Method of generating $N$-dimensional isochronous nonsingular Hamiltonian systems

TL;DR

This paper presents a new method to construct higher-dimensional, nonsingular isochronous Hamiltonian systems that can exhibit both periodic and quasiperiodic oscillations, extending previous constrained models.

Contribution

The authors generalize the $ ext{Omega}$-modified Hamiltonian procedure to create unconstrained, nonsingular $N$-dimensional systems with explicit solutions, including isochronous and quasiperiodic dynamics.

Findings

Constructed $N$-dimensional isochronous Hamiltonian systems.

Derived explicit solutions with isochronous and quasiperiodic behavior.

Extended the method from 2D to arbitrary dimensions.

Abstract

In this paper we develop a straightforward procedure to construct higher dimensional isochronous Hamiltonian systems. We first show that a class of singular Hamiltonian systems obtained through the $\Omega$-modified procedure is equivalent to constrained Newtonian systems. Even though such systems admit isochronous oscillations, they are effectively one degree of freedom systems due to the constraints. Then we generalize the procedure in terms of $\Omega_i$-modified Hamiltonians and identify suitable canonically conjugate coordinates such that the constructed $\Omega_i$-modified Hamiltonian is \emph{nonsingular} and the corresponding Newton's equation of motion is constraint free. The procedure is first illustrated for two dimensional systems and subsequently extended to $N$-dimensional systems. The general solution of these systems are obtained by integrating the underlying equations and is shown to admit isochronous as well as amplitude independent quasiperiodic solutions depending on the choice of parameters.