Necessary conditions for super-integrability of Hamiltonian systems. (English) Zbl 1223.37070
Summary: We formulate a general theorem which gives a necessary condition for the maximal super-integrability of a Hamiltonian system. This condition is expressed in terms of properties of the differential Galois group of the variational equations along a particular solution of the considered system. An application of this general theorem to natural Hamiltonian systems of \(n\) degrees of freedom with a homogeneous potential gives easily computable and effective necessary conditions for the super-integrability. To illustrate an application of the formulated theorems, we investigate: three known families of integrable potentials, and the three body problem on a line.
MSC:
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37J30 | Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria) |
| 32S40 | Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) |
| 11E72 | Galois cohomology of linear algebraic groups |
| 34M35 | Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms |
| 34M50 | Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
Keywords:
integrability; non-integrability criteria; monodromy group; differential Galois group; hypergeometric equation; Hamiltonian equationsReferences:
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