A non-linear oscillator with quasi-harmonic behaviour: two- and n-dimensional oscillators
A non-linear oscillator with quasi-harmonic behaviour: two- and n-dimensional oscillators
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DOI：
10.1088/0951-7715/17/5/019
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发表时间：
2004-06
期刊：
影响因子：
1.7
通讯作者：
J. Cariñena;M. F. Ranada;M. Santander;M. Senthilvelan
中科院分区：
文献类型：
--
作者：
J. Cariñena;M. F. Ranada;M. Santander;M. Senthilvelan
利用拉格朗日和哈密顿形式研究非线性二维系统。该模型是作为先前在经典和量子水平上研究的一维振荡器的二维版本获得的。首先证明它是一个超可积系统，然后求解非线性方程组并显式得到解。所有有界运动都是准周期振荡，无界（散射）运动由双曲函数表示。在第二部分中，系统被推广到n个自由度的情况。最后，讨论了该非线性系统与常曲率空间（二维球体 S2 和双曲平面 H2）上谐振子的关系。
A non-linear 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 super-integrable system, and then the non-linear equations are solved and the solutions are explicitly obtained. All the 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 non-linear system to the harmonic oscillator on spaces of constant curvature, the two-dimensional sphere S2 and hyperbolic plane H2, is discussed.