Differential Equations and Mathematical Physics

The oscillator's model with broken symmetry

D. B. Volov

Samara State Transport University, Samara, 443066, Russian Federation

Abstract: The equations of the oscillator motion are considered. The exact solutions are given in the form of exponents with an additional parameter that characterizes the asymmetry of the oscillations. It is shown that these equations are the special case of the Hill's equation. The equations for the three types of exponents, including having the property of unitarity are obtained. Lagrangians and Hamiltonians are found for these equations. It is proved that all the equations are associated by canonical transformations and essentially are the same single equation, expressed in different generalized coordinates and momenta. Moreover, the solutions of linear homogeneous equations of the same type are both solutions of inhomogeneous linear equations of another one. A quantization possibility of such systems is discussed.

Keywords: Hill equation, Mathieu equation, parametric resonance, Lagrangian, Hamiltonian, canonical transformation, bitrial exponents.

UDC: 517.923

MSC: 34B30, 34M10

Original article submitted 17/XII/2014
revision submitted – 13/III/2015

DOI: 10.14498/vsgtu1379

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