the Dirac magnetic monopole of arbitrary charge. The new systems have very special algebraic potential and additional integral quadratic in momenta, both in classical and quantum versions. Keywords Euler two-centre problem · Dirac magnetic monopole Mathematics Subject Classification 37J35 · 70H06
1 Introduction The celebrated Euler two-centre problem [9] was one of the first non-trivial mechanical systems integrated completely since the solution of famous Kepler problem by Newton. In its two-dimensional version, the Hamiltonian has the form H= where r1 =
μ μ 1 2 ( p1 + p22 ) − − , 2 r1 r2
(q1 + c)2 + q22 , r2 =
(q1 − c)2 + q22
are the distances from the two centres fixed at the points (± c, 0).
B
A. P. Veselov [email protected] Y. Ye [email protected]
1
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
2
Faculty of Mechanics and Mathematics, Moscow State University and Steklov Mathematical Institute, Moscow, Russia
123
A. P. Veselov, Y. Ye
Fig. 1 Position of the fixed centres in the classical (left) and the new (right) systems
In the confocal coordinates u 1 = r1 + r2 , u 2 = r1 − r2 , the variables in the corresponding Hamilton–Jacobi equation can be separated, leading to the explicit solution of the system in quadratures (see Arnold [3]). More recent detailed analysis of this classical system can be found in Waalkens et al. [26]. Its natural generalisation to the spaces of constant curvature was originally found in 1885 by Killing [13] and rediscovered by Kozlov and Harin [14] as part of a general family of systems, separable in spherical elliptic coordinates. More of the history of this problem with the references can be found in Borisov and Mamaev [5], who also discussed various integrable generalisations of this system. The main observation (in the hyperbolic case due already to Bolyai and Lobachevsky and in spherical case to Serret (see [5]) is that for the system on the unit sphere S 2 the non-Euclidean analogue of the Newtonian potential μ/r is μ cot θ , where θ is the spherical distance between the particle and fixed centre. The dynamics of corresponding natural 2-centre version with U = −μ cot θ1 − μ cot θ2 , where θ1 and θ2 are the spherical distances from the fixed centres, was first studied by Killing [13], who separated the variables in the corresponding Hamilton–Jacobi equation using the Neumann elliptic coordinates on sphere. Note that this potential U has actually four singularities on the sphere, which can be interpreted as two antipodal pairs of centres with opposite charges ± μ (see Fig. 1), and thus this system should probably be considered as a 4-centre problem with Coulomb, rather than gravitational Newtonian
Data Loading...
