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A nonlinear oscillator with quasi-harmonic behaviour: two- and \(n\)-dimensional oscillators. (English) Zbl 1068.37038

Summary: A nonlinear two-dimensional system is studied by making use of both the Lagrangian and the Hamiltonian formalisms. This model is obtained as a two-dimensional version of a one-dimensional oscillator previously studied at the classical and also at the quantum level. First, it is proved that it is a superintegrable system, and then the nonlinear equations are solved and the solutions are explicitly obtained. All bounded motions are quasiperiodic oscillations and the unbounded (scattering) motions are represented by hyperbolic functions. In the second part, the system is generalized to the case of \(n\) degrees of freedom. Finally, the relation of this nonlinear system to the harmonic oscillator on spaces of constant curvature, the two-dimensional sphere \(S^2\) and hyperbolic plane \(H^2\), is discussed.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
70H03 Lagrange’s equations
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