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<p>In this study, the Hamiltonian formalism was extended to include continuous dynamical systems involving fractional derivatives, with application to a two-dimensional asymmetric oscillator system. The results were compared with those obtained using the Dirac method to validate the accuracy of the findings related to the behaviour of this class of oscillators. The Riemann–Liouville fractional derivative was employed alongside fractional variational principles to derive the fractional Euler–Lagrange equations and fractional Hamiltonian equations of motion. The results demonstrated consistency between the Hamiltonian equations of motion and the Euler–Lagrange equations. The scope of the research was further expanded to analyse the dynamics of the two-dimensional asymmetric oscillator within the framework of fractional calculus, focussing on the stability properties and distinctive behaviours of the system. This extension allowed for a deeper understanding of the complex interactions and nonlinear characteristics inherent to such systems. Subsequently, the Taylor series expansion was used to linearise the nonlinear equations, enabling their analytical solution through eigenvalue and eigenfunction techniques. This approach helped enhance the theoretical understanding of the system's dynamics and supported the validation of the results.</p>

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Generalisation of Hamiltonian formulation using fractional derivatives and its application to the asymmetric two-dimensional oscillator: an analytical and numerical study

  • 期刊论文

  • 发表日期：

    2026年2月10日

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  • Yazen M Alawaideh

摘要

In this study, the Hamiltonian formalism was extended to include continuous dynamical systems involving fractional derivatives, with application to a two-dimensional asymmetric oscillator system. The results were compared with those obtained using the Dirac method to validate the accuracy of the findings related to the behaviour of this class of oscillators. The Riemann–Liouville fractional derivative was employed alongside fractional variational principles to derive the fractional Euler–Lagrange equations and fractional Hamiltonian equations of motion. The results demonstrated consistency between the Hamiltonian equations of motion and the Euler–Lagrange equations. The scope of the research was further expanded to analyse the dynamics of the two-dimensional asymmetric oscillator within the framework of fractional calculus, focussing on the stability properties and distinctive behaviours of the system. This extension allowed for a deeper understanding of the complex interactions and nonlinear characteristics inherent to such systems. Subsequently, the Taylor series expansion was used to linearise the nonlinear equations, enabling their analytical solution through eigenvalue and eigenfunction techniques. This approach helped enhance the theoretical understanding of the system's dynamics and supported the validation of the results.