Hamiltonian Zoo for systems with one degree of freedom

TL;DR

This paper introduces a new class of infinite Hamiltonians, called the Newton-equivalent zoo, which produce the same equations of motion for systems with one degree of freedom and include parameters affecting time scaling.

Contribution

It presents explicit forms of Hamiltonians forming a zoo that are equivalent in dynamics but differ in structure and parameters, offering new tools for Hamiltonian hierarchy generation.

Findings

Infinite Hamiltonians produce identical equations of motion.

Hamiltonians include parameters as time scaling factors.

Each Hamiltonian can generate a hierarchy of Hamiltonians.

Abstract

We present alternative explicit forms of the standard Hamiltonian for systems with one degree of freedom. This new class of infinite Hamiltonians is called Newton-equivalent Hamiltonian zoo, producing the same equation of motion. These Hamiltonians are directly solved from the Hamilton's equations and come with extra-parameters, which are interpreted as scaling factors for the time evolution on phase space. Moreover, each Hamiltonian in the zoo can be used as a generating function for a Hamiltonian hierarchy.