Reduction of the classical MICZ-Kepler problem to a two-dimensional linear isotropic harmonic oscillator

Abstract

The classical MICZ-Kepler problem is shown to be reducible to an isotropic two-dimensional system of linear harmonic oscillators and a conservation law in terms of new variables related to the Ermanno–Bernoulli constants and the components of the Poincaré vector. An algorithmic route to linearization is shown based on Lie symmetry analysis and the reduction method [ Nucci, J. Math. Phys. 37, 1772 (1996) ]. First integrals are also obtained by symmetry analysis and the reduction method [ Marcelli and Nucci,J. Math. Phys. 44, 2111 (2002) ].

Jacobi's Three-Body System Moves like a Free Particle

“…In [39] another solvable many-body problem introduced by Calogero [7], [9] was shown to be intrinsically linear by means of Lie symmetries. The Kepler problem and MICZ-Kepler problem were also shown to be equivalent to an isotropic two-dimensional system of linear harmonic oscillators in [26] thanks to Lie symmetries. In [33] Lie group analysis -when applied to Euler-Poisson equations as obtained from the reduction method [37] -unveiled the Kowalevski top [25] and its peculiar integral without making use of either Noether's theorem [34] or the Painlevé method [25].…”

Jacobi Last Multiplier and Lie Symmetries: A Novel Application of an Old Relationship

“…The admitted Lie symmetry algebra is no longer infinite-dimensional and the Lie group analysis can be deterministically applied ( [43], [57], [49], [15], [53]). A secondary outcome of this strategy is that first integrals can also be determined ( [39], [44], [35]). …”

“…We show that, if we increase the order of the corresponding system of first-order equations using the Jacobi last multiplier, then we find enough Lie point symmetries to enable us to integrate equation (3.4)à la Lie. Note that equation (3.4) has been cited as the prototype of a solvable equation with no Lie symmetries ( [17], [35]). Equation (3.4) can be trivially transformed into a system of two first-order differential equations, i.e.,…”

“…Then the admitted Lie symmetry algebra is no longer infinite-dimensional and nontrivial symmetries of the original system can be retrieved [43]. This idea has been successfully applied in several instances ( [43], [57], [47], [44], [49], [35], [45], [15], [53]). Also in [39] we have shown that first integrals can be obtained by Lie group analysis even if the system under study does not come from a variational problem, i.e.…”

“…The key is to solve the linear parabolic determining equation which comes after the reduction method has been applied; the characteristic curves correspond to first integrals. We have used this method for finding first integrals in [39], [44] and [35].…”